“n‘dv’l-‘v

711?.)
$35.51.... 5 . 15)..

. 751....4)I:§!.1Ja I
l. al.(¢.l.; u PEI-’7.»
i I" $11k.» ,.:t::.. r.

. an?

 

THESE}

\ lHlHNlNHll”Hill”!!!”I!!!”"Hill“!!!IIIHIHNI *

01555 3088

This is to certify that the

dissertation entitled

Regularity and Stability for Periodic Solutions to
Nonlinear Klein-Gordon and Schrodinger Equations

presented by

Xinming Zhao

has been accepted towards fulfillment
of the requirements for

Ph.D. Cbgnwin Applied Mathematics

2 ivi mafia? 3m

 

 

 

Major professor

Date 8/15/96

MSUis an Affirmative Action/Equal Opporluniry Institution 0- 12771

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

10W 2 01997
21"? fl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution
czbhc

WWW-1

Regularity and Stability for Periodic Solutions to Nonlinear

Klein-Gordon and Schrodinger Equations

By

Xinming Zhao

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1996

ABSTRACT

Regularity and Stability for Periodic Solutions to Nonlinear

Klein-Gordon and Schrodinger Equations
By

Xinming Zhao

In this dissertation, we first prove the regularity of the periodic solutions of non-
linear Klein—Gordon Equation on the compact manifold S3.

The L'P-Lq estimates for periodic solutions of linear wave equations have been
developed and utilized to establish the regularity for even dimensions. It turns out
to be very difficult to handle the regularity for odd dimensions due to the structure
of the kernel of the wave operator. Yet dimension three is the most meaningful case
in physical sense because 5'1 X S3 is conformally equivalent to 1R1 >< 1R3. We consider
three dimensions and look for symmetric periodic solutions. The original equation is
reduced into a one dimensional equation with singularity. We prove that under some
conditions on the nonlinearity, the smoothness of the periodic solutions is one degree
higher than that of nonlinear interaction.

We then study the orbital stability for standing waves of least energy to nonlinear
Klein-Gordon(NLKG) and nonlinear Schrodinger(NLS) equations.

Many authors have studied the orbital stability of standing waves for these equa—
tions, but their attention has been principally focused on space domain R”. We
extend stability results to bounded domains and compact manifolds. Our method
also applies on R”. The mass term and nonlinear interaction term we are considering
depend on space variable as well as on solution. The following results are obtained:
(A) If the equations have positive energy, then standing waves of any frequency of
NLKG or N LS are orbitally stable. (B) If the equations have indefinite energy, a very

sharp condition is obtained on relation between the least energy and the frequency

 

of standing waves. We apply this condition to bounded domains, compact manifolds
and whole space 1R"; in each case, we produce orbital stable standing waves.

Finally, for instability, we prove that for a certain class of nonlinearity, the steady
state of least energy(ground state) of NLKG are unstable in a very strong sense: there
is a region whose boundary ground states lie on, such that every solution starting from

this region will blow up in finite time.

 

To my mother

To my wife Rujie and my daughters Victoria and Alice

iv

ACKNOWLEDGMENTS

I am very happy to express my gratitude to my advisor Dr. Zhengfang Zhou for
his understanding, help, support, guidance, encouragement, patience and inspiration.
The special thanks are owed to Xiaodi Wang for his unselfish help. I am very grateful
to Dr. William Sledd for his kindness and thoughtfulness. I would also like to thank
Thomasz Komorowski for his interests in discussions with me and to thank Baisheng
Yang for shortening one of my proofs in my paper. Thanks go to my thesis committee
members: Dr. Dennis Dunninger, Dr. Thomas Parker, Dr. Joel Shapiro and Dr.
David Yen for devoting their time and effort to reading my thesis manuscripts. Finally
I would like to extend my thanks to Department of Mathematics at Michigan State

University for providing me with financial support through teaching assistantship.

Contents

 

Introduction 1
1 Regularity of Periodic Solutions of Nonlinear Wave Equations 7
1.1 Preliminaries and Notations ....................... 7
1.2 Boundedness of Solutions ......................... 12
1.3 Regularity of Solution .......................... 18
1.4 Small Forcing Problem .......................... 29
2 Orbital Stability With Positive Energy 32
2.1 Problem and Notations .......................... 32
2.2 Existence of Ground State ........................ 34
2.3 Orbital Stability of Standing Waves ................... 38
2.4 Outline for Schrodinger Equation .................... 40
3 Orbital Stability with Indefinite Energy 42
3.1 Introduction ................................ 42
3.2 Least Energy Solution .......................... 44
3.3 Standing Wave as a Function of Frequency ............... 51
3.4 Stability of Standing Waves ....................... 57
3.5 Nonlinear Schrodinger Equation ..................... 63
3.6 Applications ................................ 64
4 Finite Time Blow Up for Nonlinear Klein-Gordon Equation 68
4.1 Introduction ................................ 68

vi

4.2 Steady State and Weak Solution ..................... 69
4.3 Finite Time Blow Up ........................... 72

List of References 76

 

vii

Introduction

In this dissertation, we devote half of the effort to the regularity of nonlinear wave
equations and another half to the stability and instability of stationary states and
standing waves of nonlinear Klein—Gordon and Schrodinger equations.
Regularity of Nonlinear Wave Equations

We first investigate the regularity and the existence of symmetric and periodic

solutions of the semi-linear wave equation on S3 :
utt—Au+u+F(t,x,u)=0 in 5'1 X33, (0.1)

where A is the Laplace-Beltrami operator on 53. Under various conditions on function
F, the existence results for semi-linear wave equation on 51 X S" are obtained by
many authors, e.g., Benci and Fortunato [1], Brezis and Nirenberg [6, 3], Rabinowitz
[37, 38] and Zhou [56, 57], among others. The regularity results in the case of 72:] are
obtained by Brezis and Nirenberg [6] for asymptotically linear F and by Rabinowitz
[37] for super—linear F. Jerison, Sogge and Zhou [57, 21] studied and proved the
regularity results for n = 2, n = 4 and n = 6. The case of n = 3 is the most
interesting and meaningful in physics, since 51 x S3 is conformally equivalent to

R1 X R3. It was pointed out in [57, 21] that the kernel of operator

2
an: 0 _A+(

n—l

 

 

2
r on .5" X S"

0.2 )

plays an important role in investigating the regularity of the solution of the wave
equation. When n is even, the kernel of the operator C1,, is {0}, which makes it easy

to handle the regularity. In this case, we can apply the L” — Lq estimates developed

in [57], [21] which are generalizations of estimates for R x R” in [34] and [52]. If

1

 

n is odd, the kernel is infinite-dimensional, those LP — Lq estimates only apply to
the component orthogonal to the kernel of CI. The component in the kernel is very
difficult to control or estimate . Hence it is usually hard to obtain the regularity if n
is odd. Currently, little is known of the regularity for the case of odd 7?. > 1.

In this paper, we will seek a periodic solution of a symmetric semi—linear wave
equation on $3. The key observation is that the problem reduces to a standard one-
dimensional semi-linear wave equation with singularity, and the kernel for the one
dimensional wave is very well understood. Some techniques developed by Brezis,
Nirenberg [6, 3] and Rabinowitz [37, 38] can be modified to work for this case. Using

a I t o — 3 4
standard spherical coordinates, 1.e., 1f cc — ((131,:132,:r3,:z:4) E S C R ,
f

3:1 2 sin p sin ¢cos 0,

:1: =sin sin si110,
2 P P (0.2)

323 = sin p cos cf),

 

K 3:4 : cosp,

where p E [0, 7r], <5 E [0, 7r], 0 E [0, 27r], we can rewrite equation (0.1) in the form

 

Utt —

1 .
2 [(sxn2pu.)p + Aw] + u + F(t,p,¢,6, u) = 0, (0.3)

sin p
where Agu : ;.-,l,-,;l(sin dud,» + filmy] is the Laplacian of u on 52.
If the function F is independent of (f) and 0, i.e., F = F(t,p,u), it is natural to
search for solutions of (0.3) which are also independent of qS and 0. Such solutions

satisfy the equation

u” — um, — 2cot pup + u + F(t,p, u) = 0, (0.4)
and will be called symmetric solutions. If u = u(l,,p) 6 (72(51 x 53) is a solution of
(0.4), it is immediate that v(t,p) = u(t,p) sinp satisfies the equation

2)“ —— v,,,, + F(t,p,v/ sinp) sinp = 0 in S] X [0, 7r],
v(t,0) = v(t, 7r) 2 0,

(0.5)

which is a standard one—dimensional semi—linear wave equation with singularities at
p20 and p = 7r. Conversely, we will show that u = v/ sinp is also a weak solution of

Equation (0.4) for any weak solution 2) of Equation (0.5).

 

 

The procedure for establishing the regularity is the following: We first prove the
existence of an L00 symmetric solution under the condition that F is monotone in
u (i.e. Fu 2 a > 0) and asymptotically linear (Fu 3 5). We then show that
every bounded symmetric solution is actually a classical solution provided that F is
sufficiently smooth.

Stability and Instability of Standing Waves and Stationary States

In this part, we study the stability and instability of standing waves and stationary

states of nonlinear Klein—Gordon equation(NLKG)

utt—Au+u+f(:c,u)=0 in R+XQ,
u = 0 on (90 if (952 74 (D, (0-6)
u(0,x) = U(:1:), ut(0,:1:) : V(:z:),

and nonlinear Schrodinger equation(NLS)

tut—Au+u+f(:c,u)=0 in RV“ x 9,

u = 0 on 80 if 89 7f (Z), (0-7)

now) = Um,
where Q is IR”, a bounded domain in R" or n-dimensional compact manifold. Here, u
is a complex function of (1:, t) E Q X R+, A denotes the Laplace operator with respect
to space variable :1: E 9, f is a continuous function of the form f(.r,u) : g(:r, |u|)u
wheregzflxR+—>R.

Equation (0.6) arises in particle physics ([28, 32]). Special cases of (0.7) include
Hartree—type and Pekar—Chaoquard equations [9] which arises in various domains of
physics, e.g. in the study of propagation of laser beams( [24, 53]) and quantum
mechanics( [20, 30]).

Of special importance are the “solitary waves” solutions of Equations (0.6) and
(0.7). These solutions include time dependent periodic solutions(standing waves) of
the form emqfiflr) and time independent solutions(stationary states or steady states)
corresponding to w =2 0.

The search for both types of solutions leads to the consideration of nonlinear

elliptic equation of the form

 

—Au + g(:1:,u) = 0,
u=0 on an if 052#@,
It is well-known (see, e.g., [10], [50], [39] and [40]) that, there exist infinitely many

(0.8)

solutions of (0.8) apart from the trivial solution it E 0, among them a positive,
classical solution uo which has minimal energy among all nontrivial solutions. By

minimal we mean the functional
1
J(u) = f i [Vul2 + one, may,
a

where G(a:,u) = fou f(:1:,s)ds achieves its minimum at uo. We call such a solution
“ground state”.

