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Front Dynamics in Non-Smooth Ignition Systems in a Noisy
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FRONT DYNAMICS IN NON-SMOOTH
IGNITION SYSTEMS IN A NOISY
ENVIRONMENT

By

Mohar Guha

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2007

ABSTRACT

FRONT DYNAMICS IN NON-SMOOTH
IGNITION SYSTEMS IN A NOISY
ENVIRONMENT

By

Mohar Guha

Non-smooth excitable systems arise as models of combustion, nerve impulses, elas-
tic displacements during phase transitions, and proton conduction in weakly hydrated
polymer electrolyte membranes (PEMs). These systems often support front solutions
which correspond to moving phase transitions. Typically the non—smoothness arises
as an approximation to a system which is so sensitive to changes in state at criti-
cal values of internal variables, for example changes in temperature near the flash
point of a mixture, that its response may be considered discontinuous. We consider
a singularly perturbed, piece-wise smooth system with a discontinuity at an ignition
threshold for which the noiseless system supports a unique, stable front solution. At
spatial points where the front crosses criticality the system will be particularly sen-
sitive to noise, leading the solution to lose monotonicity, and engendering multiple
crossings of the ignition threshold. We identify regimes in which the front preserves
its stability, showing that noise smears the front location and modifies the front prop-
agation. In a neighborhood of the ignition threshold the system interacts strongly

with noise, the front can loose monotonicity, resulting in multiple crossings of the

ignition threshold. We adapt the renormalization group methods developed for co-
herent structure interaction, a key step being to determine pairs of function spaces
for which the the ignition function is F rechet differentiable, but for which the associ-
ated semi-group, S (t), is integrable at t = 0. We parameterize a neighborhood of the
front solution through a dynamic front position and a co—dimension one remainder.
The front evolution and the asymptotic decay of the remainder are on the same time
scale, the RG approach shows that the remainder becomes asymptotically small, in
terms of the noise strength and regularity, and the front propagation is driven by a
competition between the ignition process and the noise. The main result shows that -
the front retains its stability, but the ignition point smears into an “ignition set” over
which the front can have multiple crossings of the ignition threshold. Moreover in the
scalings for which we show the front retains its stability, the ignition set is thinner
than the front and the impact of the noise on the front velocity is a correction to the

noiseless front velocity.

T o my family

iv

ACKNOWLEDGMENTS

It is my pleasure to thank the many people who made this thesis possible.

It is difficult to overstate my gratitude to my advisor, Dr. Keith Promislow. During
these years, I have known Keith as a enthusiastic, inspiring and principle-centered
person who made a deep impression on me. His patience, flexibility and faith in me
in the past years had boosted my confidence to become an independent researcher.
Throughout my thesis-writing period, he provided encouragement, sound advice, good
teaching, good company, and lots of good ideas. I cannot thank Keith enough.

I am thankful to my committee members, specially Dr. Peter Bates, for his time
and suggestions on an early draft of my thesis.

This thesis gradually emerged amid the friendships that animated my MSU years
and provided their most lasting lessons. Rajeev helped me immensely by giving me
encouragement and friendship for past years. Pavan for providing help in Latex for
giving me such a pleasant time during the last year of my graduation.

Last, but not least, I thank my family: my parents, for educating me with aspects
from both arts and sciences, for unconditional support and encouragement to pursue
my interests. My brother for sharing his experience of the dissertation writing en-
deavor with me, for listening to my complaints and frustrations, and for believing in

me.

TABLE OF CONTENTS

LIST OF FIGURES vii
1 Introduction 1

2 Model Description and Construction of Travelling Wave Solution 13

2.1 One parameter family of outer solutions 450 ............... 14

2.2 The inner solution ............................ 16

‘ 2.3 The composite solution .......................... 18

3 _ The Rescaled Problem and its Linearization 22,

3.1 The Co—moving coordinate system ................... 22
3.2 The decomposition of the solution U and the Frechet differentiability

of F ...................................... 27

3.3 Bilinear Formulation of the Resolvent operator of L d3 ......... 42

3.4 Spectrum of the Linearized operator .................. 45

. 3.5 Essential Spectrum ............................ 46

3.6 Point spectrum .............................. 47

4 Resolvent and Semigroup estimates 56

4.1 Decomposition of the Linearized operator ................ 58

4.2 Exponential Dichotomy .......................... 60

5 The RG methodology 72

5.1 Overview .................................. 72

5.2 Control of Residual ............................ 76

5.3 The RC equation ............................. 90

BIBLIOGRAPHY 97

vi

1.1
1.2
1.3

1.4

2.1
2.2
2.3
2.4
2.5
2.6

3.1
3.2
3.3
3.4
3.5
3.6
3.7

3.8

4.1
4.2

5.1
5.2

LIST OF FIGURES

McKean’s piecewise-linear caricature of F itzhugh—Nagumo equations.

The nonlinearity 0(w, T) .........................
The anode and cathode bi-polar plates, gas plenums, membrane, and
the gas diffusion layer. The Plenums are slowly replenished from an
external feed. The black arrows indicate direction of electron flow and
of proton counterflow. ..........................
Numerical verification of experimental hysteresis from Benziger et al.

The nonlinear ignition functionalo ...................
The function g ..............................
The outer function $0. ..........................
The phase diagram for 4'),- ........................
The inner solution qiz- ...........................
The composite solution qi .........................

The coordinate system after each renormalization ............
The function ¢’(z; 20). ..........................
The graph of (,7) + W. ...........................
Measure of set E smaller that distance of E from 20. .........
The graph of (13+ W in the neighbourhood of 20 where E- is empty. .
The graph of a; + W in the neighbourhood of 20 where E+ is empty. .
The graph of 6 + W in the neighbourhood of 20 where both E'+ and
E- are empty ................................
The spectrum of L (5' ...........................

The contour C. ..............................

The graphs ofRe (y/1+-2-) = 1 and Re(‘/1+—a—57) =1 forA EC.

Evolution of front location due to the iterative scheme. ........
The graphical illustration of solution of the inequality for T0 ......

vii

5
7

14
15
16
17
18
19

23
25
32
34
35
40

41
47

58
64

89

CHAPTER 1

Introduction

Non-smooth excitable systems are important models of combustion, nerve impulses
and elastic displacements during phase transitions. To this we add our model of
‘ proton conduction in hydrated polymer electrolyte membranes (PEMs). These sys-
tems support travelling wave solutions which correspond to moving phase transitions.
When the underlying system properties depend sensitively on the system state then it
may beappropriate to incorporate a non-smooth response to the system. This thesis
studies the interaction of non-smooth systems with a noisy environment.

An example of a non-smooth model of nerve impulses is Henry McKean’s caricature
of the Fitzhugh Nagumo equations modelling nerve conduction by a tunnel diode
network [42], [30] . If a nerve is stimulated below a threshold then the signal damps
out and no information is transmitted, while if it is stimulated above a threshold then
the signal undergoes a fast change into a train of pulses. Let u(:i:. t) be the voltage
difference actoss the membrane of the nerve, then it satisfies

Bu _ 3211

E}? _ 515 + f(u(a:,s) 2 0 S s S t). (1-0-1)

The nonlinearity f in the Fitzhugh Nagumo equations has the form

f(u)=u(1—u)(u—a) 0<a<1, (1.0.2)

 

ll fl“)

 

 

 

 

 

 

Figure 1.1. McKean’s piecewise—linear caricature of Fitzhugh-Nagumo equations.

and McKean suggested a ‘piecewise linear caricature" of the cubic polynomial [19]

0 u < a '

f(u) = -—u +{ 1 u > a. (1.0.3)

A key issue in a non-smooth system of the form (1.0.3) arises from the structure of
the wave. The model studied in this thesis has a similar structure, the nonlinearity
is peice-wise linear but discontinuous with fixed jump. The problem of existence
of travelling waves for the non-smooth nonlinearty (1.0.3) has been addressed by
several authors [27],[26],[46], the final result being that model admits a diverse variety
of waveforms depending on the number of crossings of the level a. The case of
one-crossing was described by McKean [19] where he showed that, upto translation,
(1.0.1), (1.0.3) has a single waveform which rises monotonically and moves at a definite
speed. The other important problem is the existence of a threshold, that is, a level

below which no information is transmitted. This was confirmed by McKean and Moll

[20] in the next simplest case of two-crossings. In this case, there exists a standing

2

wave solution of (1.0.1), (p2, symmetrical about its peak at .r = 0 and (1.0.1) preserves
the shape of such data for t > 0. The separatrix surface, constructed in [49], divides
the space of initial data X 2, into two regions: a region of collapse for which u(:r, t) —> 0
and a region of expansion for which u(:1:, t) —+ 1. By constructing the classical solution
of (1.0.1) with nonlinearity ( 1.0.3), it was proved in [20] that any smooth, symmetric
initial data which does not collapse to 0 nor expand to 1, stabilizes to the standing
wave ((22.

To model a bilinear thermoelastic material capable of undergoing solid-solid phase
transition, Anna Vainchtein considered a non-monotone, up—down—up strain relation.
In [3], a special thermoelastic material is studied which permits analytical calculation
of travelling wave solutions for one-dimensional regularised problem. The model
includes both thermal and heat dissipation. When the nondimensional parameter
comparing the length scales associated to the viscosity and to heat conduction exceeds
a threshold value, the kinetic relation becomes non-monotone. Vainchtein, considers
an infinite homogeneous elastic bar, where u(:r,t) and T (:r,t) are the longitudinal
deformation in the displacement and temperature fields, at a point x and time t.
The local displacement of the bar is described by its displacement field and its spatial
derivative, called the strain 11) = ul.(.7:, t). The elastic property of the bar is determined

by an elastic energy density, 0(w, T), which is a function of strain, modelled as,

a(w,T)={ “w “’<"”0(T) (1.0.4)

#(w - 6T) w > w0(T),
where u is the stress modulus. The two strain intervals where stress is linearly increas-
ing represent two different material phases. In this model, the strain interval where
the stress is compressed into a single point wo (T), is where the phase transition oc—
curs. The distance between the linear branches is denoted by eT. To investigate the
kinetics of a phase boundary, the above model of 0 enables analytic calculations. In
this paper, the author derived an exact formula describing interface kinetics and its

dependence on material parameters is studied.

3

 

 

 

 

 

i' I
et W0(T)

 

 

 

Figure 1.2. The nonlinearity 0(w. T)

This thesis considers a reduced model of slow transient behaviour and proton trans-
port through Polymer-Electrolyte Membranes (PEM). The full model was introduced
by Igor Nazarov and Keith Promislow in [21]. To motivate the model studied in
my thesis, a brief discussion of PEM fuel cells is appropriate. PEM fuel cells use a
polymer membrane as an electrolyte, porous carbon electrodes containing a platinum
catalyst and need only hydrogen, oxygen from the air, and water to operate. PEM
fuel cells generate electric potential by separating the oxidation of hydrogen into two
catalysed steps performed on opposite sides of an electrolyte membrane. Hydrogen
fuel is chanelled to the anode on one side while oxygen from air is chanelled to the
cathode on the other side of the cell. At the anode the platinum catalyst causes
the hydrogen to split into positively charged hydrogen ions and negatively charged
electrons. The voltage gradient across the polymer electrolyte membrane drives the
positively charged ions to pass through it to the cathode, while the negatively charged
electrodes must pass through an external circuit thereby creating electric current. At
the cathode the positively charged hydrogen ions combine with oxygen, with the end
products of water, water vapor, and heat. The electrolyte membrane is a complex

polymer comprised of Teflon spines. These are arranged in a nanoscale configura—

 

Anode Graphite Plate

   
 
   
 

 

Fuel Plenum

 

 

 

 

V(y)

 

  

Air Plenum

 

 

 

 

 

     
 
   

 

 

 

 

 

 

 

 

 

Cathode Graphite Plate

 

Figure 1.3. The anode and cathode bi-polar plates, gas plenums, membrane, and the
gas diffusion layer. The Plenums are slowly replenished from an external feed. The
black arrows indicate direction of electron flow and of proton counterflow.

tion which facilitates the selective diffusivity of the membrane, enabling the fuel cell
to perform close to the thermodynamic limit for efficiency. The reactant gases are
distributed to catalyst sites through a sheet of porous carbon fiber paper known as
the gas diffusion layer, or GDL. Most efforts at modeling and simulating fuel cells
in the literature to date have focused on the entire fuel cell, including charge, heat,
and mass transport. Despite the complexity of these coupled models, extensive one
and two—dimensional numerical simulations yield concentration profiles that vary lin-
early through the thickness of both the GDL and PEM [13], [48], [14], [32]. Models
for convective-diflhsive gas transport in porous media have been developed in [50],

[40], [2], [11], [31], [28] and also for applications arising in a wide range of fields,

including electrochemistry [15], flow in insulating materials [39] and groundwater
transport [37],]5]. The standard approach is to couple a mass transport equation for
the mixture (typically Darcy’s law or some appropriate modification thereof) with an
equation governing intercomponent diffusion within the mixture. Similar models have
appeared in the fuel cell context, in which equations describing multicomponent gas
flow in the GDL are coupled with equations for heat and charge transport occurring
in the other fuel cell components [43], [32], [29], [35], [36] and [47].

In a seminal series of experiments J. Benziger [24] observed hystersis, slow tran-
sients, and long period relaxation oscillations in a stirred tank reactor (STR) PEM
fuel cell feed from dry inlet gases. The key features of Beneziger’s experiment are
the external resistances at which the jump from high-water content to the low-water
content is observed, the cell voltages and current at the jumps and the partitioning
of product water into the cathode and anode plenumsf Experimental work of Sone
et al [44] shows that the membrane protonic conductivity drops by several orders
of magnitude at a critical membrane hydration level.) The membrane must be well
hydrated to function, however overproduction of liquid water may saturate the sur-
rounding porous electrodes and leads to oxygen transport limitations. The control of
the motion and distribution of liquid water in both the mane-structure of the mem-
brane and the surrounding fibrous electrodes is referred to as water management, and
is critical to effective cell operation. I. Nazarov and K. Promislow, [21] proposed a
model which coupled two degenerate parabolic PDEs for membrane water content to
an elliptic equation for proton conduction. This model accounted for the impact of
membrane water content on protonic conductivity and showed that the fuel cell sys-
tem possessed two stable states, an ignited state in which the membrane has sufficient
water to sustain the electrochemical reaction, and an extinguished state in which the
high membrane resistence reduces the reaction rate to a point where the loss of water

to the dry gas environment dehydrates the membrane and extinguishes the electro-

 

Voltage verses current

 

 

 

0 5‘0 100 150 , 200 250
milliAmps

 

 

 

Figure 1.4. Numerical verification of experimental hysteresis from Benziger et al.

chemical reaction. h‘loreover they showed that the bifurcation from a uniformly hy-
drated/ ignited state to a partially ignited/ partially extinguished state could explain
the observed experimental hysteresis, Fig(1,??) and the slow transients observed in
Beneziger’s experiments. In particular the transients corresponded to ignition waves
slowly propagating by self-diffusion of water within the plane of the membrane. The
ignition waves observed in the numerical work of Nazarov et al. are initiated when
the water production within the fuel cell is so small that the approximately uniform
membrane water content approaches the critical membrane hydration level for pro-
tonic conductivity. At this point small variations in membrane water content induced
by fluctuations in the noisy fuel cell environment become nucleation points for the
igniation wave. In this thesis, following the work of Sone et a1, we model the water
diffusivity as a function of the water content U, in the membrane

0 U<ac

0(0) = { U U > 06, (1.0.5)

where ac is the conductivity threshold for the membrane.

A PEM fuel cell is a noisy environment. A key issue in this thesis is to study

7

the behaviour of non-smooth excitable systems in noisy settings. We consider a thin
membrane exposed on either side to air of prescribed humidity, which varies with
position along the length of the membrane. The relative humidity of the air produces
a local reference water content for the membrane, which is modified by the local
production of water by the electrochemical reaction which occurs only in the regions
of the membrane which are already sufficiently humidified. Averaging the water
content through the plane of the membrane, we consider the diffusive transport of

water content, U (y, t) within the plane of the membrane.
Ut : EUyy — (U _ g(y)) + ”0(Lf) + 6770/) [)7 (1'0'6)

where the ignition function a is defined as in (1.0.5) and the noise 1] E [180(11’1, 11—73).
In a recent paper by Eric Vanden—Eijenden et. al. [41], analyses the effect of small
amplitude noise on excitable systems. The authors consider a stochastically-forced
reaction-diffusion equation in a medium such that the reaction terms sustain trav-
elling pulses and the stochastic forcing is such that it acts at a neighbourhood of a
particular point. The noise initiates pulses at this point at random times and the
pulses then travel through the medium. The main result is, by appropriate scaling
limits and properly matched timescales, stochastic forcing can generate spatially-
periodic travelling wave trains. However, they avoid studying the impact of noise
on the structure of the travellig wave by restricting the noise to a localized spatial
domain.

Through detailed analysis of the shape of our wavefront in a neighbourhood of
the ignition threshold ac, we quantify the impact of noise on the wave structure,
specifically the possible number of crossings of the ignition threshold, the wave speed
and the waves stability.

This thesis is broadly divided into four sections. In the first section we provide a de-
scription of the model equation (1.0.6), a singularly perturbed, non-smooth parabolic

pde. A standard matched asymptotic expansion permits the construction of a C 1

8

front, 05(1); yo), parameterized by front position yo connecting ignited and extinguished
states traveling with a position dependent velocity v = v(y0). The front is not an
exact traveling wave solution of the underlying system, even in the absence of the
noise term, the spatial inhomogeneity induced by the equilibrium water content g(y)
renders the construction of an asymptotically exact traveling wave solution nontrivial
due to hysteretic effects. The front is monotonically increasing, but noise can break
the monotonicity, and in a neighborhood of the ignition region U (y, t) = ac, this could
possibly lead to ignition of new fronts and pulse splitting. The absence of noise for a
front'of constant shape, would allow us to apply a very developed body of invariant
manifold theorems [6], [7], [45], including the blow-up, decay, meta-stable properties
[4], [25], [9] Our front shape evolves in shape as it travels and is subject to noise.
we use a renormalization technique to study the stability of the travelling front,
breaking the evolution into a series of initial value problems, each with a frozen co—
ordinate system including a fixed linearization corresponding to frozen front position
y’(), a frozen convective frame with velocity 17 = v(y’0) and fixed spectral dichotomy

corresponding to the frozen linear operator. In the co-moving frame
Z = y - £0 - x/EL’(t70)(t — t0)

fl ,

 

(1.0.7)
the evolution equation (2.0.1) becomes
U. = F(U) = U23 + v(zU)Uz — (U — 93(2)) + MU) + 63/4-78/2773. (1.0.8)

where the scaled noise satisfies [[773]] H-“ = 0(1). The solution of the full equation
(2.0.1) is decomposed as U = ¢(z; zo) + W(z, t), where the time dependent parameter
20(t) shadows the slow evolution of U along the family of quasi-steady front solutions,
up to a small residual term W. The nonlinear evolution equation for the remainder
W takes the form,

59¢

I
Wt + 8—2020 —_- F(c,6) + Léw + (La, — Law + N(W), (1.0.9)

where L 6 is the linearized operator of F at the frozen point (I) = (0(2; 50),

9

L5 = a? + 0(20)82 — I + 'quEOOOl + “W650 (X) 550. (1.0.10)

Here the rank-one tensor products of delta-functions, 630 <8) 650,

0c
50

acts on W by

(630 (83 6§O)lV E UV, 650) 620 = l/l’(§0)d§0. (1.0.11)

The price for freezing the linear operator in the evolution equation (3.2.3) is the
introduction of the secular term, (L¢ — L (;,)W which encodes the movement of the
front location 2:0 away from the frozen value 50. A central issue is the structure of
the nonlinear operator N. The choice of the domain of the linearized operator L 43’ is
the “Goldilocks problem”. The Sobolev space’should not be too weak, so that the
nonlinearity o : H 7 —+ I] ‘(j is Frechet differentiable at (5, but not too strong that the
semigroup 5' (5(1‘.) :— S’(t) generated by L 97” ‘S'(t) : H “‘3 ——> H 7, blows up too rapidly as

t ——> 0+. The following theorem addresses the first side of the “Goldilocks Problem”:

Theorem 1.0.1 The Frechet derivative of F in HA7 for o/ > 1/2, at the composite
solution (23 is given by (3.2.15). Moreover for W 6 H7, '7 > 1/2, the nonlinearity

N(W) = Fe; + W) — F(g’>) — L 511/ satisfies:

. . 8+1 2 . '+1 2
iiN<W>liH_,3 s c (1m ll'm/ +10 “in / ). (1012)

for any 1/2 < ,3 <1.