The stability of both stationary states and standing waves of equations (0.6) and
(0.7) are physically very important and naturally have been extensively studied. We
can not expect the conventional stability of the stationary state due to the nature
of N LKG. Indeed, it has been proved that solutions of the N LKG (0.6) will blow
up in finite time under some conditions on nonlinearity f: For 9 = R", John [11]
and Glassey [12, 13], among others, studied the case f : —|u[7’. They obtained
finite time blow-up results of classical solutions of equation (0.6) for p < p0(n) E

(72 +1 + x/n2 +1072 —- 7)/(2(n -—1)), which is less than 1+ 4/(n —- 2). For a bounded

 

domain and a compact manifold Q, Payne and Sattinger [36] and Sternberg [48]
proved that any weak solution of equation (0.6) starting from some neighborhood of
stationary state will blow up in the L2-110rm in finite time.

Blow up results for solutions of NLS (0.7) have also been obtained by Berestycki
and Cazenave [2] who showed that for Q = R", under certain conditions on f, solu—
tions of NLS starting from some region near a standing wave will blow up in finite
time in Liz-norm. Under much more relaxed conditions on f, Shatah [43] proved for
Q = R” that any solution of equation (0.6) starting from some neighborhood of sta-
tionary ground state may not necessarily blow up in finite time, but its L2-n01‘m will
approaches infinity as t goes to infinity.

An interesting instability result for solution of equation (0.6) was obtained by

Keller [26]. He showed that for Q 2 IR” some stationary state no can be perturbed

into a time—dependent solution of utt + aut — Au + f (u) = 0 (a < 0) that remains
bounded in energy norm for t > 0. If f satisfies some growth conditions at infinity,
then solutions tend to zero as t —> 00. In particular, no is not stable.

A more common type of instability of solutions of equation (0.6) and (0.7) (see [45])
is that no matter how close a solution may initially be to a ground state, the solution
will eventually leave any prescribed neighborhood of the ground state. Such solutions
may not blow up in finite time, nor go to infinity as t go to infinity, nor approach
zero. In terms of techniques used for obtaining this type of instability, there are two
disparate types of instability results in the literature. The approach developed by
Strauss-Shatah [45] gives an instability criterion coming from the variational structure
of the problem; Jones’ approach [22] produced a complementary criterion related to
the difference between the number of negative eigenvalues of two selfadjoint operators
using quite different techniques. Grillakis [14] tried to combine these approaches into
one single framework.

Due to the nature of NLKG and NLS, stability in the strict sense can not be
established for the standing waves of NLKG and N LS. However a concept of orbital
stability has been proposed(for a precise definition, see the later chapters). A number
of authors(e.g. Grillakis, Shatah, Strauss [18], [19], [44], [51], Cazenave and Lions [8],
[9] and Weinstein [55]) have studied the orbital stability of standing waves of NLKG
and NLS.

It appears that almost all existing stability, instability and finite—time blow—up
results have been established for the situation where Q 2 IR".

What we shall do in this dissertation is to extend orbital stability results to the case
where Q is a bounded domain in IR” or a compact manifold and carry out the proofs
for stability in two chapters according to the behavior of nonlinearity f. Chapter
two deals with the equations with positive energy, while chapter three treats the
case in which the equations have indefinite energy. Our approach to proving orbital
stability of standing waves does not need the scaling property which is essential in
previous arguments for Q = R”, and consequently our approach applies on more

general domains Q. We also prove the orbital stability for dimension n = 2 on IR”,

 

which was left unresolved in [44] and [45]. We also present a unified approach to
get the results of finite-time blow—up of weak solutions of NLKG for all cases of the
space domain 9 that we are considering. A very important feature of our method is
that the mass m and the nonlinearity f we are considering may depend on the space

variable a: as well as on the solution u.

Chapter 1

Regularity of Periodic Solutions of

Nonlinear Wave Equations

1.1 Preliminaries and Notations

Let W = 5'1 X 53, CI = 58725 ~—- A3 + 1. We consider the solutions of the following

nonlinear wave equation on S3
Clu(t,.r)+F(t,:r,u)=0 in 14/. (1.1)
It is well—known [35] that the eigenvalues of D are
M3,!) =l2—j2, j=0,1,2---, 1:1,2m,

and the eigenfunction associated with My, l) are

ej,(t)s)m(.r), 2': 1,2, m 21,2,---,l2,
where {31m}i:=1 are orthonormal spherical harmonics of degree I — l on S3, and

6,1(t) = — cosjt, 6,2(t) = — sinjt.

fi fi

It is also known that these eigenfunction constitute an orthonormal basis of the real

Hilbert space L2(W). Thus for any u E L2(VV), u can be expanded as

2 12

u(t, :r) = Z Z Z Z ujilmeji(t)51m($)a

j=0 [:1 i=1 m=1

and

Halliz = Z Injzmilza

j,i,l.m
where “a“; = (u,u) = [W uzdwdt, and ujum is the Fourier coefficient with respect
to this basis in L2(W), and dw is the standard measure on 53.
For k 2 0 we define the Hilbert space

HEW) = {U = Z u,,-)me,«,-s)m | Halli~ = Z (1 +j2 +12)k|uy'z‘1ml2 < 00}-

j,i,l.m j.i,l.m

From this definition, we see that for H0(W) = L2(W) and ||u||0 = [[uHLz. We use the
subscript s to denote any space of functions on W which are independent of 0 and d).

For any u, w E L2(W), the usual inner product

(u,w)=/ uwdwdt.
W

is given in spherical coordinates by

211' 7r 7r 27r
(u, w) = f f f / uw sin2 p sin qfidddqfidpdt.
o o o 0

If u, w E Li(Sl X 53), this simplifies to

Zn 7r
(21,10) 2 47r/ / uw sin2 pdpdt.
o o

The restriction of D to C3(W), is the operator

 

82 d2 2 t 8+]
,——,—————— 0' , ..
an 02 C pdp

From now on, we assume that F is a symmetric function, i.e.,
F(t, :13, u) : F(t,p, u).
A symmetric solution of Equation (1.1) satisfies the equation

D,u(t,p) + F(t,p,u) : 0 on W. (1.2)

sin _lp

It 18 easy to Show that ej,(t)T is an eigenfunction of El, associated with A(j, l), and

that the set

1 ' l . .
{Wefl(t)%§7 .720719"'a [Z1727°"7 2:172}

is an orthonormal basis of LEUV). That is, for any u(t,p) E Hf(W), we have
00 oo 2
t)s__inlp
u(t : z .
P) Z Z Z ”17—" )5... p
j=0 [:1 i=1
with

Hallic = Z:(1+J'2 + (2);“ |u,~,-)|2.
3,131
Let T = S1 X [0, 7r]. Consider the set S of all C°°(T) functions vanishing near p = 0
and p = 7r. For any v(t,p) E S, we have

00 oo 2
v(t ,p) = ZZZvj-ufiefl(t)sinlp,

j=0 [:1 i=1

where v,“ is the Fourier coefficient of v with respect to the orthonormal basis

{fleflinfinlpa j:0113'°'a [21727'H’ 221,2}

Let Hg (T) be the completion of 5' under the norm

 

[liviiik = Z (1 + 52 +j2)k Ivy-m2.

3,1,1
Remark 1.1 We use ”II“ to denote norms associated with T = 51 X [0,7r], and III]

to denote norms associated with domain W = 51 X S3 or Sl{the distinction will be

clear in the context). a E HflW) if and only if?) : usinp E Hg(T).

With these definitions, IIuIIZ = 47r IIIvIIIi. and hence the identity

 

mum): amp) for v(t,p)=u(t,p)sinp,

sin p

82 i. This identity immediately implies the following

where Ell = 5; — apg

10

Lemma 1.1. u E Hf(l4/) is a weak solution of (1.2) if and only ifv = usinp E
H6‘(T) is a weak solution of
Dlv(t,p) + F(t,p,v/ sinp) sinp = 0 in T,
v(t,0) : v(t, 7r) 2 0.

(1.3)

In order to apply results from functional analysis, we extend the domain of D to

120:!) = {a E L2(W) I Z I)\(j,l)I2I’LLj.'1mI2 < 00}.

jQiil’m

For n = Z ujumejiszm E D(Cl), we define

Du : Z A(j,l)uj,-)mej.-s1m.

j,i,l,m
This definition coincides with the classical one if u E C2(W) C D(D). Under this
definition, Cl is a self-adjoint operator from D(Cl) C L2(l/V) to L2(W) with kernel

Ker(El) = {u I u(t,:r) = Z:

and the range R(D) = Ker(El)i.

ltiizmezi(t)81m($)} 7

Similarly, we can extend the domain of operators Cl. and Ell whose kernels play
an important role in the proof of the regularity of the periodic solutions(cf. Brezis

and Nirenberg in [6] and Rabinowitz in [37]). When this is done, the kernel of D1 is

N E [(67191) = [P0 +p) —r(l —/)) [P E L( (.5 )lPl E [91 P(S)ds = 0].

or in terms of Fourier series expansion

00 2
N = {v I v(t,p) : ZZUh-felim )sinlp}.

[:1 i=1

Similarly the kernel of D. is
K :- Ker(Els) : {v/ sinp I v E N},

or in Fourier series expansion

2

00 l. sinlp
K: t , .1 , .
I...) ,. {gm—— .600...>}

l::l "‘

 

11

We will prove the existence of an L2 periodic solution, we make the following

hypothesis on F.
(F1) F(t,a:,u) is nondecreasing in u for (16,33) 6 W;
(F2) There exist positive constants 771 and 772 such that

iF(t7$7u)i 2 771 [u] — 772 V (l,£E) E W;

(F3) There exist positive constants ”y < 3 and I/ such that

|F(t.:v.U)l S r IUI + V-

Then we have the following existence result.

Theorem 1.1. [fF E C(W X R) satisfies (F1) — (F3), then the wave equation (1.1)
possesses a solution u E L2(W). Furthermore, ifF is symmetric, then equation (1.2)

possesses a symmetric solution.

Proof. If we let A : Cl, H = L2(W), 6 = 3, B : F, then this theorem is a direct

consequence of the following result due to Brezis and N irenberg [7]. D

Theorem 1.2 (Brezis and Nirenberg). Let H be a real Hilbert space, let A :
D(A) C H —> H be a closed linear operator with dense domain and closed range.

Assume
(1) NM) = NUT“);
(2) A‘1 : R(A) —> R(/1) is compact.
Denote by 6 the largest positive constant such that
(Am) 2 —(1/6)||Au||2 v 6 HM)

Assume B : H ——> H is a nonlinear monotone demiconlinuousfi.e., mapping strongly
convergent sequence in H to weakly convergent sequence in H) operator and satis-
fies the condition that there exists a positive constant *7 < 6 and a constant C(w)

depending only on w such that

(Bu — Bw,u) 2 (1/7)IIBuII2 — C(w) Vu,w E H.

12

Then
R(A + B) 2 R(A) + convex hull of R(B),

where 2 means the sets on both sides have the same interior and same closure. Fur-

thermore if||Bu|| —> 00 as ||u|| —> 00, then A + B is onto.

1.2 Boundedness of Solutions

The following stronger hypotheses on F are needed to make all L2 solutions bounded.