The key step is to write the nonlinearity as in terms of the characteristic functions

0((15 + W) = (XE+ + X[§0,oo) — XL) (4? + W)~ (1-0-13)

where E- denotes the set of “false negatives” for which d)(::) > ac but 03(2) +W < ac,
and similarly E+ is the set of “false positives” for which 5(a) < ac but qf(:r)+W > ac.
Using the He'lder continuity of W allows one to show that false ignition sets are
thin compared to their distance from the ignition point 20. Although the
perturbation W may nucleate many ignition regions, this is a small, nonlinear effect
compared to its dominant, linear impact, which is to move the ignition point, as

captured by the rank-one operator 650 60 650 in L (5.

10

In the next step, we focus out attention on the spectral stability of the travelling
front. The linearized operator L 43 has a simple, small eigenvalue corresponding to the
broken translation invariance of the travelling front. All the remaining spectrum is
contained inside a closed angle lying in the left half of the complex plane.

The second side of the “Goldilocks problem” is to establish estimates on the semi-

group S (t) associated to the linear operator L (5 of the form

Ce-Vt

l mllFllH—p,

 

3(1)an g (1.014)

for F E X 50, where X 50 is the eigenspace associated to the (Oh/E) eigenvalue of L (13'
This requires estimates on the resolvent operator, (L5 - A) -1 acting on H ‘13 which
. is non-trivial since the rank—one operator is not defined on H "‘5 . We circumvent this
by decomposing

L — A = I: + 01650 (22) 530 + ling-0,00), (1.0.15)

where the linear operator L is given by,

c=a§+aaz—1.

We observe that (L6 — A) F = w + £“1F, where w solves

(L - Mu: = —om_9 (LI-1F) (20)650 —/i(£—1F)X(§0,oo)i

This reduction establishes the invertibility of Li) — A by applying the exponential
dichotomy L as obtained by Dan Henry [12].

We adapt the renormalization group method developed in [10], [17], [18], [22], [23],
[24], for the asymptotic stability of patterns. The architecture of the renormalization
group lies in the construction of a family of decompositions {(En, tn)};°,°=0, where in
is the fixed coordintate system and tn is an initial time. The decompositions permit
the initial value problem satisfied by the remainder to be broken into a sequence of

initial value problems over the time intervals [tm tn+1]. The bounds on the semigroup

11

enables us to develop estimates on the decay of W by writing the evolution equation
for W in terms of the linearized operator L <5 frozen at a point 50 in the convected
frame. As the front location zo moves away from the front location 20 there is a
natural secular growth in the estimates for W and after a finite time the control
over the decay of W is lost. We remove this secular growth by renormalization of the
evolution equations, updating the base point 2'0 through a nonlinear projection. Given
the solution U (-, tn) of (1.0.6) at any point, we choose in such that the remainder
W (-, tn) = U (-, tn) -— a(-; in) E X 5”. This decomposition when applied to the abstract
equation Ut = F(U), yields an evolution equation on each time interval [tm tn+1],
0(5 I

W: + (9—sz = R + LEOW + (L20 — Liulw + N (W) + e3/4‘78/2nsa, v.0.16)

W(§,0) = W(tn), (1.0.17)
and (5.2.2) admits a mild solution on the time interval (tn,tn+1). The long time
evolution of the original PDE reduces to the study of the renormalization map
7'72. : W(tn) ——-+ W(tn+1). The series of initial value problems generated in this man-
ner permit us to avoid the complication of studying the time-dependent linearized
operator. The nonlinear stability of the system via RG methods exploits the fact
that the evolution of the front location 20 is on a slower time scale and the secularity
L w -— L 50 is a lower order operator than either of L 30 and L 20.

Finally, for 'y 6 (é, 1) and 73 > % satisfying 7 + )3 < 2, the system relaxes quickly
in a small neighborhood of the manifold, and after that moves slowly along a thin

tube about the manifold.
Theorem 1.0.2 If the initial data U0 can be written as U(£, t) = (My; y...) + W0(y, t),

where ||W0||H7 is sufficiently small, then the solution of the governing equation can
11
be decomposed as, U(y, t) = gb(y;y0(t)) + W(y, t), where

”WM ”1344 6—”tHW0ll quit/4‘78” .
Hy Hy

and

yam = «2100) + eve/4‘78”).

12

CHAPTER 2

Model Description and
Construction of Travelling Wave

Solution

The membrane water content is governed by the parabolic nonlinear partial differen-

tial equation, U (y t) with sealed time variable t and spatial variable y
(h=FW0=%Uw-lU-9@D+udUl+m0Jl (200

The nonlinear ignition functional 0, illustrated in F ig(2), which models the con-
ductivity of membrane as a function of membrane water content, is a discontinuous
function with a fixed jump ac

0 U<oC

2.0.2
U U > ac. ( )

U(U) = {

The form of the ignition function highlights the phenomenon that the membrane is
“ignited” if the water content crosses a specific threshold, while it is “extinguished”
when water content is below the threshold. The equilibrium water content of the

membrane, g(y) satisfies the following conditions:
OUECWRL

13

 

o(u)=u

 

 

 

 

 

Figure 2.1. The nonlinear ignition functional 0

o g is monotone increasing,
0 g(y)—>9: asy—iioo

The noise function 77 E L°°(R+, H ’73), where 73,6 (%, 1). The diffusivity coefficient
6 << 1 is small and the parameter u satisfies 0 < u < 1. As a first step we construct
travelling wave solutions of the autonomous version of the second order pde (2.0.1)

via matched asymptotic expansion.

2.1 One parameter family of outer solutions do

At the leading order in e, the outer expansion for the stationary solution of (2.0.1)

takes the form,
(U-9(y))-W(U) =0, (211)

Let yo E R be the point where the solution of (2.1.1), denoted by do assumes the

value ac. The functional 0 acts on the outer solution do as

14

 

 

 

 

 

 

 

3’
Figure 2.2. The function g.
. 0 y < 310
0(990 : (2.1.2)
) { $0 y > yo,

and the solution of (2.1.1) becomes

 

g(y) y < yo. ,
‘50 : g(y) . (2-1-3)
1:7; y > 110-
This is self consistent as long as
i
g(yol < a. < 9"") (2.1.4)

1 — IL.
The self consistency condition determines an admissible range for yo, yo 6 [yo— , ya]

shown in Fig(2.2) where

yo =9_1((1_l‘)0c) and 113 =9‘1(0c)-

Fig(3) illustrates the outer solution do. The jump in do is patched by an internal
layer at y = yo. The form of the stationary internal layer is determined by the

existence of the heteroclinic solution of the inner equation.

15

 

 

 

 

 

 

 

Figure 2.3. The outer function do.

2.2 The inner solution

Using the transformation 6 = Egg—Q, we stretch the interval around yo, and freeze the

slowly varying background at yo,

Ut = U66 — (U — g(yo)) + M0(U). (2.2.1)

We construct a travelling wave solution U (E. t) = d,(§ —vt; yo) of (2.2.1) which satisfies

o’,’ + vd)’ — (a - 9040)) + WW) = 0- (22-2)

The travelling wave d,- approaches g(yo) as g ——> —00 and 21ng as 5 —+ 00. This
condition is the matching condition between the outer and inner layers. The velocity
v is to be determined. Here, yo serves as a parameter and the unknown wave velocity
v = v(yo) depends on yo. The wave speed v and the front location yo depend on the

time t. We look for a heteroclinic solution of (2.2.2), recasting it as an autonomous

cbi '_ a;
( <15;- > “ f (as. —!I(!/0)) —/w(¢i) — (l; )’ (2.2.3)

16

first order system

 

 

 

7 /(I * w.
8 y 1

/

 

 

 

 

Figure 2.4. The phase diagram for d,-

where ’ denotes differentiation with respect to 5. The fixed points of (2.2.3) are
g(yol #92
located at 62*” E 0 and Q“ E all , which correspond to thevalues

of the outer solution on either side of the jump at yo. For v > 0, the fixed points Q+

and Q" are both saddles with eigenvalues:

 

 

 

—v :l: va + 4

if = 2 , (2.2.4)
_ 2 _

A; = v i \/v 2+ 4(1 h). (22.5)

We can observe that A? > 0 and A2— < 0 where AIL and A2— are the eigenvalues cor-
responding to the stable manifold W3(Q+) of Q+ and unstable manifold W“(Q_)of
Q‘. Since the governing equation is linear on each side of ac, Fig(2.4), W3(Q+)
and W"(Q_) are exactly the stable and unstable eigenspaces of (2.2.3). In the un-
scaled variables, y, the inner solution of (2.2.3) satisfying the boundary conditions at
f —> ice is given by

g(yo) + Aoeifly—yw/fl y < yo

0% = ——
9010”) + 3032 (y--yo)/x/E y > yo-

17

 

 

 

 

 

 

 

Figure 2.5. The inner solution d,-

As in Fig(2.5), imposing the continuity of di, and that d,- attains do at yo, we solve

for A0 and Bo obtaining

+ ,_, f
(ac—9(a))th ”0W y<y0 (2.2.6)

+
91%;) + (0c — %£E)€A§(y_ilo)/‘/E y > yo-

2.3 The composite solution

The general form of the composite solution d, Fig(2.6), is given by

Afr(y-yo)/\/€

(p : g(y) + (0c — g(y0))e _ y < 310 (231)
fig/Ill + (0c - fink (y—y")/‘/E y > yo-

where Ail and A; are functions of v as defined in equations (2.2.4) and (2.2.5). To
reduce the size of the residual error F(d,t), we tune the final free parameter, the
velocity v, to render d, a C1 function. From (2.3.1) we observe that d(y;yo, v) does

not have a continuous derivative as depicted in Figure 2.4. We define E (U), which

18

 

 

 

 

 

 

 

Figure 2.6. The composite solution d

is proportional to the jump in derivative of d at yo

_ __3_\[‘-‘__ ,4
Eh) _ (0c - 9(3/0)) HO lg”. (23'?)

Lemma 2.3.1 There exists an unique v = U(yo) suCh that the composite solution

d(y;yo, v) given by (2.3.1) is C1(R). Moreover,

g .
(f) _) If“ as y —> 00
g- as y —+ —00.

 

Proof: The jump in the derivative of d at yo is given by

/\_ v ’ + v
le'lyo = ((Uc — $1103) 2\/(g) + %_(:y_93) — ((06 — g(y0)) A12) +g'(y0)) -
(2.3.3)

 

 

An explicit formula for E (v) is obtained by substituting for AIL and A2— from (2.2.4)

and (2.2.5) respectively,

_ 2x/Eitg’(y0) $29—00 , 2 _ _ _, ,/’_2 ‘
EM _ (1 - u)(0c — g(y0)) + 0c - 9(y0) l?” + (/0 + 4(1 fl) ( L + v +4)1'
(2.3.4)

 

 

 

Factoring —(v + \/v2 + 4(1 — ,u)) from (2.3.4) we obtain the following form of E(v)

 

5(2) = —(v + \/v2 + 4(1— 2))E1(v). (2.3.5)

19

where E1(v) has the form

——v + V?)2 + 4 __ 9190;} - ‘70 2\/E.qi(’!/0)fl

: v + \/v2 + 4(1— l1) ‘76 ’ 9(90) — (1 “ #lf’b’ + \/’U2 + 4(1— #llfac — 9010”
(2.3.6)

 

 

 

 

 

EMU

 

Since v + \/v2 + 4(1 — a) has no real zeros for v > 0, the zeros of E(v) are given
by those of E1(v). A simple calculation shows that E[(v) < 0 and hence E1(v) is a

strictly decreasing function of v. As v —+ 00 the first and third term on the right hand

gom_a
side of (2.3.6) tends to zero, thus lim E1(v) = ——_E—_—c < 0. Similarly as v —> —oo
v—voo (Tc-9010)

the first term on the right hand side of (2.3.6) tends to 00 and vlimoo E1(v) = 00.
The intermediate value theorem and monotonicity of E1 provide a unique solution to
E (v) = 0.

To derive the asymptotic states of d we refer to its form (2.3.1). We take the limit
y —> --00 of the left branch of d. Since A? >_ 0, the second term in the formula

vanishes and we obtain

lim d: lim g(y)=g--.
yH—w_

yH-m

Simlarly, for the right branch

lim d: lim M: 9+,
11-00 yaool-u l—u

 

since A2— < 0. I

The composite solution d does not generate an exact solution of (2.0.1). In the fol-
lowing lemma we compute the residual error, F (d), of the composite solution d(y; yo)
of (2.0.1), constructed in (2.3.1). The outer solution do is smooth except at yo. We

. o%> . . . .
introduce 6—29 to denote the second order p1ecew1se derivative of do.
:9

82d0] gyy y < 90
__ = , 2.3.7

Lemma 2.3.2 The residual F (d) is given by

__. 9e eta
17w)- tfiay +€]6y2], (2.3.8)

20

where v is given by Lemma 1.1.

Proof: Substituting the composite solution d from (2.3.1) into the expression of F

from (2.0.1) we obtain

F(G’5l = €¢yy — (9f) — 9(9)) + 110(95)-

Referring to the definition of a from (2.0.2)

_ €¢yy - (<1) — g(y)) y < yo
PM) _ { €¢yy - (d - g(y)) +_ 20 y > 110- (2'3'9)

From the formula for the outer solution do and theinner solution d, given by (2.1.3)

and (2.2.6) d can be expressed as

(o = d,- ._ Q50 (2.3.10)

We substitute this expression for d into (2.3.9), yielding

82990] at" - (95' - (20’ 1 g(y)) y < yo

Fd =€[,——— + , '1 ,' ~ . 2.3.11
( ) dy2 ee’,’ - (1 - Mez- - (1 — 1000 - g(y) y > 310 ( )
From the formula (2.1.3) for do and the (2.2.3) for d,- we obtain

8d,- 63244)
()3) 8y2 '

21

CHAPTER 3

The Rescaled Problem and its

Linearization

In this section we introduce the rescaled coordinates which form the backbone of the
RG methodology developed of Section 4. We also address the Frechet differentiabil-
ity of the nonlinearity at the composite solution and identify "the point and essential

spectrum of the linearized operator.

3.1 The Co-moving coordinate system

At a time to, we choose a fixed co-ordinate system based upon front position go
We shift to a co—moving frame with fixed velocity fiv(y‘o), for which the composite
solution d is stationary at leading order. We scale the spatial variable, removing e
from the equation, hiding it in the velocity and the function g, which are now slowly
varying. We transform the space coordinate to z where at time t = to, y = go is

mapped to z = 20 = 0

 

Z 2 y—yo— fjgtTOW-to), (3“,

22

 

 

 

 

 

NV

 

 

 

 

Figure 3.1. The coordinate system after each renormalization.

as illustrated in Fig(3.1). After the transformation (3.1.1), the front location zo(t)

at time t takes the form

300) = y()(1)— yo - £10m)“ * to). (3.1.2)

Throughout, y always refers to the spatial coordinate in the lab frame and 2 refers

 

to a co-moving frame with a fixed speed fiv(y‘o). We adapt the convention that an
overbar denotes a quantity that is fixed in time. we denote 20 E 0, 1" E 12(2'0) and
d E d(z; 2o) while ’ denotes derivative w.r.t. 2. Under the transformation (3.1.1)

after rescaling, the pde (2.0.1) inherits a drift term and F takes the form
Ut = F(U) 2 U2; + USUZ — (U — 93(2)) + uo(U) + 63/4F73173(z, t). (3.1.3)
v3(zo) and 93(2) denote scaled velocity v and background 9

vs(20) = 1‘(\/220+370+v(y"0)(t-10)) (3-1-4)

93(2) = g(fiz + 31—0 + U(ZIJOW - to))- (3-1-5)

23

Indeed, both vs and go are slowly varying functions of the front location 20 and the

spatial variable 2: respectively, i.e.

 

 

6113(20) : £81K?» = 0(\/E), (316)
2:0 620
.q’.(z> = 3]; = ONE)- (3“)

The noise term satisfies, ]]n3(t)|| H771 = 1, with the corresponding reduction in its 6
coeflicient. The shift to the co—moving frame eliminates e from the diffusive term.
The inner solution, di, (2.2.6) and the composite solution, d, (2.3.1) are expressed in

terms of the new variable 2 as

s + .
(01(2) 2 iii-[:03 gq( (Z 7 ) (318)
T_—B-+(0'C—fi)€ “0 Z>ZO
and A +
,7 _ A (z—zO)
$(Z; 20) = 99%) + (0C g3(20))€ 1 Z < 20 (3.19)

 

“195,2 + (a. — 74983595 W9) z > 20.
We show that 90921 and 53% agree to leading order in the L2 and L0o norms. From

the explicit formula of the inner solution d,- given by (3.1.8) and composite solution

d given by (3.1.9), the following relation holds

@ = _%i
820 ('92

 

+r(z), (3.1.10)
where the remainder r is given by

0 + (a. — 93(20))(Z — 20)

r z =
( ) eA2-(z_20) (_ 1 agsfzizol + (0 _ 93(3§)))(z __ ., )U—L) Z > z
T—p 320 C —/1 ”'0 20 0'

+
eA'll—(z-zo) _Bgs(:;z) (9A )
6

 

Since the rescaled velocity v3(zo) varies slowly with the front location 20, see to

(3.1.6), the eigenvalues vary slowly with 20

BAflvs) _ airmen) 31’s(20)
6Z0 — (9’03 820

 

= 0(\/E). (3.1.12)

24

 

 

 

 

 

 

Figure 3.2. The function d’(2; 2o).

and similarly
8A2— (Us (30))
(920

Combining (3.1.7), (3.1.12) and (3.1.13) we obtain the following estimate on r

= 0(,/Z). (3.1.13)

IiriziuLz +1r<c>liioo s cw. (3.1.14)

As proved in Lemma 2.1, the rescaled composite solution is monotone increasing in

the front regime and C 1 as, [[d'] :0 satisfies

[91/120 = \flllcblllyo = 0. (3.1.15)

and the relation (3.1.15) reduces to

(1.9010)

¢'(zo;20) = x/2 1_ p dy -—-

 

+ (<7¢:-9.s(20))/\i1L = - 1_ it

In the following lemma we obtain a lower bound on the derivative d', Fig(3.1), on
a bounded interval containing the front location 20. This property of d’ is used in

proving future results.