(th) there exist positive constants m and fig such that

lFtl 3H1 lu|+H2 V(t,$,lt) E W X R;

(F4) there exists positive constant oz such that a g Fu \7’ (t, a:,u) E W X R;

(F5) there exists a positive constant fl such that Fu S H V (t,:z:, u) E W X R.

Theorem 1.3. Assume that F E C1(W >< IR) satisfies (F3)-(F5) and (F3t). Let u
be any L2 symmetric solution. Then there erists a constant C independent of u, F,

V, a, [3, m and M such that

., ,Lt1+ fl +1 H2
“21le s c {—5.2— (IIFIIL2(W) + v) + 1;] -

First, to prove the theorem, we need an. estimate on solutions of the linear one-

dimensional wave equation

Dlw(t.p) +g(t,p) = 0 in T,
w(t,0) = w(t,7r) = 0.

(1.4)

Lemma 1.2. Given g E Lq(T) 0 Ni for l S q 3 00, there exists a unique solution
11) E CO’O’(T) 0 Ni of (1.4) such that for a =1 —l/q,

(1.5)

 

|Iw|ICO.0(T) S CHQI Lc/(Tp

13

where CO'O‘(T) is a Holder space with C0'0(T) = C(T) and C0'1(T) being the space of

Lipschitz continuous functions. Moreover we have an explicit representation for w

w(tap) = Wm) +p(t +p) -p(t -p) (1-6)

with

t p+:L' _
zéffldx/ (7,)xclr+a(——7r p),
t+—p 1: 7r

a:——/O7r dx/:$g (7 x) )d7 (a is a constant),

1 71'
= ,7 / My — as) — My + s.s>st.

Remark 1.2. The explicit representation was given by Lovicarovd in [33]. With such

a representation, estimate (1.5) can be easily verified.

Let u(t,p) be an L2 solution of Equation (1.2). We can write it 2 ul + U2 with
ul 6 K, U2 6 Ki. Let v : u sinp : ulsinp + U2 sinp E v1+ v2, then v satisfies
Equation (1.3) with v1 6 N, v2 6 Ni. Let p E L2(Sl) be the function such that

v1 = p(t + p) — p(t — p) and [p] E f5, p(s)ds = 0. It is easy to check that

l

—/01T[v1(y—3,8)-’Ul(y+373)ld5- (17)

My) = 2”

One can easily verify that for [,9 6 [42(9),
1
b m + mu — I’M/Mu = 5mm, (1.8)

from which it follows that

i2 = 27V IIPIIiz- (1-9)

 

 

lllvll

From the Fourier series expansion, it is easy to see that ”Di-“1H1; is equivalent to
||u1||k, and ”'vade is equivalent to |||v1|||k. In this section, C will be used to
denote various constants independent of u, F, l/, (1, fl, pl and M. For the simplicity

of notations, we denote ||F||L2(W) + 1/ by BF.

14

We will carry out the proof of Theorem 1.3 in several lemmas. As we pointed out
earlier, it is relatively easier to estimate U2 and v2. For U2, one uses Lemma 1.2 to
see that v2 6 L°°(T) from F(t,p,u) sinp E L2(T), and v2 6 00,1 from v2 E L°°(T)
and v1 6 L°° (Lemma 1.3). Consequently ug 6 L”. The estimate for ul is more
complicated. We first use the fact that F(t,p,u) E Ki or F(t,p,u)sin,0 6 Ni. By

carefully choosing § 6 N in following equation

/F(t,:v, u) sinp€(t.p)dtdp = 0,
T

we will obtain the boundedness of p in Lemma 1.3. Then Lemma 1.4 uses the test
function E E N, which is an approximation to val, to obtain Dtvl 6 L2. Lemma 1.5

uses the fact
th[F(t, x, u(t, p)) sinp]§(t,p)dtdp = 0
T

for any 5 E N and shows that p' 6 L00. Finally from Lemma 1.6 we conclude that
HUIHLOO S CHp’HLoo, and hence that u1 is bounded.

Define a function q 6 C(R) as follows. For positive number N],

f

s+M, if sS—lld,
q(s)=< 0, if (3| < M, (1.10)

 

( s — ll/I, otherwise.

Lemma 1.3. There exists a constant C such that

1 WE]?
lilv‘ZIHI/Jo < CBF, HPHLOO S 6—07.

Proof. By (F3), u E L§(W) implies F(t,p,u) E LEWV), which in turn implies that
Fsinp E L2(T). Therefore by Lemma 1.2, v2 6 00.1/‘2. In particular,
111222111... 3 6' lllFsinplllm s C llFlle : 03..

Since v is a solution of (1.3), we have

/F(t,p,v/sinp)§sinp = 0 for all 6 E N, (1.11)
T

15

CI

/(F(t,p,u) — F(t,p,u2))§ sinp = —/ F(t,p, u2)€sin p. (1.12)
T

T
Write v1 = p(t+p) —p(t —p) E vf —v1—, then 6 = q(vf) —q(v1—) E N by construction.

Note that

(F(t,p, u) — F(t,p,u2)) sinp = Fu(t,p, u*)v1, (1.13)

where u* is between it and U2. The above equality, the monotonicity of q, sq(s) Z

M |q(s)|, (F4), and (1.8) lead to the estimate

[Tam u) — Fm, U2))€Sinp
= fEtta/3.163(1)?-vf)(q(vt)—q(vf))
2 a / (vigor) + v;q<v;) — vigor) — 2).-gun)
T
2 cm] (|q(vi)|+)q(vr>l)- (1.14)
T

On the other hand, (F3) implies that

 

 

/F(t,p,U2)€sin/)) SC'BF/ (lq(vt)l+lq(v;>l). (1.15)
T T

Combining (1.12), (1.14) and (1.15) gives

.M / (qu1): + [um-)1) s cm / ((q(v1~)(+ lq<vr)l)- (1.16)

Whenever M < “PHI/>07 the integral term in (1.16) is nonzero and we can divide by
it, and therefore M S 2CBF/oz. Since N1 is an arbitrary number less than ||p||Lm, we

have HpHL00 S CBp/oz. Thus the proof is completed. El

Lemma 1.4. There exists a constant C such that

BF BF
lllvzlllco.1 3 CF’ ”U2I1Lw S 03-,

and

, 3 #1 #2
lllvulllm : c (3.8 + 1711qu g . (1.17)

16

Proof. From preceding lemma and (F3) we know that |||F sin )0”le S Cal—F. Lemma

1.2 implies that

. BF
IIIszIIcyo.1 S C IIIFSlanIILOO S 0—0:: (1-18)
from which we get
. BF
IIU2||Loo = lllvz/SlnplllLoo S C |||v2|||00.1 S 075’ (1-19)

where the boundary conditions v2(t, O) = v2(t,7r) = 0 have been used. As for v1, let
2" = (z(t +h,p) — z(t,p))/h, for z E L2(T) and h E R. Since v is a solution of (1.3),

we have
/T F(t,p,v/sin p).gs1np = 0 for all ge N. (1.20)
In the above equation, set 6 = (vb—h which clearly belongs to N, we get
/T(Fsin p)hvi‘ = 0. (1.21)
A direct computation gives

(F sin M" = F1619. u(t + p)) sin/J + Fu('t.p, u“)(vi‘ + v3). (122)

where u* is between u(t, p) and u(t + h, p), and t* is between t and t+h. Substituting
(1.22) into (1.21) and then using (F31), (F4) and (F5) yield that

 

 

l .
lllvl‘ L. S E (filelHle +|IIFt51anIIL2>
CY
S b: (flIIIv2IHC0J + #1 III’UIIIL2 + #2)»
from which the inequality (1.1.7) follows. D

Remark 1.3. In the proof of the previous lemma, we used the well-known fact that
ifz E L2(T) is periodic in t, then 2) E L2(T) if and only if there is a constant 6
independent of h such that ||zh||0 S 6 Vh E R. The test function f = (vb—h is
an approximation to va1. We did not use va1 itself since we don’t know whether

0301 E L2(T).

17

Lemma 1.5. There exists a constant C such that

 

2

llp’llLoo S C (M1 + 68F + £3). (1.23)
oz oz
Proof. It is easy to see that Fsinp E NL implies
Dt(Fsinp) = Ftsinp + Fuv) E Ni.
Actually one can easily verify that
Df : Hg(T) n Ni —> Ni; Df : 115(7) 0 N —> N. (1.24.)

Consequently, we have

 

/(Ftsinp + Fu(vlt + vzt))§ = 0 V5 E N, (1.25)
T
or equivalently
_ /(Ft5inp+ Fu’Uth = / Fuvité V6 6 N. (1.26)
T T
Using Lemmas 1.3 and 1.4, (Eat) and (F5), we have
. +
”ramp + mom... 3 #1 Hum... +172 +5 luvs”)... s 0 (“1a 53.. + it).

Write v1) = p’(t +p) — p’(t — p) E vi, — vl‘, with p E H1(S1) by (1.7). Take 5 =
q(v1+t) — q(v1_t) in (1.26), where the function q is defined by (1.10).
The right hand side of (1.26) gives

[PAWS = /T Fullaeafhvb - vil)(q(vi5) - (MM)
3 “/Tlvli‘llvb) + URI/(v.7) — mum — vfiqlvm

2 aM / (|q(v{5)l + Iqbal)-

Hence

 

a

aM/qu(vi2)l+lq(vfi)lSC(#2+“I+flBp)/qu(v1+1)l+lq(vfi)la (1.27)

 

where M is any constant less than ||p’ IL°°7 which implies (1.23). D

So far, we have shown that U2 E Loo, p’ E LOO. The following lemma completes

the proof of Theorem 1.3.

18

Lemma 1.6. For any non-negative integer k,

PI‘OOf. FIX 0' E (0,7T/4), IBIS T1 = 51 X [0,0], T2 2 SI X [7T—0',7I'], T3 2 T\(T1 UTz).

For (t,p) E T1, using

t+p
vadt. p) = p(k)(t + p) — 10“)“ - p) =/ p(k+1)(8)ds,

t-p

we get

k+l

llchvl/SinpllLOOUl) S IlP<k+UHLm 31113 IZP/ Sinpl S C I|P( )IILOO'
(USPSU)

Similarly on T2, using p(t — p) : p(t — p + 27r), we have
llval/sinpllLooa.) S Cllp(k+1)llL.. .

Finally on T3, we have

1

sin 0‘

 

IIDfUI/SiDPIILoo(T3) S “DivlllLoo S C IIP(k+1)IILoo'

LFrom these three inequalities, we get Dful E L°°(W), and the proof of lemma is

completed. CI

1.3 Regularity of Solution

In this section, we will show that every bounded weak solution of (1.2) is a classical

solution if F is sufficiently regular. We state our main result first.

Theorem 1.4. Let k _>__ 2. Suppose that F E Cfll/V >< IR) satisfies (F4) and that
u = ul +u2 E KGBKi is a bounded weak solution of {1.2}; then u) E HfflCk—Mfl I1"
and 71.2 E Hf“ fl Ck—W fl Ki for any /i E [0,1/2). In particular, if F is COO, so is u.