Lemma 3.1.1 For each bounded interval (26, 26) containing 20, there exists ao > 0,

independent of t, but depending on the length of the interval A = (26 - 26) such that

25

the derivative of the composite solution d;(2; zo) satisfies,

d/(z; 20) > do > 0, (3.1.17)
for all 2 E (26,26).
Proof: Differentiating d given by (3.1.9) we obtain

«29(2) + ma. — 93(20))6 1 (3‘20) z < 20.

 

e’(2; 20) = I _ .. ~ (3.1.18)
Thar + a. .. some» w ..
Since 9 is increasing we have
+
A+ — 2 .Al (2 0) < 2 .
d’(2;2o) 2 1“”: ififP: ( ) Z 0' (3.1.19)
A2(Uc——'1:]]—)e 2 “ 0 2>2o,
Since [2 — 20] g A, d' (2) can be bounded below as
+
We — g.(zo))e"\1A z < 20.
¢’(2; 20) Z l__ C 39(3 ) YA. (3.1.20)
A2(ac————0-‘1_#-)e 2 2>2o,
Choosing do as
+ :5 -
do = min (Afloc — 93(2o))e_’\1A,A2_(oC — 5:4 O))e)‘2 A) .
- H

we obtain (3.1.17), where do > 0 is independent of e. I

In the rescaled coordinates the piecewise defined second derivative of the outer

solution do becomes

32¢ U2gs 2 < 20
[.570] = { 62393 (3'1'21)
Z "117 2 > 20.

In the following lemma, we bound the residual in the new coordinate system.

Lemma 3.1.2 The residual R = F (d) in the convected frame is given by

 

a .- 62
Additionally we have the following bound
IlRlILz + IIRIILoo s C (fllzo — 20! +6). (3123)

26

Proof: In the coordinate system given by (3.1.9), d given by (2.3.10), when sub-

stituted in the revised form of F given by (3.1.3) yields

. _ I 02¢0] d)", _ ((41% _ 950 _ 93(9)) Z < 20
F . =1 S 2 _ I, z
“” ” ( 0”” + l 322 +[ 4,. -(1—u)<bi -(1-#)<1>0 —gs<y) z > 4.
(3.1.24)

Since d,- satisfies (3.1.8), we obtain the following expression for the residual

8d,- a?
R = —(’Us(30) — 21330052— + [1%].

The rescaled velocity is a smooth function of the front location and changes at a

O(\/E) rate. The mean value theorem yields the estimate,
[213(20) - 12450)] S C\/€']zo — 50]. (3.1.25)

From the definition of gs in (3.1.5), we see, ”931le = 0(6) which from (3.1.21)
623?
62 Lm

in (3.1.22) we acheive a bound for [IRHLoo

implies,

 

 

 

 

= 0(6). Since [195?]le = 0(1), substituting these estimates .

WWUnSCWVHm—?M+Q.

(‘ (b,-

2 .
z and [$731] decay at an 0(1) rate at infinity, we obtain similar bounds for

llRllL2- .

Since

3.2 The decomposition of the solution U and the

Frechet differentiability of F.

We decompose the solution U (2, t) of our model equation (2.0.1) in a neighbourhood

of the composite solution d as

U(2, t) = d(2; 20(t)) + W(2,t), (3.2.1)

27

where llWllH'i << 1 and W lies in a tube of width to. The width (,0 of the tube is

prescribed by

 

1 , _ 9+
w = — min ac - g , — ac , (3.2.2)
2 1 — u

+ _
where 1&1; and g- are the asymptotic states of d. One of the fundamental results
of this thesis is that the nonlinearity F is Frechet differentiable as a map from
H 7(R) —> H ‘/l(R), at the composite solution d and that the evolution for the

solution U can be written in terms of the decomposed variables 20 and W as,

Ed 7, _
(‘le HO

 

nm+ mw+LyV+nd—%MV+Nm¢ n2a

Here L d is the Frechet derivative of F : H 7 ——+ H “5 at d = d(2; 20), and the nonlinear
operator N is given by,

N = F(cB + W) — F(d) — Léw. (3.2.4)

A first step in this direction is to show that functions in H I, for 7 E (%, g] , are infact

Holder continuous.

Lemma 3.2.1 For each 7 E (1/2,3/2], there exists C > 0 independent of x,y, for

any (,0 E H7
My) — 90(-'r)l s C ly — xii-”211411117 . (3.2.5)

In particular, the LO0 norm is controlled by the H7 norm,

ll‘PllLOO S C Ilellm- (32-6)

Proof: The Fourier transform representation of 90

_ _1_ eikx A
90(x) — fl]! cp(k)dk. (3.2.7)

28

implies that

 

 

 

 

 

119(9)-19(r)l s 7_--/|fittest-"1’: 1.533141:
It
s E / e‘yk— e1“ 12<k)1dk
kg 1
2gr—x
+ 712: / eiyk— 2”":- It; ~k()]dk (3.2.8)
7f
kg 1
2hp—x

(3.2.9)

Applying the mean value theorem on all” on the interval (:17, y) we obtain the following

bound

€2yk_ eixk

 

 

_ <] [my—.11 forIklly—x]<1 (3.2.10)

2 for]k||y-—x] >1

With the estimate in hand, the two integrals on the right side of (3.2.8) reduce to

1 1
—o. < —— l. — tlc -——-— 15117119,
140)) m1 _ v27? /1 111g 4112w()1« W2; /1 1<)1r
“—11:71 ”2117—51
_}__ _ 1 . .2 ”7/2 .21V2 -
s «2.11) :1 /1 111(1+111) (<1+111) 11.410041:
kSZU—I
1 2 _ , -
— 1 1.: 7/2 1 1127/2 k dk. 3.2.11
+fi; j<+11) (<+11)1«p()1) < )
k> 1
‘2w-rl

29

Applying Hélders inequality to the two integrals in (3.2.11) we obtain

 

 

1/2
1() <)1< L 1 111.911 / “‘12 .1).
"1 — .7.‘ ’l. —.’L‘ ——-—-—-—-
9.1 so _ 271'] 7 1 (IHHQW
“are
1/2
1 2 _,
+ 72—;111011, (H111) dk
Eel—«1
y—J.‘
1/2
1 1 — 2~27
s 7513—4111121, 1k1 dis
1
“211741
1/2
1 . _~2,.
. 1
’2 ly—T '
Performing the integrations in (3.2.12) -

1 _
1am—1pm s cursing—41' l(,_,,,/,+c11.o11, 112—4107 ”/2,
9—5"

s c1y—rP—l/211211,

The L00 estimates follows from a similar estimate applied directly to (3.2.7). I

With Lemma 3.2.1 in hand, we show that F : H 7 —) H ‘3 is F rechet differentiable
for 'y E (1 / 2, 3/ 2] and [3 > 1 / 2. For notational convenience, we introduce the tensor

product
(f 63> 9)W = (W. 9) f. (3-2-13)

where f (X) g is a rank-one operator with range f. In particular, when f and g are

delta functions centered at 20

(530 (X) (550)1/1/ E (ll/,550) d = W(§0)d§0. (3.2.14)

50

30

A key step in proving the F rechet differentiability of the linearized oprator L (5’ is the
choice of the Sobolev spaces H 7 and H “’3 . The gap '7 + )8 must be large enough so
that the non—smoothness in a can be accomodated. However if 7 + B is too large the
semigroup associated to the linear operator L d1 as a map from H ’13 —> H 7, will not be
integrable in t as t --> 0+. Only the nonlinear ignition functional 0 from F contributes
to the nonlinearity. The primary challenge in proving the Frechet differentiability of
F at d is to control the impact of the perturbation W on U(d), in particular the
nucleation of new ignition sets on which d + o > 0 but ()3 < ac. We show that the
primary impact of W E H 7 is to move the location of the ignition point, which is
a linear effect defined by the rank-one operator. The spreading of the ignition point
into an ignition set is a smaller, non-linear effect, which can be ignored at leading

order.

Lemma 3.2.2 Fix 7 E (%,1]. The Frechet derivative of F, given by (3.1.3), as a
map from H7 to H-B, at the composite solution d, is given by,
L5 = 53 +138; — I +11x[: 00] + ##7520 <8 530- (32-15)
"0’ (1‘5 (20; 30)
Moreover for W E H7, the nonlinearity NOV) satisfies:
, +1 2 , +1 2
11N(1V)11,,_,, s c (1111 1153,. / +1111 111M ). (3216)

Proof : Writing N (W) explicitly using the form of F given by (3.1.3), and L d as

given by (3.2.15), we obtain the following expression

Ill

N(W) F(d + W) —— F(d) — 115W

_. 7 0C
.. (.0 + W) — 0(4)) -— 1 (new + 17—)

W(50)620)3.2.17)

To begin the study of the nonlinearity, we investigate the non-smooth term o(d+ W).
We define, the set E of “ignition points” of the perturbation d + W, to be the set of
zeros of d + W — ac,

E = {2]d(2) + W(2) = ac}. (3.2.18)

31

 

 

 

 

 

 

Nll

z=0 Zo

 

Figure 3.3. The graph of d + W.

We note that the set E is contained in the interval (2o — w, 20 + w),
[E] g 2w. (3.2.19)

Let 2 E E be an arbitrary ignition point. Since d attains ac at 2o, the following
relation holds

d(2) + VV(2) 2 ac = (9(30). (3.2.20)

Applying the Mean Value theorem to d on the interval (2, 2o) (w.l.o.g. 2 < 2o), where

d is C 1 and monotone increasing, we obtain

W(Z) = 93(50) - (13(2) = (50 - Z) M62), (3-2-21)

 

(3.2.22)

valid for each 2 E E. We denote by l(2) the distance of an arbitrary ignition point,

2 E E from the froxen front location 20, i.e.

l(2) E [Eo — 2] = 1% . (3.2.23)

 

where fiz E (2, 20). From Lemma 3.1.1 we know that

32

min d'(2) > min d'(2) Z (to > 0.
(3,50) ZEE

where do depends on the length of set E and from (3.2.19), (to is independent of 2.

Thus, there exists C > 0 independent of 2 and e such that
|l(Z)| = [Z *- Eol S C IIWHH'lw (32-24)

The ignition points of the perturbation d + W are close to the ignition point 2o of d
for ”W” H7 << 1. The elements of E are localized as illustrated in Figure3.3. This
implies the existence of a maximum and minimum element of E which we denote

by 2) and 2r, i.e., 21 < 2 < 2,. for 2 E E. Since E is closed, 2) and 2,— are zeros of

CE+W—Uc,

d(21) + l/V(21) 2 ac = d(2,~) + U"(2.,~). (3.2.25)

Applying the inequality (3.2.5) on Holder continuity of W E H 7' from Lemma 3.2.1,

with x = 21 and y = 2r, we bound the difference in the values of d on (2), 2r)

Men—84)=ww8l42mnscerweruMial 82%)

Since E Q (2o — to. 20 + 227), we have [2) — 2r] 3 1:). Then from Lemma 3.1.1 there
exists no > 0 independent of c, 21 and 2,— , such that d’(2) > (to for 2 E (21.2,) We

apply the Mean Value theorem to d on [2], 2,] to obtain
[95(Z1)— 9(a)] = Ill - Zrl [3(6)] 2 a0 |Zl - Zrl- (3227)
The relations (3.2.26) and (3.2.27) together implies
arrange—4r440nmm
This gives us an estimate of the length of the interval containing E

_1_
3 2—
[21— 2,1 g onwn/ 7. (3.2.28)

33

 

11wll,

 

.9

 

 

1 f). <
21 M40047?!

 

 

 

 

 

Figure 3.4. Measure of set E smaller that distance of E from 20.

The exponent on llWllH) varies from 1 at 7 = % to 2 at '7 = 1, so that the set E

1

1 ..
,3/2_7 >7+§,and we

shrinks super-linearly as llWllH7 ——> 0. But for ”y E ( 1 / 2, 1)

 

simplify the inequaliy (3.2.28) to
lzl — 41 s 0 1111111”?- (3.229)

However, since 21 and 2r belong to E, from relation (3.2.24), the distance of 2,, 2r

from 20 is bounded above by ”W“ [.17

120 — 41 _<. c llWllH) . (3.2.30)

120 — 2.1 s c IlWHm- (3231)
For ”W“ H7 small, the length of the smallest interval containing E is smaller than the
distance of E from the unperturbed front location 20, Fig(3.2). This key observation,
yields the form of the Frechet differential F : H 7 —> H ‘5 at d.

To study the ignition set E closely, we define the E.., the set of “false negatives”

and E+, the set of “false positives”
E- = {21¢ + W 5 0c} fl{zo g 2 g 27.}, (3.2.32)
E+ = {2]d + W 2 cc} 0 {21 S 2 S 20}. (3.2.33)

34

 

 

 

 

 

 

 

Figure 3.5. The graph of d + W in the neighbourhood of 20 where E- is empty.

We remark that 2 E E- implies d(2) + lV(2) 3 ac but d(2) _>_ ac, while 2 E E+
implies d(2) + W(2) 2 do but d(2) 3 ac. In particular, we may decompose U(d + W)

in terms of E1, and E- as follows

0(9 + W) = (XE, + x120...) — xE__) (<5 + W). (32.34)

The H ’3 norm of N (W) is formally defined as

IWlW)11H_,..—. 35p W

(3.2.35)
’UEHfl llvllHfl

We study three cases, E is to the left of 20, E to the right of 20, and E straddles 20.
Case-I: E- is empty.
As depicted in Fig(3.2), we denote by 2] E mag 2, the maximum element of E,
2E

which lies to the left of 20. So, d + W — do > O on (2?, 00). We partition E+ as

13,. = E2 Up)", 20], (3.2.36)

where E2 contains the intervals on the left of 2]", for which d + W — do > 0. Gener-

ically the interval [2]", 2o] is the largest share of E+. In this case the decomposition

35

(3.2.34) of U(d + W) reduces to

0(5) + W) = (d + W)(xEo + xplmmp. (3.2.37)

Substituting the decomposition (3.2.37) into the equation (3.2.17) the nonlinearity

takes the following form

MW) = :((>:E0 +x[~zrrzpo

if” Z:0____)_
= /1 W(X 10: X m - ) (d; + W) —O?._5(20)/1VV(§ )6: (3 2 38)
1 E2 [21 3201 ¢I(§0) 0 40' ._ .

 

For notational convenience we introduce the length scale lo
= W(Eo)
¢’(§o) '

[ the dominant part of E+. The nonlinearity

(3.2.39)

 

which we will compare to [21m — 2

N (W) from (3.2.38) can be rewritten in terms of lo as

MW) =11 (x53 + X[z]",§o]) (<5 + W) —— “totaling,

Substituting the expression for N (l’V) from (3.2.40) into the dual pairing we find

(3.2.40) .

 

1<N<W),v>1 = l (WW)(1E3+x[.;n,.,])—lo¢e (0650.2) v) (3.2.41)
2]" 21m
= ,u /(d+lV) vd2+fdvd2+[Ii/vdy—lod(20)v(io) .
E2 50 20

 

To isolate the dominant part of E+, we add and subtract l(2l )d(2o)v(2o) in

(3.2.41), where l(2lm) is defined in (3.2.23), to achieve

zlm
1<N<W).v)1 s 11/ (43+W)vdz+l143v-/ zo)dz
9.
4”
+ ,u]l(2m —lo]]d(20)]]v(io)|+u /Wvd2 (3242)

36

We denote the four terms in the left hand side of the inequality in (3.2.42) by

 

A = ,u / (d+VV)vd2, (3.2.43)
E9.
21’”
B = u fdi)—d(20)v(io)d2, (3.2.44)
50
C — H|l(2in)-lol[95(30)[|1’(50)|1 (3-2-45)
zm
l
D = u /1Vvd2. (3.2.46)
50

To estimate [(N (W ),v)| we investigate each of A, B, C and D in turn. We begin
with A, estimating d + W and v by their L00 norms

A g c [[3 + W||LOO “1.11LOOm(E$). (3.2.47)

Since IE2 Q [21,27’], using the estimate on the measure of E given by (3.2.29), we

obtain,
, +12
mlE-l) s 0 NW 117...“ .

Substituting this estimate in (3.2.47) and controlling ||v|| Loo by H (l norm we obtain

+1 2
A s 011W11},./ 11:11,“). (32.48)

_—
I

To investigate B, adding and subtracting v(20)d

50 20
B _<. l1 / 731v — v<20))dz +1» / v(50) (a) — 13(4)) dz .

l I

Since d is uniformly bounded, d E L00 and v E H (l is Holder continuous for 1 / 2 <

ll < 1, from Lemma 3.2.1, we have the upper bound
20 20
B s o / |2 _ gale-V2 ”tum, (12 + c / ”tum, (z _ Eo|]]d]]H1d2. (3.2.49)
2]" 21m

37

Performing the integrations above, we arrive at the upper bound
B s C |20 — 271“”? ||v||H/,. (3.2.50)
Since 31 E E substituting the upper bound for I20 — zml from (3. 2. 24)
B < C ”Wu/””2 n quE. (3.2.51)

The quantity C, balances the dominate component of E+ against the rank-one
tensor product of the delta functions within the linear operator. We recall that

d(ig) = ac, hence

c = MW) -lo||¢<z‘o )Iltlzo
g Ucll't’llLOO|1(31m)“10| g JCIIIUHHB [1(27“) —10| (3.2.52)

To derive a bound for II( 21’") — [0| we substitute for [(zlm) from (3.2.23) and for [0

from (3.2.39) to achieve the equality

W(zzml W(20)_“(21|Hm)c‘>’(30)_—W(~0)¢(€)|.

 

(MO <5’(50) W(é) <f3’(~0)| .

where 5 E (2)" ,Eo). From Lemma 3.1.1, 47(6) and 3(3)) are uniformly bounded

 

 

|z(z;") — 10| = (3.2.53)

below. So, (3.2.53) can be rewritten as

(mm) _ 10) g c

 

ll"(zzm)<f7(30) — Vl"(50)¢5’(€)|- (3254)

To control the right hand side of (3.2.54) by H W I] H7 we add and subtract W(20)d-)'(20),

ll(z3") — ml 3 C I (W(zzn) — W(Eol) we) + (We) — 59(5)) W<zo)|. (3255)
Applying the inequality (3.2.5) from Lemma 3.2.1 to right hand side of (3.2.55),

since W, 43’ E H 7(31’”, 50), we obtain

|l(2{”) — zol _<. 012;" — 50|7_1/2||WHH7 + c I4” — 20W”? llé’llm nwum.
(3.2.56)

38

Bounding |zlm — 20] from (3.2.24), which is valid since 2}” is a zero of (23 + W — ac,
we obtain

|l<zz") — (0| 3 C IIW'IILfil/z.
Finally, substituting for |l(zlm) — l0| to (3.2.52) yields

0 _<. c IIWIISil/Q) “qu3. (3.2.57)
To derive an upper bound for D, we estimate W and v by their LOO norms, which

are controlled by their fractional Sobolev norms,

30 2‘0
D2“, /Wvdz gc||v||H/,||W||H7 /1dz. (3.2.58)
[777. - (zl'lfll

Evaluating the integral on the right-hand side of the inequality (3.2.58) and again
using (3.2.24) we find

D s C IlvllHa ”1111’“le I20 — 2,") s c IIWIIEn llvlng- (3259)

Combining the estimates for A, B, C and D from (3.2.49), (3.2.51), (3.2.57) and

(3.2.59) respectively, the estimate on the pairing begun in (3.2.41) yields,
, 3+1 2 , +1 2
WWW ).v>| s C (uwna. / + nw uln/ ) llvlng, (3.2.60)

and returning to the definition of the H 7-6 norm of N (W) from (3.2.35) we obtain
the result (3.2.16) in this case.