Remark 1.4. As long as u E LOO, the conditions (F3), (FBt) and (F5) are no longer
needed for the regularity of the solution. The condition (F4) can not be relaxed to
a monotonicity condition. Indeed, let F(t,p,s) = F(s) E COO be such that F is
monotone increasing on IR and F(s) = 0 for s E {—1,1], then any LOO function

(p(t,p) E K with ||¢||L00 < 1 is a solution of (1...?)

19

Throughout this section, let u 2 ul + 11.2 E K EB K i be a bounded solution given
in Theorem 1.4. Let v = usinp 2 v1 + v2 with v1 E N and v2 E Ni. We know that
v is a solution of (1.3) and v1 = p(t + p) — p(t — p) for some 1) E L2(Sl) with [p] = 0.

The main idea is to differentiate the equation IC times with respect to t to obtain
ClDfu = DfF(t,p,u(t,p)).

For U2 or v2 = u2 sinp one uses Lemma 3.1 and Lemma 4.1 below to get va2 E

CO’I(T) and U2 E Hf“. For ul or v1 2 ul sinp one can consider the identity

/ Dt‘1[F(t./).U(t,p)) sinplDt+1v1(t,/))dtdp = 0.
T

which produces a good term Fulchvll2 and yields Dtkvl E L2(T). Again, Df+lv1
must be replaced by suitable approximations in the detailed argument. Consequently
ul E Hf. The facts ul E Ck‘2’“ and ug E Ck'l’“ follow from some sharp estimates on
spherical harmonics (Lemmas 4.2-4.4). It should be pointed out that Lemmas 4.5—4.9
are independent of Lemmas 4.2—4.4. In particular, one can conclude that u is a COO
solution when F E 000 without using Lemmas 4.2-4.4.

The proof of Theorem 1.4 will be carried out in several lemmas. To begin with,

some results on the solutions of linear wave equation
Du(t,x) = g(t,x) in W (1.28)
are required.

Lemma 1.7. Given g E RUE) fl Ilk, h 2 0, then there exists a unique it E R(D) fl
Hk'H satisfying (1.28) and there exists a constant C such that

llullk+l S C'llalh

Proof. Recall that

3(5) = {U I u(1,53) = 21¢.1i1m6’ji(i)81m($)} ,

1953'

and that {€ji(t)51m($)}[¢j is an orthonormal basis of 1?.(Cl).

20

The uniqueness follows from K er(l:1)i 2 12(0).

To show the existence, let g(t, x) = 23-9“ gjilmeji(t)31m($). Then clearly

g'i m
u(t,x) = fieflfiklmw) E H(D)
j?“ 7

is a solution of (1.28). Moreover

2

. lc 1 g'im
Halli... = Z<1+r+e)+ —,~7.’,

. k 1+j2+l2
= Z(1+32+l2) lgjilml2(l+j)2(l—j)2

 

 

 

j?”
2
S Cllgllk7
which completes the proof of the lemma. D

Actually we can get a sharper estimate of the solution of (1.28) by using Sogge’s

‘the best estimate on the spherical harmonics’ [46].

Lemma 1.8 (Sogge). Let H(x) be a spherical harmonics on S3 of degree t Z 1,
then there exists a constant C independent of H, p andl such that

“H

 

 

LP<S3> 5 Cl(3/7;_2)III_IIIL2(S3) VP 2 4,
where p’ is the exponent conjugate ofp, i.e., l/p + l/p’ E 1,

First, we use this result to get a sharper L'p estimate for functions in Ii". Note

that K = Ker(Ds) C Ker(D).
Lemma 1.9. The identity map
I : I-I’“+‘(W) r) Ker(El) L» HWW)

is a continuous embedding for any 1 S r < 8, where H’”(W) is the standard Sobolev

space on W consisting of all functions which together with their derivatives of order

up to k are in L’“(W).

21

Remark 1.5. The standard Sobolev embedding theorem only concludes that embed-
ding
Hk+1(W) <—> H"”‘(W)

is continuous for 1 S r S 4. Lemma 1.9 is obviously an improvement when restricted

to Ker(CJ).

Proof. It suffices to show that for 4 S r < 8, u E L" if u E H1 0 Ker(|3). Expand

u in Fourier series

)2: Z Z u1,)ms)m(x )612' (t)

with

Ilullf : Z Z<1+ 2l2) Z lulilm|2 Z 2: 2 12a?“
1 i m z 2'

where “122' = Em lumml2- Set fly-(3:) : 2m s)m(x)u1,-1m, then ”Hy-”:2 = a?,. Using

interpolation theory, one easily proves that for r > 2 and 71 + vi, = 1,

l/r’
)I L"(S‘) ‘3 C (2 Ebb") '
l i

Using this inequality, Holder inequality and Sogge’s lemma, we obtain that for 4 S

HZ: 2.5116110

 

r<8

 

 

         

2 ”21,2 le'elzlt)

= It.

1' 1 )li'

r l/r
(it) dx

 

   

Z Illicit“)
1,2'

7,/7,I l/7‘
S II‘IHITI (l.1'
tat; )
r’/T l/r’
< C Hirdu
— (2< 3' ,. 1i )
l/r’
3 C(213 ”IlHull’b)
li

22

l/r’
S C (E: 13_2r'aii)
1,2'
In ,
. 1) (22—)
1,2‘

i I
:: (jbIIZLnla

 

2—r'

757— ,
) < oo smce r’ > 8/7 and

6—67"
I

where Co = C (E l 2—r

6—6r'
2—r’

 

< —1. Hence the proof is

 

completed. B

Lemma 1.10. Suppose g E 12(0) 0 HI“, and u E Kt- solves (1.28), then for1 S r <

8, there exists a positive constant C(r) such that

[lullHkvr(W) S C(rlllgllk'
Proof. The proof is similar to that of Lemma 4 in [57], so we omit it. D
Lemma 1.11. Let M be 4-dimensional compact manifold and f E Ck, then for any
u E Hk(M) fl L°°(M), f(x,u(x)) E HWM).

Proof. It suffices to consider the case f(x,u(x)) = f(u(x)). Set h(x) : f(u(x)). It
is well-known that if M is an n-dimensional manifold, f E Ck(M >< IR) and k > n/2,
then u E Hk(M) implies f(x,u(x)) E H’“(M). Therefore we only need prove for
k = 1 and 2.

For k = 1, the proof easily follows from
IVhIZ = (mun? IWI‘Z.
For I»: = 2, we only need to check that Af(u) E L2(M). Since
Mu) = Na) IVuV + Home. (1.29)

The standard embedding theorem. asserts that Hl(lVI) ‘—> [14(M). Therefore [Vul2 E
L2(M), which combining with (1.29) shows that Vf(u) E L2(M). El

Now we come back to establish Theorem 1.4.

23

Lemma 1.12. p, E 1100(51), Dt’Ug E C0’1(T) and Hz 6 EEO/V).

Proof. First, u E L°°(W) implies v E L°°(T) and F(t,p, u) sinp E L°°(T). Since v is
a solution of (1.3), Lemma 1.2 asserts that v2 E C0’1(T). Therefore v1 = v — v2 E LOO.
Exactly the same arguments as in the proofs of Lemma 1.5 yield 19’ E L°°(S 1), which
implies that vt is bounded. Next, since the operators C11 and D) commute, we see
that Dtvg is a periodic solution of (1.3) with nonlinear term F sinp replaced by
Dt(F sin p) = F, sin p + Fuvt. From the boundedness of vt and Lemma 1.2, it follows
that Dtvg E Co’l. Finally, the fact that p’ E L00 (v1 E Hé(T)) and v2 E CO'1
implies that u E H31(W). Lemma 1.11 asserts that F(t,p,u) E Hsl(W). Therefore
U2 E H32(W) by Lemma 1.7, and the proof is completed. [3

Lemma 1.13. p” E L2(S'1).

Proof. Boundary conditions and ’Ugt E C0’1(T) yield that

|||v2t/Sin/)|||Loo < C [[thIIICOJ < 00 (1-30)

Let 5 = —(v[‘,)"h in

/Dt(Fsinp)§=0 VEEN.
T

We obtain
/(D,(Fsinp))"vf, = 0. (1.31)
F11

The mean value theorem implies

(Di (psmpw — mi.

2 F” sinp + Fmvh + 1'1qu, + Fmv.) + vatvh/ sin p, (1.32)

where on the right side, all v’s are evaluated at some point in vR E [t,t + h] x {p},
and all F’s at some point in F3 E 223 x [u(t,p),u(t + h,p)]. Hence by (F4), (1.31)
and (1.32) we have

[Ilvft[]]L2 S (1/a)I||Ftt||Loo+|lFuzl|Lm|||vhl

IlFuIILoo many. + “Falls. IllvtlllLoo lllvh/sianIIL.]. (1.33)

 

 

L"2 + IIFutllLoo HIU‘IIILB +

24

Lemma 1.12 implies that the first four terms in bracket of (1.33) are bounded by a
constant independent of h, and so are [[FWHLOO and [llvtllle in the fifth term.

As for vh/ sin ,0 in the fifth term, write vh = v? + v3. By (1.30) we have
[[[vQ/sianhm S |||Dtv2/sinp[|[L00 < oo. (1.34)
Let a E (0,77/4], let T1 = 51 X [0,0], T2 = 31 x [7r — 0,7r], and T3 = T \ (T1 U T2).

Then

h
v1

 

.2211ewsarr/jar)-

On T1, we have 0 S p S o S 7r/4, v1 : 2p folp’(t — p + 2ps)ds. Therefore we get

 

 

 

 

 

 

sin ,0

2

(rd/8iI1 p)2 = (2p/ sin p)? ([0 p’h(t - p + 2sp)d3)
S C/O (p’h(t — p + 2sp))2 ds, (1.36)

where constant C is independent of a, from which it follows that

[Al (vi/sin Pl2]% g C [foo v/OZW /01(P,h(t _ p + 25p))2dS(ud/)] %
= [AU/OI/Ozwp’h(1+(28 _ 1)/’))2dtdsdp]i

, a 2 5 . -
= c(/ llr’hHmsnd/J) =Cx/5||29”||L2- (137)

Similarly on T2, using

v1 = p(t + p) - p(t — p + 2n) = 20) — 70/0 P’U — p + 23(10 — 7r))(ls,

we obtain

1

(j) s Wampum. (1.38)

25

On T3, we estimate that

1
5 1 C
(t) — lllvi‘lllm s ”tum... (1.39)
3

Writing vi‘, = p’h(t + p) — p’h(t — p) and using (1.8) and [p’] = 0, we get

 

 

Hlvtllliz = 27llp’hlli2- (140)

Then it follows from (1.33)-(1.40) that for some constants C1, C2 and C3 independent
of h and a,

HP'hHLz S 01+ 02/3111 0 + 03% HP’hllm-
Choose a in the above inequality so that C3\/—o_ = %, we get Hp'hlle S 2(C1+C2/ sin 0).
Therefore Remark 1.3 concludes that ||p”||L2 < 00, which completes the proof. D

Lemma 1.14. 112 E H3(W), p” E L°°(.S'1), and vag E CO’I(T).

Proof. Lemmas 1.11 — 1.13 imply that F E H32(VV); therefore we have U2 E HEN/V)
by Lemma 1.7. Also [D3, D1] = 0 and Fsinp E H§(T) imply that

/D?(Fsinp)£ = 0 V5 E N.
T

We write vm : p”(t + p) — p”(t — p), let q:t = q(p”(t :l: p)), where q is defined in
(1.10). Take 6 = q+ — q" E N. Similar to the proof of Lemma 1.5, in order to show
1)” E L°°(Sl), it suffices to prove that

D?(F sin p) — Fuvm = Fuvztt + F“ sin p + 2Fmvt + vaf/ sin p. (1.41)

is in L°°(T). In fact, the first three terms of (1.41) are in L°°(t) by Lemma 1.12.
That the fourth term is in L°°(T) follows from Lemma 1.1.3 the following estimates.

By Lemma 1.12, we have

 

 

[[[vit/ Sin/)[IILoo S ”[021] Loo Illv'lt/ sin PIIILm < 007 (L42)

lllvuvzt/Sinplllroo S lllvulllLoo [IIUQ‘l/SinplllLoo < 00- (1-43)

26

Fix a E (0, 7r/4]. On T1, using Schwartz inequality and Lemma 1.13, we obtain
2 ' 1 t+P II 2
Hun/anpm222)=|22;(i_,p(sis)II
, L°°(T1)

2 t+p II H
3 mp{”/ bamb}<mvmw (Ma

sin p

 

(taP)ETI t—p
Similarly on T2 we get
Ilvit/ sin pIIL°°(T2) < 00- (1.45)
On T3 we can directly estimate
Il’Uf1/SianlL°0(T3)S C [[[vftlllLoo° (146)

Now p” E L"0 and (4.15) tell us that D?(Fsinp) E L°°(T), which implies that

v2” E C0’1(T) since to = v2“ is the solution of Dlw = D?(F sin p) in H(Cll). Cl

In general, we have the following regularity result

Lemma 1.15. IfF E Ck(W X IR)(h Z 1) is symmetric, then U2 E Hs’°+1(I/V), p(t“) E
L°°(Sl), and vag E C0'1(T).

Proof. This holds for k = 1 and 2 by Lemma 1.12 and Lemma 1.14. We proceed by
induction. Assume the lemma is true for kzj Z 2, i.e., p(t) E L°o(Sl), vag E C0’1(T)
for i = 1, - - - ,j, and ug E Hsj+l(W).

For k = j+1, we first show p(t“) E [12(5') and 212 E Hg+2(l/V). By (1.7) it suffices

to Show Df+1v1 E L2(T). By the induction hypothesis,
[FDf(F(t,/),u)sinp)£ : 0 Vf E N. (1.47)
Taking { : ((va,)h)'h E N, we get
/(D{(Fsin p))"([)fvl)h = 0. (1.48)
T
Expanding the jth derivative and using mean value theorem, we obtain

umrmmr—awmr
. Dii Dir
: Fu(Dlv2)h + Z C’Ymrthu'" .t v t v

3111/) sin. p

sin p, (1.49)

 

 

m,r

27

where Cmr’s are positive constants, and i1 + . - ~ + i,. + m = j + 1 with

Note that on the right hand side of (1.49), all v’s are evaluated at some point in
vR = [t,t+h] X {p} and F’s at some point in FR = vR X [u(t,p), u(t+h,p)]. Denoting
the second term in the right hand side of (1.49) by I(F, v, h), substituting (1.49) into
(1.48) and using (F4) lead to

|I|(va1)"llliz 3(1/a)(IIFHLoo lll<Div2r|||L.+ |||1(F,vah)|HL2)- (1.50)

In the expression of I (F, v, h), consider a typical term

D? v Dirv

sinp sinp

 

 

thur

sin p.

By hypothesis
Illthu’

 

llLoo S C. (1.51)

If r = 0, then m = j + 1 and we have no derivative on v.

If r = 1, then since i1 S j, we have
[Ilthth‘vllle S “Ft"‘ulltoo IIIDt‘vIHLm < 00- (1.52)

If r 2 2, then it is easy to check that i,._1 S j — 1. Therefore by Lemma 1.6 and

induction hypothesis we have

 

 

Dis?) _ Djs’Ul + [)2302

 

. — . , E [10°(T) for l S s S r — 1, (1.53)
smp smp smp
and _
Dirv , .00 ,_, . . . .
, Sin p E L (I) (Since i,. S g). (1.54)
8111/)

Summarizing these estimates, we obtain
Ill/(Fwahllllm S C Ill/(li’fltahllllm < 00- (1-55)

Remark 1.3, (1.50) and (1.55) yield that I){+lv1 is in L2(T). The induction hypothesis
and v1 E Hj+1 imply that u 2 ul + ltg E HgHU/V). Therefore applying Lemma 1.11

and Lemma 1.7, we conclude that U2 E HgHWV).

 

28
Next, we show p(t“) E L°°(Sl). Let qi : q(p(j+1)(t :t p)), then if = q+ — q‘ E N,
where the function q is given in (1.10). Set
I(F,u) = Df+1(Fsin p) — 71.0%.. (1.55)
The same argument as we used for I (F, u, h) shows that
[IllfFvullllLoo S 0. (1-57)

Using (1.56), 6: (1+ — q’ E N and Df+1(Fsinp) E Ni we get

fFu(p(j+1)(t+p)—p(j+1)(t~p))(q+—q‘)=/1(F,U)(q+—(f)-
T T

Then following the same procedure as in the proof of Lemma 1.5, we obtain

730+” e L°°(Sl). (1.58)

Finally, (1.56)- (1.58) imply that Df+1(Fsinp) E L°°(T). Since Dvag solves

equation (1.3) with Fsinp replaced by the bounded function Df+l(F sin ,0), Lemma.

1.2 implies that Df+1v2 E C0’1(T). [3

Conclusion of the proof of Theorem 1.4. Lemma 1.15 shows that u 2
ul + u; E (H;c n K) 619(H;c+l fl KL). Furthermore, Lemmas 1.9 and 1.10 imply that
ul + u2 E Wk”1”'(W) D K 69 W’”(W) fl KL for r E [1,8). The standard Sobolev
embedding theorem shows that ul + 112 E (Ck—2’“(W) Q If) EB (Ck—1’“(W') fl KL) for
u E [0, %). 1:]

Remark 1.6. The regularity we established strongly relies on two hypothesis:

(1) there exists a positive constant Oz such that F” 2 a;

(2) F is symmetric, i.e, F is independent ofgb and 0, when 53 is expressed in spher—
ical coordinates, which allows us to apply some results from one-dimensional

wave equation;

We strongly believe that the condition (1) can be relaxed to the condition that F
is strictly increasing in u. We also hope that our results can be extended to non-

symmetric F.

 

29

1.4 Small Forcing Problem

As in the one—dimensional vibration problem [6], we can deal with the small forcing

problem. Consider the equation

Du(t,x) + g(u) + f(t,x) = 0 in S1 X S3. (1.59)

Again we assume f is symmetric, i.e., f(t,x) = f(t,p), and look for symmetric
solutions of

BMW) + 9(a) + f(tm) = 0- (1-60)

We assume that g(0) = 0, IlfllLoo is small, and that for some positive number L,

g E C1([—L, +L]) satisfies
l9'(8)| 2 O and l9(8)| S 7 [8| V8 6 I—L»+le (1-61)
where a and y are positive constants specified in (F3) and (F4) respectively.

Theorem 1.5. Assume g has the above properties. Then there is a 6 > 0 such that
for each f E L°°(W) with ||f||Loo + [[ftHLoo S 6, there exists a weak symmetric solution
of (1.60) with [[uHLoo S L. Furthermore, ifa S g’(s) S 7 for all s in [—L,+L], then
such solution of (1.60) is unique.

Remark 1.7. Theorem 1.5 and Theorem 1./, guarantee the existence, uniqueness,

and smoothness of the small solution for the equations

Dsu + u3 + out + f(t,p) = 0 in W,

Dsu + sin u + f(t,p) = 0 in W,
provided that 0 < a < 3 and f is small enough.

Proof of Theorem 1.5. For the uniqueness, assume it and a are two solutions of

(1.60) with [[uHLoo < L, and “MIL... S L, then

/ D(u—u)(u—u)+/ (g(u) —g(u))(u—u) =0. (1.62)
W

 

30

{From the Fourier series expansion, we have for any u E D(D)
3(Du,u) + IIDullsz 2 0. (1.63)
It follows from the above two inequalities and assumption on g that
11/7) / Ig(u) — 9(2))2 3 / (g(u) — ,(—))(.. — a)
W W
s (1/3) / lD(u — (2)12: 11/3) / lg(u) — 9(a))?
W W

Hence [g(u) — 9(a)] = 0, therefore u = it.

Now let us turn to the existence. We can construct an extension 9 of g satisfying
1. g E C1(IR) and g(s) = g(s) for all s E [—L, L];
2. There exist 3 > 7' 2 ”y , 0 < a’ S a and ,8 > 0 such that

fl 2 its) 2 a’ and I§(s)l s 7’ Isl Vs e R.

Let FOL/9,11): {1(a) + f(t,x)), then Ft = ft(t.p), F1. = a’ and
W S 7’ IUI + V W S #2 with V = llfllLoo and #2 = IlftllLOO° (1-64)

Therefore F satisfies (F3), (F31) with #1 = 0, (F4) and (F5). Theorems 1.1 and 1.3
imply that there exists a solution of (1.60) such that

3+1

(CY/)2

 

llullLoo s C (I|F||1.2+V)+ ”f—j'Li . (1.65)

Note that ||F||L2 depends on ii. To get Hull”, S L, we need to estimate ||F||L2 in
terms of “fur... Since g’ 2 a.’ > 0, the Implicit Function Theorem implies that there

exists a u0(t,p) E L°°(W) such that

F(t.p,uo(i,p)) = g(uO(t./))) + f(hp) = 0.

and
l
Iluollm s 5 11/11.... (1.66)

We write u 2 ul + u2 E (L00 D K) 69(1)00 (1 Ki). Then it; satisfies

[3311.2 + F = 0.