Case-II: E+ is empty.
This analysis for this case is similar to Case-1, where E- is empty. As depicted in
Fig(3.2), let we denote by .277." the rzréig 2, so (5 + W — ac < 0 on (—oo,z,7.”]. We

partition E _ as

E- = 139 UM”, 20]. (3.2.61)

39

 

 

 

 

 

 

 

Figure 3.6. The graph of a) + W in the neighbourhood of 20 where E+ is empty.

Since, (,3 + W < 00 in E--, 0((5 + W') has the form

U(ED + W) = (X[z,m.oo) — XEQ) (d) + W). (3.2.62)

The remainder of Case—II follows as in Case-I.
Case-III: Both E+ and E- are nonempty.

As in (3.2.34), we obtain the decomposition of 0((5 + W)
0(03 + W): (“3+ + X12000) — XL) (55 + W). (3.2.63)
Substituting (3.2.63) in the equation (3.2.17), the nonlinearity takes the form

NOV) = H ((XE+ — XE_ + X[50,oo)) (<13 + W)

 

 

— , «7450) .. _
— ¢X[§0,OO)) _ I” (X[20,oo]lv + 61(20)W(z0)620) a
_ - , 55(50) _ _
— (XE, — m) up + W) — uWO, W(zowzo. (3.2.64)
Substituting for N ( W) from (3.2.72) into the dual pairing we find
(N(W), v) = p/ ((15 + W) vdz — p/ ((23 +1V)vdz — u§%?—))W(Eo)v(20)(3.2.65)
0

3+ E_

40

 

 

 

 

 

 

V

 

 

Figure 3.7. The graph of 6 + W in the neighbourhood of 20 where both E+ and E-

are empty.

We bound 6 + W by its L00 norm and v by its H 6 norm, inequality (3.2.72) reduces)

to'

WWW), 'v)| S C H6 + WllLoc lll’llH/1(m(E+)+‘m(E—))+ C |W(50)| llvllH/x-

(3.2.66)

From Fig(3.2), we observe E+ Q [21,zr] and E- g [zbzr], the estimate (3.2.29)

affords the following bound

')+l/2

mei) S Izr - Zzl S CHWllm

For any 2... E E, the following equality holds

(9%) + W(Z*) = 0c = 95(30)-
Applying the Mean Value theorem to 6 on (2..., '20) C (2), 21-), we obtain
‘ " _ _ l 2
|W(z.)| = l¢(z*) — ¢(ZO)| _<. Clz... — Zol s CIIWII'H/ .

H7

where we have used the bound (3.2.29). Now, we bound |W(§0)| as
|W(§0)| S Ill/(50) — W(ZQI + |W(Z*)|-

41

(3.2.67)

(3.2.68)

(3.2.69)

Applying the estimate (3.2.5) from Lemma 3.2.1 to the first term on the right-hand

side of (3.2.69) and the estimate (3.2.68) to the second term we obtain
IW (2 o)| < C (l5 0 — W1 ”2 IIWIIm + IIWII"+1/1). (3.2.70)
Using (3.2.67) to bound |Eo — 2...] we obtain
|W(2‘ 0>I < C (IIWII§;~,1/1‘1+1/11+ 1+ IIWII’IH/1‘)< cIIWIUJV2 (3271)

since HWHH‘Y < 1. Returning to (3.2.66), we bound 171(13):) and (14"(20)! from (3.2.67)
and (3.2.71) in (3.2.66) to obtain

|(N(W) v>| < CIIWIIW2 II vlng. (3.2.72)

which leads to (3.2.16).

3.3 Bilinear Formulation of the Resolvent opera-

tor of L4;

The bilinear form B [ , ] : H1 x H 1 ——> R, associated with the linearized operator

—L& from (3.2.15), is given by:

B[u,.u] E (—L¢3u,v) = (””22 — 1 3(50)uz+ —;LX[:0 00)” + #6’0 (:0) u(§0)620 (8) 650,11) .
(3.3.1)
Integrating by parts on uzzv we obtain,
00
B[u, v] = / (uzvz — vsuzv + uv — ”ya—)MEUWGOWEO — I‘U(3)1’(3)X20,oo)) dz.
—OC
(3.3.2)

The evaluation operator 650 is continuous on H 1 and we have the inequalities:
- 2 2
l?t(zo)l S IIUH00 S C||U||H1HUIIL2 (3-3-3)

42

We see that B is a bounded operator since,

Im _ _
|B(u,v)| S llitllgill’vllHi+(1+/t)||"||H1||v||H1+$,—(gg—)lu(zo)v(zo)l

S CIIUHHl Ilvllgi

for all 11,1) 6 H 1. To see that B is coercive, we evaluate B[u, u]. The convective term

vsuzu vanishes upon integration by parts.

00

B[u,u] = / (113 + U2 —- ,uu(z)2X[2-O,oo)) dz

—00

Mac

—=—- u E 2.
MO) (0) (334)

 

We note that the coefficient (9'09 ) of 11(‘z'o)2 is positive. We obtain lower bound for
30

B ['u, it] given by (3.3.4), by applying the inequality (3.3.3),

 

 

oo —oo oo
.0'
B[’u.,'u.] = / “312+ / 712dz — lt/11.2(.Z — 66(50)“(30)2
~00 —oo 20

IV

2
llullHl — H |lu||L2 — a IIuIIHI Hulle.

where a = 3%. Applying Young’s inequality to the last term of (3.3.5), we obtain,

2
2 1 2 C 2
Bow) 2 IIUIlHl — u IIuIIL2 - ~2-llullH1- —2— llul|L2
2
a , 2 1 , 2
BIu. u) + (7 + H) IIuIlL2 2 2 IIuIIHl.

This inequality demonstrates the the coercivity of B, i.e. there exists 7 > 0 such that
B 2 > 1 2 3 3 5

(uiu)+7llullL2— llullHl' ( ° ' )

The Lax-Milgram theorem provides an upper bound for the Resolvent operator. Con-

sider the following problem for u E H 1,
L6” + /\u = f. (3.3.6)
The bilinear form corresponding to (3.3.6) is

BAl“: 1)] = B[u, v] + A (11., '1') (3.3.7)

43

Since B is coercive B A satisfies the Lax-Milgram conditions each A 2 7 and for each
f 6 L2, which gives the existence and uniqueness of a weak solution in H1. Hence,
(AI — L (5) is one-one and onto for /\ 2 '7. The resolvent operator RA : L2 —» L2 is
defined by

1w := (AI — L,;,)‘1f.
We have shown that, for each f 6 L2, there exists u E H 1 such that B A(u, v) = (f, v)

, for all 1) 6 H1. Defining u = R A f , then the coercivity of B7 shows that,

:3 Ian; 3 3701,11) 5 BAMU) = (M) s llf||L2||u||H1-

1Ne deduce that,

llRAllHl S (3 llfllL2-

44

3.4 Spectrum of the Linearized operator

The spectrum of the linearized operator L (5, denoted by 2L, contains the singularity
of the Resolvent. The spectrum of L55 is a. disjoint union of the essential spectrum
2333 and point spectrum Ept which consists of all isolated eigenvalues with finite
multiplicities and hence is discrete. While determining the point spectrum requires
a detailed knowledge of the full nonlinear front 6, the essential spectrum is entirely
determined by the linearization of F about the asymptotic states of 6 determined in

Lemma 2.3.1. We formulate the eigenvalue problem for 1,0 6 ”7(R) as,
L306 = A6. (3.4.1)

Integrating the eigenvalue problem (3.4.1) about a small interval [50 — 6, 50 + 6], L 9;
given by (3.2.15), yields

50+(5 20-1-6 50111-6 30-1-6
/ tbzz + vsufizdz -— (1+ A) / t'bdz + u (Adz + 11.7—0— 1/( (50) /6 6ZO(~) )-dz __
50-6 5.0—5 50 ;0_5

Taking the limit 6 ——> 0 we obtain

 

[lit/DEG - (1+ A) Iii-11:0 = -uq3,((7:0 )1! (26). (3.4.2)

Since 6 E H '7, it is continuous at 20, so [(11450 = 0, and the relation (3.4.2) reduces

to a jump condition on 6’.

lit/"Ila, = wage—112(50)- (3.4.3)
20

An eigenvector d; solves (3.4.1) if and only if it also solves

 

7/)22 + 733$}: — (A + U161 ‘1‘ [NI/1247090) = 0: (3'4-4)
II _ _ _ ac _
[[10 H50 — #¢,(§0)¢(Zol (3-4-5)

45

3.5 Essential Spectrum

Since the linearised operator is defined on R, we do not expect the spectrum to
consist of purely point spectrum. We define essential spectrum as the set of {A E C}
for which A — L 6 : H 7' —1 H ‘6 is not a Fredholm operator of index zero. We refer to
the following theorem from Dan Henry [12], which is the most compact criterion in

determining the essential spectrum.

Theorem A.2
Suppose 11/1 (1:) N(:1:) are bounded real matrix functions and M (1:),N(:1:) —>

.Mi, 1121: as .1: —> $00, and suppose D is constant symmetric and positive. Define,
Lu(.r) =2 —Dum + 1’11(1‘)11x +N(:1:)u, -—oo < :1: < 00. (3.5.1)

Let St = {Al det(7'2D + ’iTA/Ij: +Nj: — AI) = 0} for some real 7', where 00 < T < 00.
Let P denote the union of regions inside or on the cureves S +, 3-. Then the essential

spectrum of L is contained in P, and in particular includes 3+ U 5..

Comparing —L(,-5 given by (3.2.15) with the operator L (3.5.1) from Theorem A.2,
we obtain, D = 1, M = Us and N(:1:) = (I — (“1501000 (:13). The asymptotic states
of N are N- = I and N+ = (1 — 11)]. Hence the essential spectrum of L 6 lies in the

union of the essential spectra of the limiting operators

L- = —uzz+vsuz+u.

L+ = —uzz + vsuz +(1— p)u,
i.e., regions inside or on the curves Si where,

s_ = (A! — k2 — 531k — 1 = A}, (3.5.2)

5+ = {1| — k2 — 531k — 1+ 11 = A}. (3.5.3)

46

 

 

 

 

 

Figure 3.8. The spectrum of L43.

Furthermore, we observe that larg AI —+ 71 as [kl ——> oo, in each connected component

of the essential spectrum.

3.6 Point spectrum

Having determined the essential spectrum of L d3 we now consider its point spectrum.
we show that the with exception of a single, simple eigenvalue the entire spectrum

lies an O( 1) distance inside the left half complex plane, Fig(3.8).

Lemma 3.6.1 Let L6) be as given in (3.2.15).

a) The point spectrum of Lg; is real.

b) Ept(Lq-5) (Ml—p, oo) 2 {AP} where Ap = —\/EA1, where A1 > O is given by (3.6.27).
The principal eigenfunction 1)) corresponding to Ap, normalized w.r. t. the weighted

L3 norm generated by the inner product defined in (3.6.13), is given by
1 “(é—50) 5 < a
, 935’ — ~0»
1' = - 3.6].
The coefi‘icients a, b are given by (3. 6. 6), (3. 6. 7) and the constant 91 is defined as in
(3.6.28).

47

Proof: We seek to determine whether there is any point spectrum satisfying, ReAp
2 —1 + ,u. Suppose 6 is an eigenfunction with eigenvalue ReAp 2 —1 + 11. Thus, 1))

satisfies

 

9/1" + My - (1,, +1)'¢’+ ufoo. ooh/2 = 0. (3.6.2)
[ll/1'11 50 = - (320)24501 (3.6.3)

The equation (3.6.2) has constant coefficients with associated characteristic equations:

Thus—(1H,.) = o z > 20 (3.6.4)

T2 + U37. — (1'1" AP) '1" I1: = 0 Z < 20 (3.6.5)

Equation (3.6.4) has a positive root a, and (3.6.5) has a negative root b given by

 

a, + 11§+ 4(1+ 1p)

. __z 3.6.6
(I . 2 7 ( )

;1,1‘9 —- \/’U3 ‘1' 4(1 + Ap — it)
b = 2 . (3.6.7)

 

 

 

The eigenfunction '6 takes the form

{ ’I/J(Z0)€“(€'30) E S 50,

10(6) = ¢(§0)eb(€—Eol g _>_~ 50.

(3.6.8)

We turn to the proof of part(a), motivated by the example on page 130 Dan Henry
[12], where he transforms the non-self—adjoint eigenvalue problem into a self-adjoint
one. Examining the characteristic equation for the limits 6 —+ too, we will show that
any eigenvector 112 must actually tend to zero at least as fast as 0(6‘1’3lg-50l/2), when
15 —2 :l:oo. Indeed, we show that t/J(£)e1’3(€-§0l/2 vanishes as 6 ——> :too. From (3.6.8)

we have

vs(£-Eo)/2 _ 6(50)e“(5‘§0)el’3(5‘20)/2 = .v9/)(§0)e(a+vs/2)(E-§0) g S 50,
Mile _ ¢(50)eb(€-50)evs(€-50)/2 = ¢,(§0)e(b+vs/2)(€-30) E Z 50,

(3.6.9)

48

while from (3.6.6) and (3.6.7) we see

 

 

 

 

 

 

 

 

+113 _ —v3 + (fig + 4(1+ 1p) + ,,8 _ Jug + 4(1+ 1,) > 0
a 2 " 2 2 _ 2
b v. _ -vs- v3+4(1+/\p) + v. _ ,/vg+4(1+),,) < o
2 — 2 2 _ 2
Defining
w(r)=w(€)e ”8‘5 201/161-610)

we see that w decays at :too and moreover, w satisfies

. 2
w” — (~35 +1+ AP) w + ”ml-([3060) = 0, (3.6.11)
I (TC _
w : = —;1:10 2: . 3.6.12
ll 11.0 6’(30) ( (I) ( )

This shows that the linearized operator is self-adjoint in the weighted space LEAR) ,

with the exponential weight V(z) = e1’3(z‘“50) and inner product defined as,

(111,11)V = /(31"9(z—20)'11.(:)11(z)dz, (3.6.13)
R

for any 11, v E H '7. Hence, all eigenvalues of L 6 must be real and they form a countable
sequence. The point spectrum is determined by a jump condition (3.6.3), which in

light of (3.6.8) reduces to,

0'0
6’ (20).
Substituting for a and b from (3.6.6) and (3.6.7) in (3.6.17) any eigenvalue AP 6

 

b — a— — —11 (3.6.14)

{—1 + 11,00) satisfies,

 

 

2_,u_ac
6—_’(Z0)

We substitute for 6’(EO) from (3.1.16) into the right-hand side of (3.6.15) to obtain

(613+ 4(1+ 1,, —— ,1) + 62 + 4(1+ 1,): (3.6.15)

the relations

 

 

2110C

(ac _ 93(2-Oll)‘:

49

I1— fdg<y11l+ 01611316)

 

((113 + 4(1+ AP — )1) + (fig + 4(1+ AP):

By the continuity of 6’ at 20, from (3.1.16) we also obtain the relation

 

 

 

21401: 619(11'0) 21406 \/€ (196171)
— 6——-— O c = — _—
(Uc "' 98(50))’\_1+(1 f dy + ( )) (06 - T8712»; [1 1" “ dy
+ 0(6)] (3.6.17)

Substituting for A? and A; from (2.2.4) and (2.2.5) the leading order terms in (3.6.17)

satisfies

 

213+ v§+4 _ vs - \/v§+4(1—11)
(0c - 94230)) — (0c(1— u) - 9390))-

The left-hand side of (3.6.17) is monotonically increasing for AP, the equation has

 

 

(3.6.18)

atmost one, simple root. We COnsider an asymptotic expansion of AP
= 10 — £11 + ‘ (3.6.19)

Then substituting for Ap from (3.6.19), into the spectrum equation ( 3.6.17), we obtain :

the leading order equation for A0

 

 

2110c

 

6,? + 4(1+ )0 -- 11) +' v2 + 4(1+ )0) , (3.6.20)
\/ \/ 8 (ac "‘ 9.5450)»: 1

211(7C _ 2uo‘c

(06—93(20))Af’ " (ac—.9464

Substituting for AT and A; from (2.2.4) and (2.2.5) in the above, we obtain

 

 

(3.6.21)

 

 

4
\/vg+4(1+A0—11)+ z,)§+4(1+,\0) —_— (ac Iuoc +\/;§_(33)5.)22
3

‘93( (30)) (—’U3+

 

Rationalizing the square root terms in the denominator of the right hand side of

(3.6.22) yields

0_ 0 uoc(v3+\/v2+4 4)
([3 +4(1+) 11)+(/v§ +4(1+,\)= (a..- _gs(20)) (3.6.23)

 

 

 

50

Now, we show that A0 = 0 solves (3.6.23). Plugging A0 = 0 into the left-hand side of
(3.6.23) we obtain

—v3+\/v2+4(1—11)+vs+,(/v§+4
= _((1(_C ”)Ug(zo)()20) )(v3+v1/323++4)+v (/v§+4 (from (36.18)),

110663 + WE 4)
(ac “593( 0))
= RHS. (3.6.24)

 

LHS

 

 

Hence the leading order term A0 in the asymptotic expansion (3.6.19) of Ap is zero
and we obtain,

1,, = —\/E/\1 + 0(6). (3.6.25)

To determine the sign of the coefficient A1, we equate the \/E terms on both sides in

the spectrum equation (3. 6. 17) and solve for A1. ' Utilizing the expansion

(/y+\/'x=f+f2f+(),0

in the left-hand side of (3.6.17), we obtain

 

 

 

 

 

1 1 2lmc (100/6)
(2%13 + 4(1—11.) 21/113 + 4) (7c - 93(20) (111 ( )
Solving for A1

 

 

(ac — Minna/E + 4(1— ,1) + «123+ 4) 43/
We observe that A1 > 0 since 9 is strictly increasing and 06—93(20) > 0. We normalize

the eigenfunction 1,1! from (3.6.8), w.r.t the inner product defined by (3.6.13), defining

61 as
6% E / ev'9(z_20)1/12(z)dz, (3.6.28)
'12
and the corresponding eigenfunction satisfies,
1 obi—20) < -
9—6 g — 01
/ = _ 3.6.29
11(6) { 91161716720) €220. ( )

We expand the expressions of a and b given by (3.6.6) and (3.6.7) to express them in
terms of the eigenvalues A?" and A2— respectively. Substituting for Ap from (3.6.25)

into the expressions for a and b

—v,g+ v§+4 Al/E

 

 

 

 

 

 

 

 

 

 

a = — —— + O 6 ,
2 21g+4 ()
b = —vs— \/v§+4(1-—11) A1\/E 0(6).
2 mfig+M1—m
We recall the definition of A? and A; from (2.2.4) and (2.2.5) to obtain,
My;
a: Mum— +01, 36%
l 2\/1:§+ 4 ( ) ( )
M .
b = A2— —— W + 0(6). (3.6.31)
2\/v§ + 4(1—.11)
We define the coefficients of \/E in a and b as ll and 12,
M '
1 = ——___., , 3.6.32
1 2m 1 1
[\1
12 = (3.6.33)

2¢fi+4u—y)

The principal eigenfunction corresponding to the eigenvalue —\/6Al + 0(6), is given

 

by,
+ ~__.:
9160, —¢e‘11+0<e))<1 ~o) z s 20
1)) = 1 _ /_ ~ _ (3.6.34)
%6(A2 —V€l2+0(€))(2—20) Z 2 20
I
The adjoint eigenvalue problem is given by,
1111113 — 1131/1: —— (A1, + 1)’U')1L +11le0, DOW/'1 = 0. (3.6.35)
11 __ 0's f _
14 __,__w z. 36%
[[ 120 1'60) < 0) ( )

where 1111 is the eigenvector corresponding to A1,. The adjoint of L 6 is denoted by L1.

6
We define the following inner product with the exponential weight Vl(z) = e—”3(Z_§O)
(114))” = /e‘v51z—50111(z)v(z)dz. (3.6.37)

R

52

Lemma 3.6.2 a) The point spectrum of L3) is contained in (-oo, —1 + 11] except for

a single, simple and real eigenvalue Ap which has the following expansion
1,, = 41¢; + 0(1),

where A1 is given by (3.6.27).

2
u

b) The principal eigenfunction of L; normalized w.r. t. the L T norm, induced by the

inner product (3.6.37) is given by,

1 Call-(£9-50), E < 2;

61(6) = 11 _ ‘. 0 (3.6.38)
1 W5 ~
516 ‘~())_ 5 Z 50

where a1 and b1 are defined according to (3.6.40) and (3.6.41), and satisfies

[11.111112 = 1'. 1 (3.6.39)
R .
Proof: The proof of part a) directly follows frOm Lemma 3.6.1. Since Ap is real, the

’1
6

to (3.6.35) has roots a1 and b1 given by.

point spectrum A; of L satisfies, A; = A1,. The characteristic equations associated

 

12,, + 112 + 4(1+ AP)
0.1 = 3 2 , (3.6.40)

b = 2 , (3.6.41)

 

 

 

and the corresponding family of eigenfunctions are,

1 1115-20) 2;

e , 6 5 1
111(1): 6 ,1 .. 0 (3642)
We (6—70). 152 50
Comparing the forms of if) and 61 given by (3.6.29) and (3.6.42), we find

6—1’312‘2011/21 = 1)). This shows that 1,01 is normalized w.r.t L21L norm
V

I613. = lemma.

z / 611305-50) (.1)2 dz = 1. (3.6.43)
R

53

Now, we prove the orthonormality of 1/2 and W, by studying their dual pairing

<11“ wT>L2

/ mavens,
ev3(z—30)v’)(z) (e_v3(Z—50)wl(z)) dz,

evsk—EOW’Rfi/Kfldz,

u
~w\ w'\ :u

I
In the following lemma we prove that there is a nonzero angle between (530 and the

eigenvector 215(2; 30).

Lemma 3.6.3 There exists 60 > 0, independent of e, such that,

6(50) E <86“)

()30

L2

for any 30 E R.

Proof: Referring to the form of the composite solution from (3.1.9), the angle
between 2315(3) and the eigenfunction w(z;50) (using the form given by (3.634)),

denoted by 6(30) is given by,

6(50)

III
A
NI I Q.)
o“ ‘94

S
V
h
to

 

.. + Us I ..
= i / emi‘fi’lxflm ((Uc-9(50))(—Afr+(z-50)8A1( h-aeo)

 

 

01 830
1 °° c) an ) f '(‘>
_ (2A2——¢E12)(z—20) _ g 20 _ _ _ - 2 vs _ 69 Zo
+ 61/6 ((06 1—#)( A2 +(z .20) 820 ) I‘ll ).
30

Performing integration and referring to relations (3.1.7), (3.1.12) and (3.1.13) we

obtain the closed form of 6(20) upto leading order,

9(5”)
__ : ac — _
6 _ = —}\+——UC g(VO) + /\_————H +0 ‘ . 3.6.45
(Z0) 1 2A:- —- fill 2 2A2— — \/E12 (fl) ( )

54

Simplifying the expression in the right-hand side,
6(20) = £292). + mm, (3.6.46)

which is uniformly bounded away from zero for all 20 E R.

55

CHAPTER 4

Resolvent and Semigroup estimates

Having described the spectrum of the linearized operator, we study the resolvent
operator, (L$ — A)—l : H"? —+ H7 for 7,6 6 (%, 1]. We prove that the linearized
operator L 6 is sectorial following the definition from Dan Henry [12]:

A linear operator L in a Banach space X is sectorial if it is a closed densly defined

operator such that, for some 6 E (0, 7r / 2) and some M >_ 1 and real a, the sector
Sa,0 = {M9 S larg(x\ - a)l SW. A 72 a}

is in the resolvent set of L and

M
IA - al

 

”(A _ Lrlll 5 VA 6 Sa,0- (4.0.1)

We introduce the space, X 20 = {U I ”U H H7 < 00 and TEEOU = 0}, where the spectral

projection is defined as,

U, at
«ECU _ ( Z?) wgo, (4.0.2)
(W50, W20)
which reduces to,
woo = (U, plow/1.0, (4.0.3)

by the scaling of ‘1120 and \Ilgo. We denote the complimentary projection by
fro = I — 7r0. (4.0.4)

56

The space X50 corresponds to the L20 spectral set which is uniformly in the left-
half complex plane, while X 50 = ROG-0) is the eigenspace associated to the O(\/E)
eigenvalue arising from the broken translational invariance. We fix the contour C in
p(L&)) with argA —> 21:6 as IAI —+ 00, for some 6 in (772‘, 7r). The branches C1, C2 and

C3 of contour C are given by

61 = {AIAT g —1,,\,- = 471),}, (4.0.5)
C2 = {AIAT = -—l, —ml S A,- g ml}, (4.0.6)
C3 = {AI/\r g —-l,/\,- = mAr}, (4.07)

where Ar and A,- denoted the real and imaginary parts of A E C. As illustrated in
2
Fig(4), a — ,u > l > 3% determines the position of the contour and m determines the

opening angle of the branches. 'We denote by a E R
'02
a 2 1+ —‘-9-. (4.0.8)

We employ a to obtain the decaying estimate (4.0.1) of the resolvent operator. For
A E C, IA + aI2 denoting the square of distance of C from ——a can be expressed as a

map (1
d : R+ —> RJr
(a — [)2 + 1‘2 0 _<_ :1: S ml (4.0.9)
(a — %)2 + .162 ml g :1: < 00

We note that

(1(1) _>_ 11:2 + (a — 02, a.- e R. (4.0.10)

which provides a lower bound for IA + al, A E C,

IA + a) 2 a — l > 1. (4.0.11)

57

 

 

 

 

 

 

 

Figure 4.1. The contour C.

4.1 Decomposition of the Linearized operator .

For F E 11—3, the resolvent (L -— A)‘1 finds the associated u solving

(L— A)u = F,

where L E L 6' In this section we introduce the notation

1

 

a=u— >0,

¢>’(§0)
and L from (3.2.15) can be written as
.2 .
L = dz + 'Usdz — 1 + 01650 (X) 650 + #X(30100)'
We decompose L — A as
(L — A) = I: + (1650 (X) 630 + :U’X(50,oo)i
where the linear operator 1C is given by

c = a? + v.0, — 1.

In the following lemma we derive estimates on .C‘IF for F E H ’6.

58

(4.1.1)

(4.1.2)

(4.1.3)

(4.1.4)

(4.1.5)

Lemma 4.1.1 Fix 7 + )6 < 2. There erists c > 0 such that for all A on the contour

C and F E H‘6 the following estimate holds
“5 1FIImg C(IA+a.I) (”(+13 2) V2 |IFI|H_,3 (4.1.6)

Proof: Referring to the definition of L given by the equation (4.1.5), [Eu = F is
equivalent to,

UZZ + U321; — (1 + A)“, = F. (4H17)

A

Taking the Fourier transform of (4.1.7) and solving for 11(k) = £‘1F(k), we obtain,

11(k.) - F(k)

_ . 4.1.8~
—lc2—ikv3— (1+A) ( )

 

To compute the H 7 norm of L—IF we multiply both sides of equation (4.1.8) by

(l + 1:2)7/2 and take the L2 norm,

(1 +13)1 I1fi‘(/.')I2

 

2
Held! .2
H7R|k2+1rms+1+xl

2 )7+/} . 2
= (1+ " (1 + 11:2)’-'3/2F(k) 111:. (4.1.9)
RIk2 +ikvs+1+AI2

 

Applying Holder’s inequality to the integral on the left hand side of (4.1.9) with

p = 00 and q = 1, we Obtain,

_ 2
I]: ‘Fll... 3 (flag...) (Fug-.. (4.1.10)

where,

1 k2 7+6
9,: ( + l 2. (4.1.11)
|k2+ik63+1+A|

We refer to S_ given by (3.5.3), the leftmost branch of the essential spectrum of L 5’

 

and observe

I13 + m, + 1+ AI = dist(A, —s_).
From F ig(4) since S _ separates C from —a we see that
dist(A, —S_) 2 dist(A, —(1), (4.1.12)

59

and relation (4.0.10) yields
Ik2+1I g Ik2+(a—l)2 g IA+a.I, (4.1.13)

for A E C and k E R, since a — l > 1. Thus, we obtain the following bound for

 

 

(1%(90,
max(g ) < C (4 1 14)
keR k — IA + 0.12-7-(3’ ' '
and substituting for (nag g), in inequality (4.1.10) we obtain,
‘E
-1 . v (7+fi—2)/2 .
IIc rIIm g c (A + (1|) |IF||H-B.
I
4.2 Exponential Dichotomy
Referring to the formulation (4.1.4), we decompose u solving (4.1.1) as
u=£*F+w m2n
and refering to (4.1.4), to solves
(L — 1)..) = F — (L — A)£‘1F = —a(.C—1F)(50)650 — )1 (L’IF) mom)- (4.2.2)
For notational convenience we define
It s —a (L‘IF) (30) e R, (4.2.3)
and
(I a —,1(£—1F) X(50:°0) e 1.2(R) n 117(50, 00), (4.2.4)

where G has support in (50, 00). The expression (4.2.2) can be rewritten in terms of

nandG'as

(L — A)'w = K650 + G. (4.2.5)
In the following lemma we derive an upper bound for w in L2 and L00 norms.

60

Lemma 4.2.1 For 7 + 6 < 2, BC > 0, independent of 6, such that for all A on the
contour C and F E H-fl, w solving (4.2. 5) satisfies

anLoo +1st s 0 IA + aim/"‘2”? “NIH—6. (4.2.6)

Proof: We construct a continuous solution 10(2) of (4.2.5) which decays exponentially
as 2 —+ :too. We note that w’ has a jump at 20 which can be prescribed by integrating
both sides of (4.2.5) over a 6 - neighbourhood about 2’0 and then taking limit 6 —+ 0,
which defines the jump as,

IIw'IgO = K + aw(20), (4.2.7)

We rewrite the ode (4.2.5), along with the continuity and jump in the derivative at

50 as,
w" + vsw’ — (1 + A)u; = 0 for z < 20, (4.2.8)
w” + vsw’ — (1 + A — ,u.)w = G for z 2 20, (4.2.9)
(11)]3250 = 0, (4.2.10)
[Iw'II 2:50 2 n + ow(§0). (4.2.11)

For 2 < 20, the solution of (4.2.8) is given by
+ ~_: — _-
10(2) = AleVI (‘ ‘0) + Bleul (Z 20), (4.2.12)

where

 

:t = —'US :l: 123 +4 +4A
1 2 ’

for A E C. To determine the subspaces E; and E; we study the signs of the real

1/

(4.2.13)

parts of the eigenvalues 11?. Using the definition of a from (4.0.8),
Re(1/1i) = —\/a —- 1:1: fiRe< 1+ é).
a

A

We consider the principal branch-cut of the multi—valued complex function 1 + a,

and its real part has the form

,\ 1 A A. 1/2
1 — =— 1 — 1 —— . 4.2.14
Re( +0) fi<I+aI+ +a) ( )

61

From (4.0.6), A E C2 satisfies A,» = —l and —ml g A,- g ml, and from (4.2.14) we

A - 12 ,\ 2 11/2
14(11): ((1-) (1)
a a a a

obtain

 

 

IV
a 31‘ 3%

I w]
A

1—‘

1
{DIN-
V
L—_.l
R

 

where the lower bound is well-defined as l < a. For, A E C2, l < 1, implies

. Re(uf)=—\/a.—1+\/5Re I/1+3) Z—VG.—1+VG—l>0, (4.2.15)

A

. Re(uf)=—Va—1—\/6Re((/1+2)S—Va—l+\/a—1<0. (4.2.16)

For A E C1 U C3, substituting for A,2 = m2A3 from (4.0.5) and (4.0.7), in (4.2.14) we

obtain,

p..( .3) -7 “(1131,.(iffy/2.1-1.441”. (4.2.17)

From Fig(4.2) we see that

Re((/1+§) 212 “—1, (4.2.18)
(1 a

for A E C1 U C3 provided, A, 3 if. For A E C1 U C3, AT is negative and bounded

m

 

away from the origin by, —00 < Ar < —l. Hence, if we impose the condition

—4a

m 2

 

—a§—l<

, (4.2.19)
then from (4.2.18)

Re(1/il) = —\/a—1+\/6Re(IIl+-2)_>_‘\/——\/a—1>0, (4.2.20)

Re(I/1') = —\/a—1—\/c—1Re( 1+2>£—\/fi+\/a—1<0. (4.2.21)

62

Wefixu=\/a—l—\/a—1<f—\/a.—1,then

Rem/1+) > u, (4.2.22)

Re(u1—) < —1/. (4.2.23)

for A E C provided (4.2.19) holds. Now, refering back to the solution (4.2.12), we
solve for the arbitrary constants A1 and B1 by expressing them in terms of the initial

data (41(20).w’(20‘ )),

16(20) = A1+Bl, (4.2.24)

w'(26) = VII-A1 + Vl—Bl- (4.2.25)

To construct a solution that decays as 2 —+ -—00, we set the coefficient B1 of the
growing part of the solution (4.2.12) to zero, which reduces the solution (4.2.12) and

(4.2.25) to

+ =
16(2) 2 111(30)cV1(Z-"0) for z < 30, (4.2.26)

w'(ZO—) = Ul+w(20). (4.2.27)

For 2 > 20, we solve the inhomogeneous equation (4.2.9) to obtain
Z
12+(z—E ) V—(z—E ) . .
111(2) 2 A26 2 0 + 826 2 0 + R(z,§)G(£)d§, (4.2.28)
4
where R(z,§) is defined as

 

R(::,§) = _ _ , (4.2.29)

and the eigenvalues of are given by

 

Vi _ —z-.. :t \/v§+4(1+A —,1)
2 _

 

A
2 =—\,/a—l:l:(/a—u 1+—. (4.2.30)
a - M
We note that ,u < 1 < a. To examine the signs of the real part of V35, we follow the
A

a—ll

 

similar steps as before. The real part of the principal square root of 1 + is given

63

 

Re (1+A/a)=1 Re (1+A/a—u )=1

Ar —4a(1—p)/m2

 

'3 -l—4a/n12

 

 

 

 

 

Figure 4.2. The graphs of Re (“I + 2%) =1 and Re( 1+ a1») = 1 for A E C.

 

 

A 1 A A. ”2
R. 1 —— =—— 1 1+ . 4.2.31
e( +a-u) V50 +a-u + a-u) ( )

From (4.0.6), Ar: —l > —(a — 11) and —ml S Az- 3 ml, for A E C2 yields

A 1 z 2 A- 2 1 ”2
Re( 1+a_u) = Ellw—a—fl) +(a_’#) +1—a_#] ,

 

 

 

 

 

 

 

2 1 —
a. —- [1
Hence A 6 C2 satisfies
Rec/g) > —\/a — 1+ ./a. :— 1— ——> 0 (4.2.32)
Re(u2—) 3 Va — —\/a — 1 —- —< 0. (4.2.33)

64

For A 6 C1 U C3, substituting for A? = 7712).? from the definition of C given by (4.0.5)

and (4.0.7), into (4.2.31) we obtain
2 2 1/2
Re( 1+ A )=_1_ ((1+ Ar)+m2( Ar)) +1+ Ar
a—u \/§ a-# 61-14 61-14

From Fig(4.2), for A 6 C1 UC3

A
Re(1+- )21,
a—p

. For A E C, the real part of A satisfies, —00 < Ar < —l.

1/2

 

 

 

 

(4.2.34)

 

provided Ar 3 logy—i)-

171

Hence from (4.2.34) we can achieve

Hal/2+) 2 —\/a — 1+ $17, (4235)
Re(1/2_) S a —T — Va. —— [1, (4.2.36)
if we impose
—4 1— ,.
—a. + p. g «P: g ——3(—2—fil (4.2.37)
m

which is less stringent than the condition (4.2.19) imposed on 1. Also note that for

—4a(1—p)
m2

(4.2.37) to be consistent we require —a + 11 < , which gives a lower bound

on the slope m of the branches

m2 2 M. (4.2.38)
a - u
Wefix V’=—\/a—1+ a—u—l> —\/a—l+\/a—}1>0, then
Bang) 3 11’, (4.2.39)
Re(1/2_) 2 —z/’. (4.2.40)

Thus, for any A E C, satisfying the condition (4.2.37) and (4.2.38) we obtain the
relations (4.2.22), (4.2.23) and (4.2.39), (4.2.40) on Re(z/1i) and Re(u§h). Refering

65

back to the solution (4.2.28) for z > 20, substituting the expression for the kernel

R(z, E) into (4.2.28) and combining the decaying and growing parts, we obtain

 

 

_ z l
W) : 326u§<z—20> + / eu§(z-4)__G_(€_):d€
- 1’2 — l’2
- 20 _
_ 2 C . _
+ z + _ ,
+ A262 (Z—bo) _ [6"2 (Z €)—:—(—€)—:d§ . (4.2.41)
- V2 _ l’2
_ 30 -
Since 10 is continuous at 30, we see that,
111(20) = A2 + 82. (4.2.42)
Evaluating the derivative of 10(2) given by (4.2.41) at 50,
10,63) = AgzxgL + Bgl/g. (4.2.43)

Substituting for 11.1726) and 11/(56L ) from (4.2.27) and ( 4.2.43) into the jump condition
(4.2.11) we obtain,

2421/2Jr + 321/2— = K +(I/1+ + a)w(§0). (4.2.44)
Solving for the arbitrary constants A2 and 82 from (4 2.42) and (4.2.44) we obtain,

(V1+ — u; + a)w(§0) + n

 

 

A2(w(20)) = + _ , (4.2.45)
1’2 - l’2
_ uv(20)(1/+ — 11+ + a) + K.
B2(w(::0)) = 1 + 2 _ . (4.2.46)
1’2 ‘ ”2

it remains to tune 11420) so that 10(3) —> 0 as 2 —+ 00. This is acheived by balancing

the coefficient .42 against the inhomogeneous term in (4.2.41)

4201450)) = —_—1—.: / e‘”3“‘50)0<4>dg. (4.2.47)

20
Since Re(zxé+ > V), the integral in the right hand side of (4.2.47) is convergent and we

denote its value by M

00

. —v+(£—2 )

M: c 2 OG(§)d£, (4.2.48)
20

66

4
A (4.2.49)

which modifies (4.2.47) to
4201450)) = ._ +-

1’2 ‘ 1’2
(4.2.50)

||G||L2

WI S ——- _ ,
(/Re(1/;) V
(42.51)

Usrng (4 2 39) M is bounded above by the followrng
< llGllL2 .
Substituting (4 2 45) into (4 2.47), the solution w ( 0) of (4 2 47) is given by

* _ 11/! + H
w (20) = _ +

V2 — V1 — (1

(4.2.52)

From (4 2 22) (4 2 23), (4.2.39) and (4.2.40) we obtam
)—a| 2 u’+u+a.
(4.2.53)

II/2- —— Vil— — al> IRe(1/2) ——Re(1/
IVS" -— Vi" lRe(1/3") — R-e(1/2-)| 2 211’.