 

31

Taking L2 scalar product with u, we find that
(Usug,u2) + (F,u) = 0. (1.67)
The Fourier series expansion gives
3(Clu2,u2) + [[DU2[[:2 Z 0. (1.68)
LFrom the definition of u0(t,p), we have
F(tap,u)(u - 11o) = |F(t,p,11)| |u — 11012|F(t,p,11)ll11|-|F(t,p,11)lluOl,
from which we obtain
|F(taPIU)| IUI S F(ta/JWW + 2 [FUR/0116)] Ivol- (1-69)
Estimates (1.67)-(1.69) together with “Usugllig = ”F”; imply that
[W IFI Iul s 11/3) ||F||i2 + 2 [W Ira/1.11) |11o|- 11.70)
Equations (1.64) and (1.70) yield
11/7') [W W (IFI — u) 3 11/3) ”311:. + 2 [W IFI IuoI.

which gives us

3 ’ 1/
”FM/.2 5 217, (; + 211101....) 313. 1111)

Together, (1.65), (1.66), and (1.71) show that there exists a constant C depending

only on 7’, 0/ and 2 such that

HUIILoo S C(llf

 

 

Loo + 111111110) - (1.72)

We can, therefore choose 6 so small that [[fIIer. + Hf,

 

 

Loo S 6 implies Hull”, S L_

 

Chapter 2

Orbital Stability with Positive

Energy

2.1 Problem and Notations

In this chapter, we assume that Q is a bounded domain in IR” or a n-dimensional
compact manifold. We shall prove a sharp stability theorem for a wide variety of

nonlinearity f for ground state standing waves of NLKG

utt—Au+f(x,u)=0 in IR+><Q,
u = 0 on (99 if 811% 0, (2.1)
1110.3) = U13). 11110.12) = V13),

and NLS

iut — Au+ f(x,u) = 0 in IR+ x Q,
u: 0 on 811 if 811 75 (b, (2.2)
u(0,x) = U(x),
By a standing wave we mean a solution of the form u(x, t) = elw‘<b(x) with w a
real parameter, called frequency. The nonlinear interaction f we consider here is very

general, and has the form f(x, u) = g(x, |u|)u and satisfies the following conditions:

32

 

33

(F) There exist constants l > 2 and C > 0 such that for all x E 11
|F(x,s)[ g csl v s 2 0, (2.3)
where F(x,gb) = F(x, [45]) = 0M f(x,s)ds.
(FG) For t specified in (F), one of the following is true

1. F(x,s)/s2 —-> 00 as s —> oo uniformly on (1 ifl < 2n/(n — 2)

2. g(x,s)/s’"2 ——> A for some A > 0 as s —> oo uniformly on (1 ifl Z

2n/(n —2).

We will carry out the details for stability of NLKG (2.1). The investigation of
orbital stability for NLS is very similar, and is outlined at the end of this chapter.
Note that the search for standing wave of NLKG leads us to following nonlinear

elliptic equation

—A¢_w2¢+f($a¢):0)
$20 on 811 if 851%0.

Any solution of (2.4) is a critical point of energy functional

1 2 2 2
1111):, [Q IWI —w |</>| + / F133).

Note that by (FG) the functional Jw is bounded from below. Since we do not need
the explicit use of dependencies of (f) and .I on frequency a), we suppress the subscript
w for notational brevity.

Next we introduce some notations which will be used throughout the rest of this
dissertation. H = [11(9) (1 L’(f1) if 11 is a n—dimensional compact manifold, and

H 2 H661) fl Ll(f1) if 11 is a bounded domain in IR" and L” = L”(f1). As usual, we

use

(u,v): [ubdxy

f1
1
P
1111,. = (f |qu 13) .
Q

Ia2=fuwr+um.

 

34

to denote L2 inner product, LP norm and H norm in space variable. Next we introduce
the notion of weak solution [41] [49] of NLKG (2.1). A weak solution of (2.1) is a

function u(x, t) defined a.e. in S1 for each real t such that

(A) u(resp. at) is a weakly continuous function of t with values in H (resp. L2).
(B) f is locally integrable function on f1 >< IR.

(C) utt — Au + f(x, u) : 0 in the sense of distribution.

We now are ready to specify our problem. From assumption (FG), J is bounded
from below for any frequency 1.11, so it makes sense to define the minimization problem

to search for the least energy solution.

3 = $21,113) 12.3)

In next section, we show that d is actually achieved at some gbo(x) 2 0. Any
minimizer of (2.5) is called “ground state”, and corresponding eiWIqb(x) called “ground
state with frequency w” or ground state for short. For fixed frequency to, define S to be
the set of all minimizer of minimization problem (2.5), i.e., S = {ab E H I J(q5) = d}.

Now we are in a position to state main result in this chapter.

Theorem 2.1 For any fixed frequency w, the standing waves of NLKG (2.1) with
frequency w is orbitally stable in the following sense: for any given 6 > 0, there exists

(1 6(6) > 0 such that any weak solution it of (21) with initial data satisfying
inf.(||U — (Fill-I ‘1‘ [1V —1W(k1121 < 6
(1)633

has the property

[2190111111 — (kill-I 1‘ [1111(1) ‘10”(21121 < C V 1 Z 0

2.2 Existence of Ground State

In this section, we prove the existence, positiveness and regularity of ground state,

and give some characterization of ground state set S.

 

35

Lemma 2.1 Every minimizing sequence in H of problem (2.5) has a convergent sub-
sequence in H. In particular, d is achieved at some d. Moreover the minimizer can

be chosen nonnegative.

Proof. Let {dk} be a minimizing sequence. We divide the proof into two cases

Case 1: l < 2n/(n — 2). By (FG), there exist constants C1 and C2 that
J13.) 2 01131.12 — 02 V 3 e H

from which it follows that there exists a do and a subsequence of {dk}, still denoted
by {dk}, such that d)c ~—\ do weakly in H and d], —> do a.e. in 11 and strongly in LP
for any 1 S p < 2n/(n — 2). Thus from (F) and weak lower semicontinuity of [[1le

we have
£2.00 / 1112,33,.) = / F1330),

.1121]... / |¢1l2= / I312,
1,113,311 f 1w»)? 2 / Wot. 12.6)

It is easy to see that if a strict inequality held in (2.6) then we would be led to

following contradiction:
d S J(do) < limian(dk) : (l
k——+oo

Therefore we have dk —> do strongly in H1(f1), thus in H by Sobolev embedding
theorem.
Case 2: l 2 2n/(n — 2). (FG) implies that there exist some constants C1, C2 and
C3 such that
113).) 2 0.111113, + (1211311?— 03

which implies that there exist a do and a subsequence, also denoted by {dk} E H
such that

d), —‘- do weakly in Hl’(f1),

(1,, _. (to weakly in [1(9),

dk ——+ do strongly in Lp(f1) Vp E [1,2n/(n — 2)),

d;c —> do a.e. on $1.

 

36

Assumption (FG), F atou’s lemma, boundedness of f1 and weak lower semicontinuity

of ”NH yield that

liminf/F(x,dk) Z /F(x,do), (2.7)
k——+oo
limiande|§ 2 |V¢o|§. (2.8)
k—>oo

A strict inequality in either (2.7) or (2.8) would lead to contradiction
d S J(do) < limian(dk) = d.
k—too

Therefore equalities hold both in (2.7) and (2.8). An equality in (2.8) implies
die ——2 do strongly in H1(S1). It now remains to show that d), —> do strongly in

Ll(f1). Assumption (FG) implies that there exists some constant C such that
A
Fean—§MJ+C23
which combined with Fatou’s Lemma gives that

A A
liggf/ (“33,4511— 5 [(151211) 2 / (FUCK/>0) — 51%|!) -

Hence we get

k——+oo
A
2ggg/hmwn—gm0+qggf§mt
A A
/([P(x,(bo) — E— [(2011) -1-/—2—I(150[l
I fFf-tfiko).

From this expression, all the inequalities are forced into equalities, therefore we obtain

liminf/[dicll Z/I‘koll»
k——>oo

which implies d;c —> do strongly in Ll(f1) by a theorem in [4]

fF(x,do) = liminf/F(x,dk)

IV

The existence of a non—negative minimizer follows from the fact that

J(|¢|) S J(¢) V 11 E H.

 

37

Lemma 2.2 Assume that f is Lipschitz continuous on (1 X IR, then every real mini-

mizer of problem (2.5) is a classical solution of equation of (2.4).

Proof. Case 1: l < 2n/ (n — 2). The lemma is a direct consequence of standard
elliptic theory.
Case 2: l 2 2n/(n — 2). We use some kind of bootstrap argument. Let d be a real

minimizer of (2.5), then we have

(Vd, Vv) — w2(d,v) + (f(-,d),v) = 0 Vv E H. (2.9)
For a > 0, set
t HhflSm
v0 2 —0' If d < —0',

o if d > 0
Then for q = l — 2 > 0, v = [valq v0 E H and Vv = (q + 1) [valq Vva. Substituting
them into (2.9) yields that
a+1> Iauan—wflfItrt—wflf MHIM
12., 9., mac

9+2 q+l ”C : .
{Lngamnf¥f umww)3 1mm

where

fb=l$€Q|WWHS0}

Our choice of q and assumption (F) imply that each term in (2.10) is well defined.
Choose 0 large enough so that the last term to the left hand side of equality sign in

(2.10) becomes positive due to assumption (FG). Therefore the following estimate

holds
f]a”mwmswfmm=wwm3 1mm

On the other hand, using (FG), there exist positive constants C, and C2 independent

of a such that

mwamw2ajaar42/ur? am
no 9., Do

 

38

Combining (2.11) and (2.12) yields for some positive constant C3

(Iwrsawua

which by arbitrariness of 0 implies that d E LH’q. Thus the regularity of d is increased
by an order of l — 2. Repeating the above procedure, we are able to improve the
regularity of d so that d E L” for all 1 S p < 00. Therefore, by elliptic theory,
we have d E H2Ip(f1) for all 1 S p < 00, which implies d E 0110(9) by embedding
theorem. From the Lipschitz continuity of f, we have f E Ca(f1). Hence d E C2'a(f1)
by Schauder theory, and the proof is completed.

To conclude this section, we give some characterization of the set of ground states.

The proof for the following lemma follows the same idea as in [9].

Lemma 2.3 For any d(x) E 5', there exists a non—negative function do(x) E S such
that d(x) 2 eigdo(x).
Proof. Let d E S and d = d1 + id2 where d1, d2 E H are real—valued, then
d: [d1] + i [d2] is still in S, and this yields that

_A¢>7 _ ‘0sz]. '1' g(CL‘, [d)[)¢j = 0)

—AI¢>’I -w2 WI +g($,l<t>|)|¢7| =0-
where j : 1 and 2. This shows that 1.122 is the first eigenvalue of the operator

—A — g(x, Idl) and d1, d2, [d1] and [d2| are all multiples of the positive normal—

ized eigenfunction of —A — g(x, [d[) and the proof is finished.

2.3 Orbital Stability of Standing Waves

In this section, we will prove Theorem 2.1. Let u(t) be a weak solution of NLKG
(2.1) with initial data U and V.
Define the total energy and charge as

mam=§(/wwt+/wqu+/Fewwwa

620105)) =Im(u,(1),u(t)).

 

39

It has been shown [41, 49] that energy inequality holds for weak solutions of NLKG
(2.1) for the nonlinearity f we are considering, i.e., E(u(t1) S E(u(t2) for all to 2
t1 2 0. From the arguments in [41, 49], it is easy to show that the charge identity
holds for weak solutions of NLKG (2.1), i.e. Q(u(t)) E Q(u(0)) for all t 2 0.