Usmg and (4 2’0) and (4 2 52) |w*(§0)| given by (4 2 51) satisfirs the bound

| *(son 3 0 (101,2 +141).

(4.2.54)

The mequahtles (4 2 50) and (4 2.53) provides a bound for A2 given by (4 2 49)
(4.2.55)

I442| S C HGHL2-
A2, and using (4.2.55) and (4.2.54) we have

Denoting B* = B2(w*(50)) = “C(50) -
(4.2.56)

the following bound
|B*| s C (“01,2 +141)

Substituting for 42 from (4 2.47) into (4.2.41), the solution of (4 2 9) for z > 20, IS

w(z) : B*eV§(z—§0)+/eu2—(2—£)____ 0(6) -——-—dE
_ ”2 ‘1’2
_ 30
.00 fig
+ fe-V2(€—Z) 7d€_ je—V2(€—2) __(—G€)+d€ ,
_ V2 —V2
.30
Z
1. z z — 0(6) +5 0(6)
= Be"2( (1)/12'2““ _ +d€ +/ ”’2‘ z) _ ————+d€
_ V2 1’2 z 1’2 1’2
. 20
(4.2.57)

 

67

Hence the solution of (4.2.5) is given by

 

 

 

111*(20) Z S 20
we) = 3.615e—20) + jeuge—e) _G(6+ d6 + fe-ugre—z) f(€)+ 4g Z 2 50
2 V2 —'V2 Z V2 —V2
0
(4.2.58)
Computing the L2 norm of 11), we find
IIUJIIQ <I’LU*( (Z0)I IB*I (“GHLQ llGllL2
L2
<I/Re( V1) +I/Re(—1/2) +I/Re —1/2 )sz2 — 1x2 I +I/Re(1/1+) II/2 — V2 I
(4.2.59)

Using relations (4.2.22), (4.2.23), (4.2.39), (4.2.40) and (4.2.53) determining the lower
bound of the real parts of Vi}: and 113‘, and substituting for Iw*(§0)| and |B*| from

(4.2.54) and (4.2.56) into (4.2.59) we obtain

141,2 _<. C (101,2 +141) . (4.2.60)

From Lemma 4.1.1 we obtain the following estimates for “GI! L2 and IKI,

IIFII _
I = —1 r _ < C 'H ’
1w ”(a 4)..
IIFHH—a
IA + aIQ-w—m/Q’

 

 

 

avs (5‘11?) (20)I g C

lfil =

which yields an upper bound for ”w“ L2

l/\ + aI(2-‘r-1‘3)/2'

 

||wl|L2 S C (4.2.61)

From (4.2.58), we compute IIwIILoo. Using the lower bounds on the real parts of Vi}:

and 113: we obtain
”6|le 3 C (w) + |B*| + ||G||L2). (4.2.62)

Substituting for |w*(§0)I and |B*I from (4.2.54) and (4.2.56) into (4.2.62) we find

lIFllH—fi
l4 + aI(2-7-H)/2'

 

s C (lnl + IIGHL2) s c (42.63)

ll'wIILoo

68

I
We define the semigroup S 63(t) E S (l) generated by L (5’ as an operator S (t) : H ‘fi ——>

H '7 which maps F E H "'5 to the solution at time t of the initial-value problem

wt = ngw, (4.2.64)

W(O) = F. (4.2.65)

If L 43 is sectorial, then for F E H ‘43 , S(t) has the following representation

_ 1 A. -
S(t)F _ 2— /e ’(L — A) leA. (4.2.66)

m
C

‘By the aid of the previous lemma, we ‘obtain the bound on the H 7 norm of the

solution a of the resolvent problem (4.1.1).

Lemma 4.2.2 For all 7 +13 < 2 3 C > 0 such that for all /\ 0n the contour C and
F e H‘”

 

(IA HUB-(W? ' '

where u = [I—IF. Referring to 77rd defined in (4.0.4), the semigroup action

ll'l‘llH’)‘ S

 

 

S(t)7~r0F||H7 satisfies

~ Ce—Vt
l|S(t)7T0FllH7 S W llFllH—fia (42-68)

for all F e H-3.
Proof: Refering to the decomposition (4.2.1) of the solution a of (4.1.1)
u = C—lF + w,

we obtain an upper bound ”74ng using (4.1.6) and (4.2.6),

C llFllH—B
(|/\ +al)<2-'v-fi>/2'

 

IIUI|L2 s Ils-‘FllL2 + IlelL2 s (4269)

69

Note that IILI‘IFIIL00 S II£‘1FIIH7. Using (4.2.63) and (4.1.6), ||u||Loo can be

 

bounded as
0 NF“ —3
< 'lFII < H ' . 4.2.70
quILoo _ Ill: H, + “114le _ (a + al)(,_,_,,,/, ( >
From formulation (4.1.4), (L — A)u = F can be written as,
£14 = F — au(§0)6§0 — #uX(20,oo)- (4.2.71)
Inverting [1 yields
u = £—1F — (12450171650 — u£_1u)((5000). (4.2.72)

Now, taking the H 7 norm of u given as in (4.2.72) and using the estimate (4.1.6)

from Lemma 4.1.1,

 

 

 

 

 

, . -1 , -_; +1 _ -1 __
Ilttllm 3 Hz: Film + alu—(~o)1II£ 6.0 H, + 11 IL UX[:0,66)IIH,,
c IIFIIH_); (.‘(r IIuIILoo II5§OIIH_/3 c IIuIILg
I/\ + aI)(2—’7—4’3l/2 l/\ + aI(2-’7-[3)/2 IA + aIl‘Z-‘r-Wfl

Using the bounds from (4.2.69) and (4.2.70) we obtain

CHFHH—6
A. < ' _

 

for all /\ 6 C. As per definition (4.0.1) introduced in the begining of this Section,
L <5 is a sectorial operator and the semigroup S (t) generated by L 5 can be espressed
as (4.2.66). Taking the H 7 norm on both sides of (4.2.66) and using the bound for

II'ullHy from (4.2.67) we obtain,
IISUWOFHHV S C/Ie’VI II(L — A)"17F0FIIH7 dA,
C

llfOFIIH—a

. ,At _
g (,/I(. I |A+a|(2‘7‘”)/2d)" (4.2.73)
C

 

We prove that there exists constants u > 0 and w > 0 such that,

Ar 3 —(1/ + w IAI), (4.2.74)

70

for /\ E C, where Ar = Re(/\). We define u = g and w = 2 11 2. For A E C1UC3,
+n1

 

where C1 and C3 are given by (4.0.5) and (4.0.7),

1
A +w/\ A +———- A2+Ni2A2
" " " 2WV " "

2 a: _<_ :1 = _V,
2
For /\ 6 C2, from (4.0.6) we obtain
Ar+w|A| = —l+-—-—1-—- l2+/\2
2V1 +7742 2
1 —l
g —l + ————\/l2 + 772212 = -— = ——1/.
2v1+m§ 2

(4.2.75)

Using (4.2.74) to bound /\ in the integral (4.2.73), we obtain

é—wMH

l)‘ + ale—2%),"

 

IISUWOFHmSCe—Vt / 261A llfoFHH-s-
.c

For large time the integral is uniformly bounded. For t < 1 we set 5‘ = MM and the

integral transforms to,

e—wA

)(Q-a-fil/Q)

e-aA

r14; ||7F0F||

 

H-fi’

fisl)’:

C

IISWEOFIIHs g are /
(a+(

(}€-_Vt
t(7’+l‘ll/2 6/(at(2“y—/5)/2+S(2—7—3l/2)

 

d3 Madman

For 7 + ,8 < 2, the integral is uniformly bounded since the singularity at t = 0 is

integrable. We also have

IA

 

 

IIFHH—a + |(F- 41.0“ ““40

C llFllH—y,

-F _ ,
“70 HH .4 41—4

IA

Hence, we obtain the semigroup estimate (4.2.68) for F E H ’3 and .3 > 1 / 2. I

71

CHAPTER 5

The RG methodology

We employ the modification of RC method developed in [1, 33, 34, 38] for asymptotic
decay of dissipative pde.. In this section we employ. the smoothing bounds on the
semigroup S (1‘) from Section 4 to obtain decay estimates on the remainder I/V given

by (3.2.1).

5. 1 Overview

W'e decompose the solution of the initial value problem (IVP)

Ut = F(U), (5.1.1)
U(O) = UO, (5.1.2)

where F is given by (3.1.3), as
U(z,t) = ¢(z; zo) + W(z, t), (5.1.3)

where the front position 20 = 30(t) is time dependent. The IVP (5.1.1) becomes

 

2;, = F(a) + LZOW +N(W), (5.1.4)

W(2) 0) = U0 - 45(2; 20(0)). (5-15)

72

The operator L30 is time-dependent. To eliminate this obstacle, we recast the IVP
(5.1.4) as a series of IVP’s on [tn, tn+1] each with a fixed coordinate system determined
by En. Given the solution U at time tn, we first determine the frozen front location
in so that

Wn(0) E U(-,tn) — (bfzizn) 6 X2..- (516)

IV" now evolves according to

 

')
W(Z, tn) = ‘4’," (5H18)

with 26 chosen to enforce W(l) E 73(7rgn) for t E (tn, in“). Note that the front loca-
tion 20 is discontinuous at the renormalization times. The diagram Fig(5.1) shows
that 20(t) jumps from 20(t; ) to in at t = tn.The RC equation is then the map from
”"1, ‘——> Wn+1. The nonlinear stability of the system via RG methods exploits the fact
that, the evolution of the front location 20 is on a slower time scale and the secularity
L20 —— L50 is a lower order operator than either of L20 or LEO' In this manner we
attain uniform decay estimates without uniform smoothing bounds on the semigroup
S(t) for t > t0.

Throughout this section a subscript En indicates a quantity associated to the coor-
dinate system centered about the base point z z: in. For example, \Ilgn = \Il(z; Zn)
denotes the principle eigenfunction of Lin where L5,, denotes the linearization of F

at o(:; 2.”). We assume at time t = t0, the initial data U0 satisfies,

for some 25 E R, where 6 > 0 will be specified later. The following proposition allows

 

<sz — UOllm g 5, (5.1.9)

us to choose our base point 30 about which we develop the local coordinate system

for the RG method.

Proposition 5.1.1 Fix (5 << 1. Assume U0 and 25 E R satisfy ||W...||H~, _<_ (5, where

the remainder W... E U0 -— (1536. There exists M > 0, independent of U0 and 25, and

73

 

 

 

 

 

20(1)
20(t 3')
2
(t 7)

2/20 2

/ZO(t 13
20

l l l ‘

t0 tl t2 t3

 

Figure 5.1. Evolution of front. location due to the iterative scheme.

a smooth function H : X50 —+ R such that 50 = 25 + HOE.) satisfies,

W0 2 U0 — 4250 6 241.0. . (5.1.10)

'Moreover, if H}. E X50 for some 20 E R then
lfo-liil S Molfo-Eolllwarllm- (5.1-11)
Proof: Substituting for U0 2 0,536 + IV... in the definition of VVO given by (5.1.10)

we obtain

W0 = l/V... + $26 -- (D50. (5.1.12)
The condition (5.1.10) is equivalent to the orthogonality of WO to ‘1’20
”EOWO = 71'50 (W... + $26 — (2550) = 0. (5.1.13)

We prove by the Implicit function theorem that, for given W .. there exists 50 satisfying

, ~" 0’ ~0 .

74

we observe that A(E() = 26, W... = 0) = 0. The partial derivative of A w.r.t 50 is given
by,
1,
04/20
85.0 ,

 

—,\1/:0 ' =1: — :7

and refering to Lemma 3.6.3, the inner product of and \Iljlj0 is uniformly

83
cf” NI

bounded away from zero for all 20 E R. Thus

 

0A {M530 T
=(0Z0 , WEO)I 20—:* “7* _0 —60 > 0 (5.1.15)

The implicit function theorem guarantees the existence of a smooth function ’H which

820 20:20 W*=0

provides a solution of (5.1.13) for UO'in a neighbourhood of the manifold,

M = {670:0 e R}. (5.1.16)

In addition if W... E X 50 then (Vl'"...,\I!:-,O) = 0. By the result above, there exists
30 E R such that

._ t _
(w. + 45.6 - 6.0.11.0) - 0.

Applying the mean value theoem to (1)50 on the interval (26, EU) and observing that

II6<15EL — _O(1) and II\III~0IIL2 = (9(1) we find,

6:09,
|(mio)l (6.44220),

(M425) W) )
a g ’ 20 1
Zn

_ _ 2
= 6(zg)|23—20|+0|25—-0| . (5.1.17)

 

 

*

= 130 “ zOl

 

 

where 25 E (20,26) and from Lemma 3.6.3, 6(z5) 75 0. Alternatively we estimate

I(W..., W20” as,
(mu

{I

Kit/.210 — 111,)l s also — fol “mum. (5.1.18)

75

This gives the desired bound,

136 — 501$ C150 — EOIIII'V*11H7-

In application of Proposition 4.1 at time t = tn we take

50 =- En—l: (5.1.19)
3* = Z0(tn.—l)a (51-20)
and construct 20 = 2... (5.1.21)

5.2 Control of Residual

We now estimate the decay in the remainder over the interval [tn_1. tn]. Without

loss of generality we consider n = 1. For the underdetermined decomposition,
U(z, l) = 96(2; :50) + VI/(z, t), (5.2.1)

the evolution for the remainder W takes the form,

as ,

 

1.1/Mayo: = R+L20W+AL W+N(W)+e3/4—“I's/2ns(g,t), (5.2.2)
W(£,O) = W0. (5.2.3)

where 20 and hence W0 are provided by Proposition 4.1. The term AL = L 30 — L 50
describes the secular growth implicit in 20 sliding away from the frozen front location

20 and takes the form,

W(Zo) 6 _ W(EO)

AL W = (116(20) - 1's(50)l3z”’7 + 00(¢’(z0' 2:0) “:0 <1)’(§0' 50)

650) + lu'X(50,Z0)VV'
(5.2.4)
The residual R is given by (3.1.22). The decomposition (5.2.1) is made determinate

by requiring W E X 30 which is achieved by imposing the non-degeneracy condition

76

7150 Wt = O, which is equivalent to the following equation,

 

(94sz T I __ 3/4_7 /2 i
(620 .WEO)L2 Zo - (R+L20W+ALW+N(W)+6 s 77.s(',t).‘I/50)

(5.2.5)

Solving for 26 we obtain the front evolution equation,

, (R + L50 + AL w + N(W) + 63/4—1'8/2n..(-, t), 111,0)112
20 __. , (5.2.6)

(5631’ wifllfl

which is well-defined since 6 = (59%,11IIE0)L2 is uniformly bounded away from zero

 

by Lemma 3.6.3. We observe that,
_ 1 _ 1 1 _ ,, 1 1 __
(LZOW, 111.0) _ (W, Lzoxpzo) _ (1V,,\0\11,0) _ 0,

since W E X 50. This reduces equation (5.2.6) for front velocity to,

(R + AL 11/ + MW) + 63/4~-1s/?ns(.,1), )1]; ) 2
I 0 L
.20 = . . (5.2.7)

642 i ‘
(fi’wiolfl

To estimate the terms in the inner product we examine the residual R. From the

 

 

 

 

 

 

 

bound on the residual given by (3.1.23) and the observation that I ‘1'le L1 = 0(1).
we find,
|(R, 111:0)I 3 ”3|le | 11-. L1,
S C (\/Elz -— fol + 6). (5.2.8)

From (5.2.4) we see that of the projection of secularity AL W onto \I'IEO takes the

form,

I(ALW,\I/,1,O)I g (63(20)-v.(20)|I(aZW,xIJ;O)I

 

 

M 1 ., _Wliol 1 ;
+ ”C 4(4)) 50““) 412612431

77

To estmate the first term in the left hand side of (5.2.9) we integrate (BZW, \III) by

parts and estimate the velocity terms from (3.1.25) and rewrite this term as,

 

 

 

 

 

lv.s(20) — v.(56)l |(a.I/v,1fi)| s 0% I2 — zoI (W. 8.14.0)I
From the form of 4120 given by (3.6.38) we observe that (934/20 L1 = 0(1), and
we achieve the following estimate,
11’s(30) “ Usi£0ll I(8zVVa \I'I)I S 61¢;le — 501 lllVlIH‘Y- (5-2-10)

The second term on the right of (5.2.9) involving the point evaluations can be ex-

pressed as,

 

 

, f _ 1110(4))
S I” (20) — W (Zollm
LIA/(EON-
95130; 50)

+ ‘IW'IZOWEOGOII

 

 

+

 

1111.426) — 114066)] (5.211)

¢’(20) - 03’(§0) I
<1)’(Zo;zol¢’(50;§0) '

we apply the inequality (3.2.5) to the three terms on the right hand side of (5.2.11),

 

 

since 42’, Ell-,0 and W E H ‘5. Also from Lemma 3.1.1, (12’ is bounded away from zero

in any bounded interval containing 20, reduces (5.2.11) to

111‘; (26) 11*. (26) 2
W zO)—-:0—— — W 30 —“9—_— g C :3 — 2 7‘1/ w . 5.2.12
< “mm, ( beam I0 4 IIm.( )
The last term on the right hand side of (5.2.9) has the following bound
20

I(X(50»20)Ww‘1’:‘0)I s / |W|

.—
~
,

0

1111.0

 

 

cusm—amwud

 

1110

 

 

LmSCW-mmwhr

(5.2.13)
Finally substituting the estimates developed in (5.2.10), (5.2.12), (5.2.13) into (5.2.9)

we obtain

l/\

I(AL W,‘1’§0)I C (x/EIZO — Eol + I26 — 26W”2 + (20 — zol) (Iii/(1...,
< C (Izo - 50'1—1/2) IIWIIm. (5.2.14)

78

where we have neglected higher powers of Leo — 50]. Addressing the nonlinearity

N (W), from (3.2.16) with B = 7, we obtain the following estimate

 

 

 

 

 

 

 

 

[(MW), 1111-0)] s IIN(W)llH—v I 112,, m,
s c (Iii/“$1”. (5.2.15)
Finally, for the noise term
C3/4—78/2 (7,30, l), (111:0)I S c3/1'11-“153/2 III]3(t)IIH._73 \I’EO H7,
3
S Cf3/4_78/21177.9(tllIH-‘rs ° (5-2-16)

Turning to (5.2.7) we see that the denominator is uniformly bounded away from zero
by Lemma 3.6.3. Employing the estimates developed in (5.2.8), (5.2.14), (5.2.15)

and (5.2.16), we obtain an upper bound'on the front velocity,

- —'—12 7 ,. +12 _..2
IzhlSC(\/5|30—20l+|26-zoll / IIW llH7+||II “117/ +c3/4 .s/ ll’lSIIH—as+€)-

(5.2.17)
To control the front dynamics we introduce the quantities,
710(7) : sup les-‘OMIMSMI, (5.2.18)
t0<8<t0+T
T1 (T) = sup IZO(S) — 50I. (5.2.19)
t0<3<t0+7

where the (9(1) exponential decay rate V is motivated by the semigroup estimates
(4.2.68). Using the definition of T 0 and T1, for to < s < to + 7', the expression for

front velocity 2:6 given by (5.2.17) reduces to

 

36(8ll S (1" (W710) + Tl(Tll—1/2To(Tl<-1'V(3_t0)

+ T0(r)1+1/2e-V(7+1/ 2“8-10) + 63/4-13/2 (Ins(s)IIH_,, + @6220)
Throughout this section we follow the convention that To E T0(r) and T1 E T1(r).
In the following lemma we develop a bound for the change in position of the front

location, T1, in terms of T0 and the length of time, 7', that the coordinate system is

frozen by.

79

Lemma 5.2.1 Fix 7 E (é, 1). For 6 and T0 sufliciently small there exists C > 0,

independent of r and e satisfying
fir << 1, (5.2.21)

such that
T. g C (TOWI/2 + 63/4““18/2r) , (5.2.22)

is true.
Proof: Integrating 26 over the interval (t0, to + r), and using definition (5.2.19) for

T1, and the estimate (5.2.20) we find

t0+T T

/ I33(.9)I4.=/(.5(10 +6163
[0 0
T

C/(x/ZTl +Ti‘r-1/2TUC—vs+T8+1/2c—V(7+1/2)s
0
1523/4719” IIn.(.s>IIH—.. + .) ds.

7-1le

|/\

|/\

We evaluate the integral on the right hand side of inequality (5.2.23), obtaining
T1 3 0 (war. + :rf‘l/QTO + T3+V2 + 63/4-18/2II1..IIL66 T + .7) (5.2.23)

Here, for notational convenience, 1].; ,oo denotes 773 00 + _. . Applying
L L (R .17 ”S(Rl)

, . - 1 7‘1/2 ' __ l — 1 .
Young s inequallty to (/T1 T0 w1th p —— (_1 /2 and q — m we obtain,

 

1

1/2T0 g $1". + (CTO)372-1, (5.2.24)

CT]—

for 7 E (1 / 2, 1). Substituting (5.2.24) in the estimate for T1 in (5.2.23) yields

 

1 1
T1 3 0.577114. —2-T1+(CT0)3/2‘7 + my”2 + c (63/4—18/2 IInSIILoo + e) T.
(5.2.25)

80

Since we have rescaled the noise term so that “773” L00 is 0(1), we combine the two
terms in the coefficient of r in the right hand side of the above inequality (5.2.25).
We neglect higher order powers of T 0 and solve for T1

T7+1/2 +e3/4‘78/22

T1300 1
2'—(7VQT

Since the constant C depends only on 7 we have

 

(5.2.26)

so long as the following conditions hold
\flr << 1. I

To estimate the decay of the remainder IV, we return to the evolution equation (5.2.2)

and apply 77' to both sides. Since 710”". = 770L2- lIV = 0, 5.2.2 reduces to,
0 0

W. = i2 + LEOW + 710 (AL W + N(W) + 63/4‘1‘1/2ns(5. 1))

1’,V(€ t0) = 1V0 (5.2.27)
where we have introduced the reduced residual R defined as,

R = 71.. (R — 26%) . (5.2.28)
30

Using the frozen coefficient semigroup S = S (r; 20) associated to L50, introduced in
Section 4, the variation of constants formula applied to (5.2.27) yields the solution

W(E, to + T) = S(T)l’V0
t0+T

+ / S(to + T — s) (1‘? + 710 (AL W + N(W) + 63/4‘119/2n,(g,s))) ds.

t0
(5.2.29)

81

Changing the limits of integration and taking the H 7 norm on both sides of equation

(5.2.29) we obtain,
“W(to + TlllH7 S llS(T)WOIlH’T

+f ”5 <1 — s) (11 1 from W +N<W> + 61/477244. 8») IIH. 15-2-30)
0

In section 4 we derived estimates S acting on F E H “"3 . In the following lemma

we develop semigroup estimates on the terms in the right hand side of the inequality

(5.2.30).

Lemma 5.2.2 : For \/ET << 1 we have the following estimates on each of the terms

on the right-hand side of (5.2.30).

((1) Semigroup estimate on the reduced residual H. is:

~ _ _, . 7-.. i , .--12 , 1—1/2
II.s(T_.<.~)RIImgIIIe ”(T ”((5714.63’4 18/24IIWIIH. (qu / +||W||}n- ))

 

 

(5.2.31)
(b) Semigroup estimate on the secularity AL(W) is:
. 7-1/2
~ . _ _ 6T1
“S(T — s)7r()ALW||H~, g Me ”(T 3) IIWIIH. d??? + (,1- 3)7 (5.232)
(c) Semigroup estimate on the nonlinearity N(W) is:
~ e—V(T—S) 744/2
(d) Semigroup estimate on the noise term 773 is:
-u(7'—s)
“S(r - shronsllm S M (5.2.34)

82

(e) Semigroup estimate on the initial condition W0 E H 7' is:
||5(T)W0||H7 S Me—W “Well.- (5-2-35)

Proof (a): We estimate the action of semigroup on the reduced residual 17. Since

it E H7, substituting ,8 = —7 in (4.2.68) we obtain,

 

 

 

 

 

 

 

IIS(T —- s)fiII S Ce_V(T_S) FII . (5.2.36)
H7 H7
From the form of it we obtain,
- 61¢
RII < C R C 2’ ' .
l] H. ._ II Hm + I 0I 8,0 m

We estimate for IIRIIH", from (3.1.22) and I:(,I from (5.2.20) to obtain,

IIRIIHi S C (film _— EUI + €3/4n73/2)

_ _ _' , —12 ,
+ C(IZO-zoIHzo—zow l/2+II11II},./)IIIIIIm. (5237)

Using the definition of T1, and neglecting higher powers of T1, in (5.2.37) we obtain

the following estimate,
Ilse — as“, s Mew-8) (7271+ 9/478” + llWIIm (734/2 + mum/2)).
(b) To prove part (b), using (7?)

”S(r — s)7~rOAL WllH’Y S “S(r — s)AL WIIH’7' . (5.2.38)
refering to the secular term AL from (5.2.4) we obtain,

“3(T — SlAL Wllm S It’s(20) - vs(50)| ”S(T—Slfioazwllm

 

 

 

 

 

 

~ W(Zo) W(Zo)
+ ”C S (7—8)“)(19’00) “’0‘ 44261530 m
+ uIIS(r—s)7~I'0X(§0,ZO)VVIIH7 (5.2.39)

83

Since W E H7, BZW E 117—1 substituting for [3 = 1 — 7 in (4.2.68) we have the
following bound,

6—1/(7— s)
VT —3

The formal pulse speed us is continuous and varies slowly in space satisfying the

”S(T — .13)a.w “H. < M (IazwIIH,_. (5.2.40)

bound (3.1.25). The derivative operator D is continuous as a map from H 7 to H 7‘1,
and we have ||Wz||7,_1 _<_ IIWIIT Substituting these estimates and using the definition
of T1, in the first term of the right hand side of (52.39) we obtain

PHI/(T s)

l"s(30) —113(30)|||S(7— )(le ll7< M—fi—T—S'WTI llWllH’i‘ (5241)

To obtain the semigroup estimate for the second term in (5.2.39) involving the differ—

ence of delta functions, we compute its H '7 norm. By definition,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M5, "11325- (p
W(Zo) WGO) . o’tm) “0 ¢’(50) ”0’ L2
‘/ ~ 30— I : '—§06 '— Sllp A
(9 (~10) 7) (:0 l H“? (.0 E H7 ‘1’ H7
19 # 0
W’ 2 1V 5 _
#3,) (”0) - #1900)
= sup (5.2.42)
, E H. “Him
<79 75 0

Since ¢’(z) is uniformly bounded away from zero on (20,2130) by Lemma 3.1.1), the

numerator of (5.2.42) can be bounded by

 

 

 

WW 7-0) WOW |¢(20)90(Zo) + W(zo)-¢'(50)IW(§0)90(20)
W(ZO )¢’(Zo)| |¢’(Zo)¢'(50)|
Mzol- <P(20)|W(30)¢'(30) C ,, -2 7—1/2 ,, ,
+ I¢I(ZO)¢I(§O)I S 1‘10 01 llWllH'V ll‘r’llHW -

where we have used the Holder continuity of W and (p E H 7. Recalling the definition

of T1 from (5.2.19), we arrive at the estimate

W(Zo) ~ _ W(50)6_
<P’(~"0) i0 4’90) 20

84

—1 2
32,1 / llWllm- (5.2.43)
[1‘7

 

 

 

Substituting 73 7 in the semigroup estimate given by (4 2 68) we obta1n
W(Zo) W(Eo) ) 6"”(7’3) 7—1/2
S 7—3 -— z — _ 65 < Ill—T W . 5.2.44
II ( W444.) 0 71(4)) 0 —s>1 1 ” ”H” ( l

_ 7
Turning to the third term on the right-hand side of (5.2.39), the characteristic function

 

 

 

H A]:

)W E H 7 plugging B = 7 in semigroup estimate (4 2 68) gives

 

 

 

 

 

                          

 

 

X(§0,ZO
W(T-Sl
_ __ _ /
II( S( T S)X’~'(0130)7WII < M (I _ I), X(,0I,O,u Hfl. (5.2.45)
We obtain the followmg bound for IIX .. "WWIIH—‘r
I X(50~20)W"p 2I
IIX(:0IZ (”W II_ = sup ( . . )L
I E H7 ”SVIIH'I
99 75 0
f’IWIISPIdZ
g sup ~0 .
99 E 1]", Iii/“H7
19 75 0
_<_ I20 — 7 (5.246)
Substituting this estimate into (5.2.45) we achieve
—I/(T— s)
IIS(T- 6))“,0 0,14 7 < AIWTl IIWII,. (5.2.47)

Combnnng the estimates (5 2.41), (5.2.44) and (5.2.47) into (5.2 39) we the secularity

 

 

35,
—1/2
T T
_ A < I —l/( T—S «E l __ 1 ,
use s) W I... A e IIWIIm m + (T, _ 3., (T _ 8,,
(5.2.48)
Neglecting the higher order terms we obta1n
—1 2
x/ETI f /

 

_ A .< I —z/(1r—s) ,7
Ilse 4) LWIIH._4e IIWIIm 77—“ (7-.)7

(c) To derlve semigroup estimate on the nonlinearity we use the estimate (3 2 16)

85

and substitute [3 = 7 in (4.2.68) to obtain,

—l/(T-3
IIS(T — S)N(W)|lm s MEET} ”Mail/2.
(d) For the noise 773(-, t) 6 11-73, substituting {3 = ’73 in (4.2.68) yields,
—l/(T—S)

e
“S(T — 5)7lsIII-I’T S MW llnsIIH—7s~

Since II7)3IIH_A,S = 0(1) we obtain,

—l/(T-S)
”S(T - 5)773”H87 < (II(—:32]:-
(e) For W0 6 H7 from (4.2.68) we obtain,
IIS(T-—5)W0II _ <Me_"( T’s )IIHO ”7' (5.2.49)

I
Combining the estimates developed in (5.2.36)-(5.2.49), the solution W given by
(5.2.30) satisfies

T
“W(lo + T)|IH7 S Ale—W III'VOIIH‘I' + M/e—V(T_S) IJETI + 63/4—73/2
0
+1 2 ~_1 2
IIWII}, / + IIWIImTI /

_I.

—1 2 l 2
JETl Tiy /+IIW”H7/
+ IIWIIm WTS+ (7-5))
€3/4-fl3/2
I s

(7' — s)73

 

 

+ (”(7—3) (5.2.50)

To estimate the decay of IIWII H‘Y we multiply the above equation (5.2.50) by e’” and
use the definition of To from (5.2.18)

7.

“W(to + T)|lm s M “Wollm + M f I (m. + «234-722)
0
1 2 —l 2
+ “II/Hm (uwnm/ +237 /) (5.2.51)

 

 

m1 T3"1/2+IIW||7 ”2 ”3.3/44,22 d
«Ti—3+ (7.35 +8 (__‘w— 3

+ IIW||H7

86

We take the supremum of both sides of (5.2.52), over T E (0, T) to obtain,

7'

 

 

T0(T) S 111T0(t0) + 61/ I6” (V2711 + 53/4‘"/s/2)
0
+ T0 (6274/3373 _1/2 + T17 —1/2) (5.2.52)
+ T x/ETl 7117—”2 + e‘(”"1/2)"3T07_1/2 + W 63/4-73/2 d
0 VT—s (T—SW e (T—s)78 8

Performing integration on the right hand side of inequality (5.2.52), we obtain

To(T) S M [T0(t0) + 6'” (‘v/ETI + €3/4J/S/2 (1 + 71—78))

+ (T8+1/2 + 710711,)-1/2) (T + ”Fl—7) + fiToTl]. (5.2.53)

1_.

Since T > 1, T satisfies T > T 7"", T > T1‘7 and the constant term 1 is dominated

by T, which reduces the right hand side of (5.2.53) to

T0(T) g M IT0(t0) + e” (fin + 63/4“‘IS/27-) + (T3 +1" 2 + T0T17"1/ 2) r + (EFTOTII .

(5.2.54)
From the inequality (5.2.24), the term TOT: M1/2 can be bounded above by,
T0T3‘1/2 g C(T1 + Tg/(3/2‘“).
Now substituting for T1 from (5.2.22) we obtain,
TOTY—l/Q S C(TJ‘I'1/2 + Tol/(3/2—7) + E.3/4-‘78/27.)3
g C(TJH/Z + 63/4—78/2r), (52.55)

for 7 E (1/2,1). Bounding T1 from (5.2.22) and 71’071721/2 from (5.2.55) in the right
hand side of (5.2.54) we obtain,

T0(7‘) _<_ [I] ITO(tO)+€'/T (fl(Tg+l/2+€3/4—73/2T)+63/4-78/27)

+ (7344/2 + (3/4—73/2T) T + WT0(T8+1/2 + c3/4_73/2T)I(5.2.56)

87

We combine the terms in (5.2.56) to obtain
T0(T) S M IT0(t0) + \/EeVTT8+1/2 + ewe3/4-73/2T (\/E + 1+ e—WT + e‘VK/e—TTO)
+ Tg'+1/2(T + Jam] . (5.2.57)

VT

The term T6_ is bounded above for T > 0. From the condition (5.2.21) on T,

\/ET < 1 and, neglecting higher orders terms in T0, we simplify (5.2.57) obtaining,
3/4-78/2 VT VT 7+1/2 7+1/2
T0(T) S ll] T0(t0) + E 8 T + \fle T0 +T0 T . (5.2.58)
We impose a second constraint on T, requiring
7T8"'1’/2 << 1, (5.2.59)
. 7+1/2 T . _
we can make M TT0 < -§. The estimate (5.2.58) for T0 reduces to,
T0 3 III (T0(l0) + 63/4—IS/QCWT) + fie”T8+L/2. (5.2.60)

To simplify 5.2.60) further, we note that T g e”, provided l/ 2 11171, which reduces
(5.2.60) to
T0 3 III (T000) + 63/4-78/26'”) + (fleWTJH/z. (5.2.61)

\Ne solve the above inequality (52.61) for T 0 with the aid of the following lemma. The
motivation of the lemma being, if the increasing function f (11:) = a + bx“, 1 < a < 2,
is small initially, a << 1, and grows slowly, i.e. f’ ((1) << 1, then it intersects y = .2:

exactly twice with one fixed point lying in a neighbourhood of a: = a.

Lemma 5.2.3 Fix 7 E (1/2, 1). Let T0 : (0, T) —+ R+ be continuous. Then ifT0(t0)
is sufi‘iciently small and T0 satisfies (5.2.61) then

T0 g M (T0(t0) + 63/4—7'8/2eW). (5.2.62)

for all T satisfying

T s I3 <3/4 - 73/2) ““6'

 

(5.2.63)

9
V

for any [i < 1

88

 

f(x)
."yzx

a Y-i/z
..--"E[(Y+1/2)g) : )

 

I

 

 

 

 

Figure 5.2. The graphical illustration of solution of the inequality for T 0.

Proof: To simplify the computation we define the constant term and the coefficient

of TJH/Z in the right hand side 'of (5.2.61) as,

M (T0050) + 63/4T73/2eWT1—7) , (5.2.64)

9
'l

b = flew, (5.2.65)
which reduces the inequality (5.2.61) to,
T0 g a + 523* 1/ 2. (5.2.66)
We fix T and investigate the fixed points of the function
f(:c) = a + 5557+”? (5.2.67)
We note that

1
f’(x) = b(7 Jr-1/2):I:'l_1/2 >1, if :1: > (W)7_1 2, (5.2.68)

as illustrated in Fig(5.2). The function f (2:) has a fixed point at
.‘L‘0(T) = a. +l)(17+1/2 + 0(b2a27+1), (5.2.69)

89

and another at

l
1 7—1 2
—— . 5.2.70
“(7) > (be +1/2)) ( )
Thus T0 satisfying (5.2.61) also satisfies

To < .1‘0(T) or To > 2:1(T). (5.2.71)

Thus if a << 1 and b << 1 there is an excluded region, either 0 < T0 < 2:0 or
211 < T0 < 00. Since 2:0(T and T0 are continuous and T0(t0) < (150(0), then T0 < 2:0(T)
so long as 10(T) < 1T1(T) which is true if (1 << 1 and b << 1. This holds only if
T satisfies (5.2.63). This condition on T prevents the secularity from dominating in

the linear operator, in particular it is a stronger condition on T than imposed after

(5.2.21) so long as, (5.2.59) holds true, i.e.

T0 << 1/ IlneI. (5.2.72)

5.3 The RC equation

The definition of T0(t) in (5.2.18) we substitute ”W(tllle _<_ e_V(t—tO)T0(t), so that

(5.2.62) reads
IIWWHH‘I S M (IIWUIIH'Y 6—"(t_t0) + 53/4‘73/2) , (5-3-1)

for t E (to, to + ,I3(3/4 —- 73/2) “I’ll—(l). Defining t1 as

t1 = 150 + 5 (3/4 — 73/2) Hit/3| (53-?)

we find

”W(tllllIn

|/\

M (5,3(3/4—73/2) lll’VOIlH’Y + 63/4-73/2.)

Ale/3(3/4_73/2) (”u/0”!” + E(1-13)(3/4-1Is/2)) . (533)

l/\

90

To complete the RG procedure, we fix 6 < 1 and T = ,6 (3/4 —7/2)|1%l. The

renormalization times are then defined sequentially by,

tn =tn_1 + I3 (3/4 — 78/2) “T'- (53.4)

 

We break R+ into equal disjoint intervals In = [tn,tn+1I as shown in Fig 5.1.
On each interval In, we solve the initial value problem (5.2.27) with the initial data
Wn E W(tn) 6 X2”, with the quantities To,” and Tl,n defined in (5.2.18) and (5.2.19)
over In. The renormalization map g takes the initial data Wn_1 for the initial value
problem on interval In_1 and returns the initial data Wn for the initial value problem

on the interval In

gwndl = W", ' (5.3.5)
Argueing inductively, the initial data and the new base point in are obtained from
W(t; ), the end-value of the evolution of W over [n-1, by applying Proposition 4.1.