Proof of Theorem 2.1: Suppose that there exists a frequency w for which standing
waves are not orbitally stable. Then there exist co > 0, sequences of {tk}, {U k}, {Vk}
and {uk} such that

.12th — 11.. + |le — 11111.) ——> 0 12.13)
and
yen; 1111111) — 11.. + ”11111) — 11111.) 2 11 12.14)
A direct computation shows that
E1111 )) —— 111211 =§)/ I11) —1wu1 11)I + 111111)) 12.13)
E(u(0)) — wQ(u(0)) = 21/ IV — inI2 + J(U). (2.16)

By Lemma 2.1, S is a compact set in H, thus we may assume that Uk ——> do in H,

Vk ——> iwdo in L2 for some do E S . Hence using (2.16) we obtain
E(uk(0)) —wQ(uk(0)) ——> d. (2.17)
On the other hand, (2.15), energy inequality, charge identity and (2.17) give rise to

qukftkl) S EfukUkI) 7 wau ”I k.» S EIU [col )1— WQ(11k(0))—> d: (2-18)
which implies that {ltk(tk)} is a minimizing sequence of problem (2.5). Lemma 2.1
implies that there exists a {/3 e s such that

uk(tk) —> 13 strongly in H, (2.19)
J(uk(t’“)) —> d. (2.20)
Using (2.20) and (2.15), we have

uf(tk) ——> iwd strongly in L2. (2.21)

Combination of (2.19) and (2.21) is a contradiction to (2.14), and the proof is com-
pleted.

 

40

Remark 2.1 First ifJ has unique positive minimizer gb for some frequency to, then,

by Lemma 2.3, the ground set can be characterized as
S = {1% l 0 e R},

and we have “real” orbital stability: For every given 6 > 0, there exists a 6 > 0 such

that
22,1; (HU —- 119111.11 + “V — 1111191112) s 6
implies
33,1; 1111111) — 1161111 + H1111) — 1111101112) s e V12 0-
For it = R”, the minimization problem {2.5) usually is either NOT defined or
only has trivial solution. 50 minimization is taken over some hypersurface in H. In
this case, the NLKG or NLS may not possess standing waves for all frequency 11), even

they have standing waves for some frequencz, the standing waves may not necessarily

be orbitally stable [9, I8, 19, 44, 5]].

2.4 Outline for Schrodinger Equation

In this section, we illustrate the proof for the orbital stability for standing waves of

NLS. The associated nonlinear elliptic equation used to seek the standing waves is

_A¢_w¢+f($v¢):07
$20 on 81211012710,

(2.22)

By same argument, we see that the nonlinear elliptic equation (2.22) has non—negative

ground state (1511 Z 0 for all frequency 112. Let
S = (d) E H I .]((/)) = d}

where

d = inf J11) = 2 / 1th —w W) + / F1111).

1156 H

We have the stability theorem.

 

41

Theorem 2.2 The standing waves of NLS offrequency w are orbitally stable in the

following sense: given 6 > 0, there exists a 15 > 0 such that if
‘ f U — < 6,
1125 H ¢HH

then

. _ .> 2
22911111) 111;; <e\11_0

The proof for this theorem is similar to that for stability of standing waves of N LKG.

Here we can use
1211) = / 1111112.
111111)) = 2/ W + [1111)

to replace E and Q for NLKG. The details are omitted.

 

Chapter 3

Orbital Stability With Indefinite

Energy

 

3.1 Introduction

In this chapter, let it be R“ or a bounded domain or a compact manifold. We shall give

a sharp condition for orbital stability of standing waves of nonlinear Klein—Gordon

equation(NLKG)

u” — Au + m(:c)u + f(x,u) = 0 in 1R+ x Q,
u=0 on (99 if 897MB, (3.1)
u(0,:1:) = U(ac), ut(0,:r) = V(:z:),

and nonlinear Schrodinger equation(NLS)

iut — Au + m(.r)u + f(zc,u) : 0 in R+ x it,

u =2 0 on (99 if (99 # 91, (3.2)

u(0,:t) = U(:I:),
where m is a real bounded function such that the lower bound 111 of the spectrum of
the operator -—A + m is positive. In this chapter, the nonlinear interaction f is very
different from that in last chapter, consequently new techniques must be employed.

Again f(x, u) = g(x, |u|)u and is imposed on the following conditions:

42

43

(f1) f 6 01(0 X IR) and f2(:1:,s) and f(x,s) —> 0 uniformly as s ——> 0.

(f2) There exist constants 2 < l < 2n/(n — 1) and C such that |f,’,(a:,s)| S 051-2 for
large 5 > 0 and for all .7: E 9.

(f3) f,',(:c, s) < 0 for a.e. :2: E Q and all s > 0 and there exists a constant 19 > 1 such
that sf;(w,s) g 19f(:r,s) for all :1: E Q and all s > 0.

Remark 3.1 It follows from the assumptions (f1)—{f3) that the following statements

are true:
1. F(x,s) = f03f(:1:,'r)dr g 0 for all :1: E Q and all s > O.
2. sf(:r,s) S 0 for all :1: E Q and all s > 0;

3. For any given :c E Q;
1

0+1

is a non-decreasing non-negative function ofs on (0,00).

F(zc,s) —

 

sf(:c,s)

4. For any non-negative v E H,
/v(:r)f(:1:,v(.r)) : 0 <=> v(.r) : 0.

We will carry out the detailed proof for NLKG. The proof for NLS is similar and
will be outlined at the end of this chapter.
As in last chapter, search for standing waves of NLKG (3.1) leads to the following

nonlinear elliptic equation

—A1 — 111111) — 11211 + 111.11) = 0.
$20 on (90 if 1')Q#91.

If we define

11111) = 2 / Ive)? + 2 f (11111) — 112)I<1.12+ / F111...)

where qbw is a least energy solution among all solutions of (3.3), then we have the

following main result:

 

44

Theorem 3.1 If d”(1.oo) > 0 and tag < 111, then the standing waves offrequency 1.110

are orbitally stable.

It should be pointed out that although our main result is similar to that in [44],
our result applies to both it = R” and Q : compact manifold or bounded domain,

and our method allows nonlinearity f to depend on space variable :1: as well as on u.

3.2 Least Energy Solution

In this section, we shall prove the existence of a positive least energy solution of (3.3).

Since we include the case 9 = R”, we need to redefine Hilbert spaces H and L2

as follows
H}(Rn) if Q = R”,
H = HMO) if Q is a bounded domain,
H1(Q) if Q is a compact manifold.

L2 = L3(R”) if Q : IR",
L2(Q) if Q is a bounded domain or a compact manifold.
where subscript r indicates that the corresponding function space consists of only
radially symmetric functions. When 9 = R”, we also assume that m(:1:) = m(|:1:|)
and 111.11) = 111111.11).
It is easy to see that every solution 17) E H of equation (3.3) is a critical point of

energy functional

1.111) = 2 / (|W1l2+ 1m11)—112)I<1)2)+ / F1111).

It is also easy to verify that every solution of (3.3) satisfies the functional identity
11.11) = / 11W + (11111) — 112) W) + / 111111.111) = 0.

Therefore, it is natural to search the nontrivial least energy solution by solving the

following minimization problem.

d(w) = inf 1,111). (3.4.)

d’EMw

 

45

where surface Ma, 2 {9b 6 H | Kw(<,b) = 0, 1b 75 0}.
Indeed we will show that for every 102 < 111, d(w) is achieved at some nontrivial d)
and all minimizer of (3.4) are least energy solutions of equation (3.3).

First we define functional

1

Iw(¢1 Z Jw(¢)— mlfiikb)
(9 —1 2 2 2
: am/(‘W' +(m111)—w1|15|1

+ / (1:11.111) — fi 11111111110 .

and set
MJ=f¢€ H | [{w(¢1307 45750}-
Next we give several lemmas to lay foundation for existence theorem of ground

states. The first lemma is about equivalent H—norm

Lemma 3.1 Let ,u < 111, define

B111) = 33,1; (/ 11%)? + 11111) — 11) 1112). 11111,. =1},
then 3(a) is a positive decreasing function ofu.

Proof. 8(a) is a decreasing function since the integral is a decreasing function of a.
For positiveness, we prove by contradiction. For ,11 < 111, suppose that there exists

a sequence {vk} such that
11113. = / (le1|2 + 1112) =1. 13.5)
(111— 11)] |ka2 S /(|Vv;,|2 + (m(.r) — ,11) Ivklg) ——> 0 as l: —> 00. (3.6)

From (3.6), |ka2 —> 0 as k ——+ 00. By boundedness of m and by second part of
(3.6), we obtain
/|Vvk|2 ——> 0 as ls —+ 00. (3.7)

Therefore a combination of (3.7) and Ivk|2 ——> 0 leads to a contradiction of (3.5),
which completes the proof.
Next let us prove that minimization problem (3.4) is equivalent to a very useful

minimization problem.

 

46

Lemma 3.2 For any 1112 < Al, Ma, and M; are non-empty, and
d(w) = inf Iw(qb).
¢€MJ

Furthermore, Iw(qb) > d(w) if Kw(qb) < 0.

Proof. It easily follows from assumptions (f1) and (f3) imply that M; is non—empty
for all 1.1). The non-emptiness of Mu, is a consequence of the following arguments.

Consider any function v E H such that Kw(v) < 0. Let va(:1:) = ozv(:1:), then

2

K11.) = 2 f 11%)? + 1111 — 112M111?) + a 111 11.1.11 11)).

Now for a = 1, Kw(v1) = Kw(v) < 0 and for a close to zero Kw(va) > 0. Therefore
there exists an are E (0,1) such that Kw(vao) = 0. Remark 3.1 and definition of [w

imply that
Vv E H, Iw(sv) is an increasing function of s on (0, 00),

which yields
d(w) g [w(vao) = w(ozov) < Iw(v).

Hence we get

1111.)) g inf 1,111).

¢EMJ
But by definition

d(w) = inf Jw(¢) : deli/i“, Iw((b) Z inf Iw(</)).

fJEMw (bEMJ

which concludes our proof.

Lemma 3.3 For 11:2 < 111, Ma, is a Cl-hypersurface in H, and both MW and [V]; are

bounded away from zero.

Proof. (f1)-(f3) imply that Kw is a C’l-functiona.l in H which in turn implies that
Mw is 01 hypersurface.
For any small 6 > 0, from (f1) and (f2) there exists a C(e) > 0 such that

111111. 11)) 2 —e 1112 — C11) I111. 13.8)

 

47

Using (3.8), Lemma 3.1, Sobolev embedding theorem and LP interpolation theorem,

we have that for e < Al — 1112

2/UV1”+(m-w1W—1|11)/ul
BWZ‘HWma—O 1)/u

Z Clii¢iiH — C2||¢||ir

K1115)

IV

IV

which implies that Ma, and M; are bounded away from zero, and the proof is com—

pleted.

Remark 3.2 For it 2 IR”, if the mass term m and nonlinear interaction term f
are independent of space variable :13, then any nontrivial solution v E H of nonlinear

elliptic equation {9.3) also lies on another C1 hypersurface

M1. = {15 e H l 111111) = 0,11% 0},

where

 

Kuu) : ng2/IVu|2+n/ [$(m—w2)|u|2+F(u)].

To prove this, we need to use the scaling property of function in HI(R”). Let u 6
H3011") be a solution of (3.3). Put u,,(:1:) = u(re/u), then

1 l

2/wwt+2/1m— W)u1+/an
71—2 71

: #2 /|Vu|2+%/(m—w2)(u|2+u" [F(u).

Since u is a solution, d(.]w(u,,))/d,u = 0 at [t = 1. An easy computation shows that

/[1m—w)ut+rufl.