We see that W(t;) E in_1 and so from (5.1.11) we have

Ian - mm 3 Mo III-Vamllm [2023) — all 3 Mo IIW--’(t;)||m T1,n._1(~r).
(5.3.6)
Substituting the estimate of T1,n—1 from (5.2.22), we bound the z jump, see Fig 5.1,
by

|/\

(2., _ 2(I;)| Mo IIW(I;)IIH, (Tag-142 + .3/4—73/27)
s No IIW(t;)IIm (73,2142 + 63/4—73/2 lln(e)|) (5.3.7)
The solution at time t = in is independent of the decomposition
U(In) = (2(th + wag) = is. + Wm (5.3.8)
so we can bound the jump in W at each renormalization by controlling the jump of

$7

”W(tfi) — Wnllm = - 22,.

 

 

 

 

¢z(t;) m S C '2" _ 2(t;)|

M0 ||W(t;)||H, (Taxi/12 + 53/4—73/2 Iln(e)I) (5.3.9)

|/\

91

Observe that the jump in the front location and remainder W are controlled by
the same estimate. From the definition of T0 using the equality T0,n—1(tn— 1) =

IIWn_1II H71 and the inequality (5.2.62) we have the estimate,
T0,n—1(T) 3 MI (lan—1Ilm + e(3/4‘Is/2>e"<T—‘n-I>) (5.3.10)

for T E Itn_1, tn]. Since the length of each renormalization step satisfies (5.2.63),

which yields eV(T_t’n—1) S (Illa/4’73”), and (5.3.10) satisfies
male) 3 MI (Ill/'Vn—Illm +I<1—3><3/4—IS/2>) (5.3.11)

To derive an upper bound for IIW(t.; )II H'Y’ we refer to (5.3.1) and (5.3.4) to obtain

|/\

I'M/((DIIHV A] (e-I/(tn—t-n—ll Ill’l‘fn-nIIIH‘I + 6(3/4—73/2l)

|/\

I'll (03(3/4—78/2) IIan—IIIHV + (QM-23(2)) . (5.3.12)

To obtain a bound for QW'n_1 = W”, we apply triangle inequality and (5.3.9),

IIQWn—1llm S IIWUE) - WnIIH7 + IIWUDIIW,
3 MO IIW(I;;)|IH., (T821? +' (_3/4—73/2 (111(1)|) + ||II~'(I,;,)||H,,
g III-1"(t;)IIH7 (1+ IIIngffl/f) (5.3.13)

using 53/‘1‘73/2|1n(e)| < 1. Substituting the estimates for Tan—1 from (5.3.11) and

IIW(t;)IIH~, from (5.3.12) into (5.3.13) we achieve an estimate for IIan_1IIH7,

”awn—um s M (53(3/4-78/2) |IW,,-1||H7+((3/4-2’s/2))

.. . +1 2
(1+ MIIH/2 I(||Wn—1||H‘I + ,._(1—II)(3/4—7.<;/2))I'7 Q3314)

Neglecting higher powers of I|Wn_1I| H7 and positive powers of 6 within the second

parentheses on the left-hand side of (5.3.14), yields

IIgl'Vn—IIIH'Y S [W (65(3/4—73/2) IILVn—IIIH7 + 6(3/4—73/2l) . (5.3.15)

92

 

We define the renormalization series Tn which bounds IIW (tn)|I H7 as,
Tn = M (5(3/4‘73/2) +€,II(3/4-7s/2)Tn_l), (5.3.16)

We solve the recursive equation (5.3.16) for Tn

1 _ (MEN/ewe)"
1 __ II..[C/5(3/4-“Is/2)

 

r.,, = 1163/4778/2 + (5114(3/4—78/2l)"r0 , (5.3.17)

where 70 = “ll/()IIH'Y- For 6 small enough so that M 613(3/ 4‘7'3/ 2) < 1 then IIWnII H7 S

Tn, implies the estimate

IIWnllm s. 5163/4722 ( (1765(3/4-78/3)” ||Wo

 

 

m +1) . (5.3.18)

The term M " may be interpreted as a logarithmic correction to the exponential decay
rate V. This correction arises from the time dependent modulation of the linearized

operator induced by the slow fold on the manifold. Defining U] as

 

Vlnll/l
l/ = , , (5.3.19
1 3(3/4 - 73/2)|1nc'| ‘ )
for t > tn, we can bound M n by
M" g e”1(’—‘0). (5.3.20)

From (5.3.4) in — 1,0 = ”5(3/4‘38/2ll‘ml, which along with (5.3.20) modifies (5.3.18)

 

to

(III/"((113, g Me3M—73/2 (e-V(tn-t0>e"1<*-to) ”mm + 1) , (5.3.21)

for any time t > tn. From (5.3.1) for t E In = Itn, tn+1], IIW(t)|IH7 can be bounded
by,
”W(tmm g M (Ia-"(Mn ||Wn||m + 391/4772”) . (5.3.22)

Substituting IIWnIIH‘I from (5.3.21) into (5.3.22) we obtain,

“W(tlllH‘r

|/\

A] (e—V(t—$11)e—V(tn—t0)eu1(t—tO) IIWOIIHT + 6(3/4—73/2)) ,

< III (e_<V-V1><t—t0> IIWUIIHA, + 6(3/4-78/2>) . (5.3.23)

93

Defining the new rate I/ as,

 

In M
u'zu—V =V<1— )>0. 5.3.24
1 [9’ <3/4 — 73/2) lln cl ' ( )

we obtain an uniform bound for IIW(t)II H7 for t > to,
llW(t)Ilm s M (ea/“70> “Wollm + 43/4722)) . (5.3.25)

To recover the asymptotic pulse motion, we consider the situation where t satisfying
(5.2.63), is sufficiently large that IIWII H7 < M 6(3/ 4’73/ 2). In this regime we see from

(5.2.23), T1 3 53/4—73/2 IlneI. From (5.2.14), we obtain the bound
I(AL W, 1111.0) IL2 g CT] “II/Hm 383/2778) IlneI. (5.3.26)
The inequality (5.2.15) yields, 1
I(N(W), 11:10) IL2 g ce<1+1/2>(3/4"7/2). , (5.3.27)

Substituting these estimates in the equation (5.2.7) for :6 we get,

[2,1111 '
2,3: ( z0)L2 +O(e(5/4—Is/2)). (5.3.28)

(5222’ Wgolrfl

From the form of residual R in (3.1.22) and the relation between the derivatives of

 

(I), and 52 given by (3.1.10), we obtain

(70520
820

 

(R, 2120)!) = (113(4)) - 12350)) ( + 7(2), ‘11:;0) + 0(6),

and (5.3.31) reduces to

+ 0(e(5/4—7-9/2>). (5.3.29)

 

26 = (113(20) - ’L‘3(50))" (113(ZO)_ ”8(50» (

Using (3.1.25) and (3.1.14), we bound the second term in the left-hand side of (5.3.31)

as

Ivs(£’0(t)) — 03070”

0.550(1) — fol llrlle I 3.0 12’

g (1171(7) g (.7/4—73/2I1ncl. (5.3.30)

 

 

 

 

A
a
A
N
v

'6
611-0-
0
V
b1
[\D
|/\

which modifies the expression (5.3.31) for .26 to
26 = (123(20) - 11420)) + 0(e(5/4_73/2)). (5.3.31)

To complete the proof of our main result note that the curve 20(t) given by (5.2.6)
is smooth except for the jumps from 2,-(ti-) to 2. We may replace 20(t) with a smooth
enough curve 20(15), which is close enough to 20(t) such that the estimates on W are

not effected. For M > 0 and 20(t) which satisfy
Izoe) — 20w 3 M (Ia—“HM HW0||H7 + 43/4772)) . (5332)

we define W(z, t) = U(2, t) — (15(2; 5005)). The following holds true

IIww) — II”<..I>|IH,< _ C IZo( )- 20w. (5.3.33)
for (.7 > O, which leads to the bound
/ __ 7.7 . ' , ~ —l/,( (l— to) 7 (3/4— 73/2))
III III” I II . II ( ,t)IIH7+I|W um g M(e [Mg] m + e

 

 

(5.3.34)
for some M > 0. In order to be able to choose such a curve :50 to be smooth we need
to verify that the jumps suffered by 20(t) at t = tn, lie below the error bound (5.3.32).
This holds true since the jump in the front location and remainder W are controlled
by the same estimate and in the asymptotic regime, the jumps in 20 at t = tn are
0(e(5/4‘73/2)) with an 0(Iln(e)I) time interval between the jumps. Thus we may
choose the smoothed curve 50 in such a manner such that I20 — {0| = 0(c(5/4_73/2)),
for t 75 ti, i = 1, 2, Dropping the tilde notation we may then write the evolution
for the smoothed curve 20 as in (5.3.31).

To revert back to the original coordinate system we introduce the rescaled H 3 norm

which uniformly controls the L°° norm,

1/2

[III/“Hg = (15-1/2|W|2+67—1/2|DTW|2dy . (5.3.35)

Hence we have proved the following theorem,

95

Theorem 5.3.1 Given 7,73 6 B, 1), then for e sufiiciently small there exists posi-

tive constants M and to such that for all initial data of the form

Uo(y, t) = ¢(y;y*) + Woe/I t),

where III/VOIIHJ S “—367, the solution of the governing equation (2.0.1) can be decom-

posed as
U(y. t1) = @(y;yo(l)) + W(y. t) for t> to,

where the residual l/V satisfies
I!WHH7 s M (6-5.50) “WollHi + (.3/4-78/2) .
ll ' y

In particular, after the perturbation W has decayed to 0(63/4—73/2), the pulse evolu-

tion is given by the reduced equation

:55) = 721150) + 0<e5/4-1S/2I.

96

BIBLIOGRAPHY

[1] A. Doelman ; T.J . Kaper ; K. Promislow ; [2005] Nonlinear asymptotic stability of
the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model. SIAM
J. Math. An., 38(6) 1760—1787.

[2] A. Friedman and S. Kamin, [1980], The asymptotic behavior of gas in an n-
dimensional porous medium, Trans. Amer; Math. Soc., 262 pp. 551-563.

[3] Anna Vainchtein [2003], Non-isothermal kinetics of a moving phase boundary,
' Continuum Mech. Thermodyn. 15(3)‘1—19

[4] A.Vanderbauwhede, G. Iooss, [1992], Center manifold theory in infinite dimen-
sions. Dynamics Reported: Expositions in Dynamical Systems. Springer, Berlin,
pp. 125163. '

[5] B. E. Sleep andJ. F. Sykes, [1993I,Compositional simulation of groundwater con-
tamination by organic compounds. 1. Model development and verification, Water
Resour. Res., 29 pp. 1697-1708.

[6] Bates P. W. and Jones C. K. R. T. : [1989] Invariant Manifolds for Semilinear
Partial Differential Equations, in Dynam. Report. Ser. Dynam. Systems App].
2, Wiley, Chichester, UK, pp. 138.

[7] Bates P. W., Lu K., and Zeng C. : [1998] Existence and persistence of invariant
manifolds for semiflows in banach space, Mem. Amer. Math. Soc., 135.

[8] Benziger, J., Chia, E., Moxley, J., Kevrekidis, I., [2005]. Chemical Engineering
Science 60 (6), 17431759.

[9] C. Eugene Wayne.[1997] Invariant manifolds for parabolic partial differential
equations on unbounded domains. Arch. Rational Mech. Anal, 138(3):279306 .

[10] Chen, L-Y., Goldenfeld, N., Oono, Y.,[1994] Renormalization group theory for
global asymptotic analysis, Phys. Rev. Lett. 73, 1311-1315 .

[11] D. L. Koch, R. G. Cox, H. Brenner, andJ. F. Brady, [1989], The effect of order
on dispersion in porous media, J. Fluid Mech., 200 , pp. 173-188.

97

[12] Dan Henry, [1981] Geometric Theory of Semilinear Parabolic Equations,
Springer- Verlag 140-142.

[13] D. M. Bernardi and M. W. Verbrugge, [1992], A mathematical model of the
solid-polymerelectrolyte fuel cell, J. Electrochem. Soc., 139 pp. 2477-2491.

[14] D. Singh, D. M. Lu, and N. Djilali, [1999], A two-dimensional analysis of mass
transport in proton exchange membrane fuel cells, Internat. J. Engrg. Sci, 37,
pp. 431-452.

[15] F. J. Mancebo andJ. M. Vega, [1992], Weakly nonuniform thermal effects in a
porous catalyst: Asymptotic models and local nonlinear stability of the steady
states, SIAM J. Appl. Math, 52, pp. 1238-1259.

[[16] 'Goldenfeld, N., Martin, 0., Como, Y.,[1989] Intermediate asymptotics and renor-
malization group theory, J. Sci. Comp. 4, 355-372.

' [l7] Goldenfeld, N., Martin, 0., Oono, Y., Liu, F.,[1990] Anomalous dimensions and
the renormalization group in a nonlinear diffusion process, Phys. Rev. Lett, 64,
1361-1364.

[18]Goldenfeld, N.,[1992] Lectures on' phase transitions and the renormalization
group, Addison- Wesley, Reading .

[19] HP. Mckean, [1970] Nagumo’s equation, Adv. in Math, 4, pp. 209-223.

[20] HP. Mckean, V.Moll [1986] Stabilization to the Standing Wave in a Simple Cari-
cature of the Nerve Equation, Communications on Pure and A ppld. Mathematics
39 485—529.

[21] I. Nazarov and K. Promislow (2006) Ignition Bifurcations in a Stirred Tank PEM
Fuel Cell, Chemical Engineering Science 61 , 3198-3209.

[22] J.Bona, K. Promislow, and C. Wayne, [1995], Higher order asymptotics of decay
for nonlinear, dispersive, dissipative wave equations, Nonlinearity 8 vol. 6.

[23] J.Bricmont, A.Kupiainen, G.Lin ,[1994], Renormalization group and asymptotics
of solutions of nonlinear parabolic equations, Comm.Pure.Appl.Math., 47, 893-
922 .

[24] J. Bricmont and A. Kupiainen,[1995], Renormalizing Partial Differential Equa-
tions, Constructive Physics,83-115, ed. V. Rivasseau, Springer.

[25] J. Carr. Applications of centre manifold theory, volume 35 of Applied Mathe-
matical Sciences. Springer-Verlag, New York, 1981.

98

[26]
[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

J. Evans, N. Fenichel, and J. Ferroe, [JE], Double impulse solutions in nerve
axon equations, SIAM J. Appl. Math, 42, pp. 219-234.

J. Feroe, Temporal stability of solitary impulse solutions of a nerve equation,
Biophys. J., 21, pp. 103-110.

J. L. Vazquez, [1992], An introduction to the mathematical theory of the porous
medium equation, in Shape Optimization and Free Boundaries, NA T0 Adv. Sci.
Inst. Ser. C Math. Phys. Sci. 380, pp. 347-389.

J. M. Stockie, K. Promislow, andB. R. Wetton, [2003] , A finite volume method
for multicomponent gas transport in a porous fuel cell electrode, International
Journal for Numerical Methods in-Fluids, 412577-599.

J.Nagumo, S.Arimato, and S. Yoshizawa, [1964], An active pulse transmission
line stimulating nerve axon, Proc'. IRE, 50, pp. 2061-1070.

J. Salles, J .-F. T hovert, R. Delanna’y, L. Prevors, J .-L. Auriault, andP. M. Adler,
[1993], Taylor dispersion in porous media. Determination of the dispersion tensor,
Phys. Fluids A, 5 pp. 2348-2376.

J. S. Yi andT. V. Nguyen, [1999], Multicomponent transport in porous electrodes
of proton exchange membrane fuel cells using the interdigitated gas distributors,
J. Electrochem. Soc., 146 pp. 38-45. I

K. Promislow,[2002], A renormalization method for modulational stability of
quasi-steady patterns in dispersive systems, Math. analysis 33, No. 6, 1455-
1482

K. Promislow and J. Nathan Kutz,[2000], Bifurcation and Asymptotic Stability
in the Large Detuning Limit of the Optical Parametric Oscillator, Nonlinearity
13 , 675-698.

K. Promislow, J. M. Stockie, [2001], Adiabatic relaxation of convective-diffusive
gas transport in a porous fuel cell electrode SIAM Journal on Applied Mathe-
matics, 62(1):180-205 (2001).

K. Promislow, J. M. Stockie and RR. Wetton,[2003], A finite volume method
for multicomponent gas transport in a porous fuel cell electrode , International
Journal for Numerical Methods in Fluids, 41:577-599.

L. M. Abriola andG. F. Pinder, [1985], A multiphase approach to the modeling
of porous media contamination by organic compounds. 1. Equation development,
Water Resour. Res., 21 pp. 11-18

99

 

[38] Moore R and Promislow K: [2005] Renormalization group reduction of pulse
dynamics in thermally loaded optical parametric oscillators, Physica D 206 62-
81.

[39] N. E. Wijeysundera, B. F. Zheng, M. Iqbal, and E. G. Hauptmann, [1996], Nu-
merical simulation of the transient moisture transfer through porous insulation,
Internat. J. Heat Mass Transfer, 39 pp. 995-1004.

[40] R. E. Cunningham andR. J. J. Williams, , [1980], Diffusion in Gases and Porous
Media, Plenum Press, New York.

[41] R. E. Lee DeVille and Eric Vanden-Eijnden,[2007], Wavetrain response of an
excitable medium to local stochastic forcing, Nonlinearity 20 51-74

[42] R. Fitzhugh, [1969], MAthematical models of excitation and propagation in
nerve, Biological Engineering, H. Schwan, ed., McCraw-Hill, New York, , pp.
1-85.

[43] S. Ahn andB. J. Tatarchuk, [1997], Fibrous metal-carbon composite structures .
as gas diffusion electrodes for use in alkaline electrolyte, J. Appl. Electrochem.,
27 pp. 9—17.

[44] Sone, Y., Ekdunge, P., Simonsson, D., [1996], Journal of Electrochemical Saciety
143, 1254-1259.

[45] T. Gallay, [1993], A center stable manifold theorem for differential equations in
Banach-spaces, Comm. Math. Phys. 152, 249-268.

[46] WP. Wang, [1988], Multiple impulse solutions to McKean’s caricature of the
nerve equation. 1. Existence, Comm, Pure Appl. Math, 41, pp. 71-103.

[47] V. A. Paganin, E. A. Ticianelli, and E. R. Gonzalez, [1996], Development and
electrochemical studies of gas diffusion electrodes for polymer electrolyte fuel
cells, J. Appl. Electrochem., 26 pp. 297-304.

[48] V. Gurau, H. Liu, and S. Kakac,[1998], Two-dimensional model for proton ex-
change membrane fuel cells, AIChE J.,44 pp. 2410-2422.

[49] V. Moll, S. Rosencrans, [1990], Calculation of the threshold surface for nerve
equations, SIAM J. Appl. Math, 50, pp. 1419 - 1441.

[50] Y. G. Chirkov, V. I. Rostokin, andA. G. Pshenichnikov, [1996], The influence
of the porous electrode structure on the gas generation modes, Russian J. Elec-
trochem., 32 pp. 1009-1016.

100