Note that for n = l and n = 2, M1, is not bounded away from zero, and the mini-

Jw(u,,)

 

(“L/(“1111b
411

                  

mization problem can not be defined.
Now we are ready to present our existence theorem for ground states.

Theorem 3.2 Let 1122 6 (0,111). Then,

 

48

1. d(w) is positive;

2. Every minimizing sequence ofproblem (3.4) possesses a convergent subsequence.

In particular, d(w) is attained at some dw;
3. This minimizer qbw can be chosen positive;

4. Every minimizer of problem (3.4) is a solution of equation (3.3) and is called
the ground state.

Proof. Let {pk} be a minimizing sequence in Mu, for problem (3.4). Remark 3.1 and
Lemma 3.1 imply that there exists a constant C(w, 6) > 0 such that for all (b 6 [WW

C(w,9)||¢lliz s ,ij3 / W2 + (m — w?) W s W) = W), (3.9)

which implies that a3}, is bounded in H. Thus by Sobolev embedding theorem (if
Q = R" we need corresponding embedding theorem developed in [50]), there exist a

9250 E H and a subsequence, still denoted by {gbk}, such that

4),, —> do weakly in H,
<sz —> do strongly in LP(Q),

(bk ——> (to a.e. on Q.

where 2 < p < 2n/(n — 2) if Q 2 IR”, 1 < p < 2n/(n — 2) otherwise.
Next we want to get strong convergence of sequence {qbk}. To that end, let 0 <

a = %()\1 — w2) and rewrite Kw((b) as follows:

nu) = so) + PM» (3.10)
with
Bu) = / IWV + [on —— w? —- a) W,

and

PM) = a j W + / lam, |(/2l)-

 

 

49

By Lemma 3.1, ‘/ S (<15) makes an equivalent norm on Hilbert space H, therefore after
selecting another subsequence of {fl}, we get that ¢k —\ (150 weakly in H under the

new norm, and by weak lower semicontinuity of the norm x/S(-), we have

1mg / IV¢kl2 + /<m —w2 —a) W 2 / weal? + /<m —w2 — 0) l¢ol2- (3.11)

Without loss of generality, we assume the existences of lim f [qfiklz and limP(¢k).

In (3.8), choose 0 < 6 < a, we have that for some positive constants 01 and 02

PM) 2 01 / W — 02 lastl’,

which by Fatou’s lemma implies that

1.13:. [fiance / W] 2 P(¢o)+02 / I¢ol’.

Since qbk —) (to strongly in L” for 2 < p < 2n/(n — 2), we immediately obtain

)1;anka P(¢>o)- (3.12)

(3.11), (3.12) and Remark 3.1 yield that

Iw(¢0) S lirninflw(qbk) = d(w), ‘ (3.13)
[g(eso) g limjanwwk) = 0, (3.14)

A strict inequality in (3.11) would imply a strict inequality in both (3.13) and (3.14)
which in turn would imply (b0 ¢ 0, and thus by Lemma 3.2 would generate a contra—
diction
d(w) < Iw(¢o) < d(w).

Therefore we obtain the strong convergence of {$16} to (1)0 under the equivalent norm
which implies the strong convergence under the original norm ||-.||

Lemma 3.3 and (3.9) show that (1)0 aé 0 and d(w) > 0.

The existence of positive minimizer <25“, follows from Jw(|¢|) 3 me), Kw(|¢|) s

Kw(¢), Lemma 3.2 and strong maximal principle.

 

 

50

Finally to show that (1)0 is a solution of equation (3.3), we have by Lagrange
multiplier method

6Jw(¢0) = A6I{w(¢0),

01‘

—A¢0 + ("1(73) '— W2)<150 + f(% $0)
= A [—2A¢0 + 2(m — w2)q5o + f(m, $0) + ¢0fl($a |¢Ol)]-

Taking inner product with (be on both sides and using Kw(¢0) = 0 lead to
0 = A / (|on + (mm — w?) |¢o|2 + W f’(:v, |¢ol))-

Using Kw(¢o) = 0 again , we obtain

 

0 = A / |¢o| (f(x, nasal) — label? f’(:c, lea»- (3.15)
From (f3), it follows that

/ l¢ol (f(x, |¢ol) — l¢o| m, |¢ol))

2 <1 —0) / Mammal)

= (a — 1) / [Ivar + (m — w?) |¢o|2l > 0, (3.16)
which implies that A = 0 and the proof is completed.
Corollary 3.1 Every minimizing sequence of the minimization problem

inf Iw(¢), (3.17)
¢eMJ

has a subsequence converging to a 4)“, 6 MW. In particular, (15w is also a minimizer of

minimization problem (3.4).

We conclude this section with the definition of the set of ground states

5.. = {</> 6 Me | Jw(</>) = d(W)}~

 

51
3.3 Standing Wave as a Function of Frequency
In this section, we prove that standing waves are smooth functions of frequency.

Lemma 3.4 d(w) and ||<bw||H are uniformly bounded for w2 on compact subsets of

(0,A1).

Proof. The uniform boundedness of d(w) follows from the fact that given we 6

(0, A1), there exists $0 6 Mg, hence there exists an e > 0 such that
Kw(¢o) < 0 for w 6 (we — e,wo + 6),
from which and Lemma 3.2 it follows that
d(w) S Iw(¢0) S C for w 6 (we — 6,000 + 6),

By Remark 3.1 and Lemma 3.1, we have

d(w) = Jae...) = 1.1a) 2 % / [Ivar + (m w?) 141.12]
Bow — 1) 2
_ 2(6—+1)H¢allya (3-18)

which implies the uniform boundedness of Howl)”
Lemma 3.5 d(w) is a decreasing and continuous function ofw form 6 (0, \//\1).

Proof. Let 0 < wl < wg < \/;\_1 and d(wl) = le((bwl), then

1

[{W2(¢W1): [(w1(¢w1)— 5(a); _ wf) / |¢m '2 < 0'

Therefore by Lemma 3.2 we have

(1(a)?) S IW2(¢W]) < le(¢u/1): d(wll'

which concludes the proof for monotonicity of cl.
Next to show continuity, let we 6 (0, fl), and we will show cl is left and right

continuous at Loo.

 

 

 

 

52

For left continuity, let 0 < w < we and va(a:) : a¢wo(:c), then

g(w,a) E Kw(va)

is a smooth function of a and (.0. Moreover, we have

g(WOa 1) : 03

and by (3.16)

912(0)“ 1) = / (|¢wol2 fl($i¢wo) _ |¢w0| f($a¢wo)) < 0 (319)

Therefore by implicit function theorem, there exists a neighborhood of too and a 01
function a = oz(w) in this neighborhood such that oz(w0) = 1 and g(w,a(w)) = 0.

Hence we have

d(wo)<d(w)<1(a(w))¢eo
a, 0—
(012)21(0+1)( ‘00— UAW-’0' +Iwo(a (“0951110)-

Let w —> (do, then a(w) —) 1 and [wo(a(w)¢wo) —> d(wo). Hence d(w) —> d(wo) as
w ———) too, which concludes the proof for left continuity.

For right continuity, select (.02 such that we < 1.02 < m and let we < 1.0 S (.02. To
show that

11m N“ )= d(wol-
w—Wo
It suffices to find a function a(w) such that
Kwo(a(w)¢w) : 0 and d(w) —> 1. as w ——> 023“. (3.20)

In fact,

(“(#0) = JW0(¢WO)<JWO(Q (: l¢W)
W(()¢w)+ 0/2 2W)(2w_wo)/¢2

d(w) + [a (aw...) — was] + 0‘ ,“(wz — at) / 453.-

 

 

 

53

Hence d(w) ——> d(we) as w —) wg, since ||¢w||H is uniformly bounded on [we,w2].

Now we come back to find a(w) satisfying (3.20). Set
_ [{wo(a¢w)

9012,01) 2
0’

= / |V¢el2 + (m — «in: + g / Mme).

where w E [we,w2] and a > 0.

Note that we can write
g(w. a) = Mac. a) + QM
with
have) = / Ivar + (m — one: + g / eta-east),
ow = (w? — wt) / 413..

It is easy to see that h(w,1) = 0, Q(w) > 0 for w > we and that Q(w) —) 0 as
w ——> we by uniform boundedness of ||¢w)|H. We can find the derivative of h with

respect to a as follows:

h’.(w.a> = 01/ [maritime — areflwwwfl-

Hence h(w, oz) 2 ha(w,fl)(a— 1) for some fl between a and 1. Using (f3), Kw(<bw) : 0,
Remark 3.1 and Theorem 3.2 we have

0

 

awn s 7,1 [Bane/1a) < 0. (3.21)

/3

Therefore we can find an a : u(w) satisfying

g(w, 0) = (0 _1)h0(wvfl)+ 62(10): 0,

 

i.e.,
a = 1 — $52) (3.22)
Note that by Remark 3.1, we have
d(w) = W.) s g / (Ivar + (m — mac/>3.)- (3.23)

 

l

 

54

Using (3.21), (3.22), Remark 3.1, Kw(¢w) = 0 and (3.23) we arrive at

        

       

 

 

he(w,fl) S (21,391.)
g f(x,¢..)
: (:1 ((vewl +(m —w2)¢3.)
S %1—?flW)S2(1:gflWl

Since Q(w) —-> 0 as w —> we, from (3.22) and (3.24) it follows that to show that
a(w) —) 1 as w —) wg, we only need to show that a is uniformly bounded on

[we,w2]. Suppose that this is not true, by uniform boundedness of (be on [we,w2],

there exist a sequence {wk} 6 [we,w2] such that

wk —> Li) 6 [000,w2],
a = a(wk) —> oo,

¢k = abwk —\ gb E H weakly,

and

aiinomkd’k) = /lv¢kl2 + (m — we)¢i + aik/mflfflaakqn) =

Notice on one hand, by uniform boundedness of “(15ka we have
lim sup/ [IVcka2 + (m — wg)<b,2,] < oo,
k—aoo
On the other hand, we define for s 2 l

=§/eaaeat

then
7/[ [S¢2f’ (IE S¢w) _¢wf($7s¢w]
Eli/(Hem (e squ) (byfg)

21.0(8)

8

 

(3.24)

 

(3.25)

(3.26)

(3.27)

55

Therefore we have

 

 

since C(S) < 0 if s 2 1,

from which it follows that
G(s) g G(1)S"‘1. (3.28)

Next let us estimate 0(1). By Kw(¢w) = 0 we have
0(1) = / arms.)

— / (lat + (m — Mei)
—2d(w2) < 0. (3.29)

l/\

Therefore (3.27) (3.28) and (3.29) yield

liminfik/pkf(x,akgbk) 2 —oo.

k—mo a
which combined with (3.26) is a contradiction to (3.25), hence the proof is concluded.
If we assume the minimization problem (3.4) has a unique positive solution, then

we have the following theorem regarding the continuity of Q, on w.

Theorem 3.3 Forw near we, let (be, be the unique positive solution of problem (3.4).
Suppose that zero is not an eigenvalue of the linearized operator £e = —A + m —w§+
f’(-,¢wo) at (two acting on L2 {real valued). [ff is a Cl function, then w —> (be, is a

Cl mapping from a small neighborhood ofwe into H.

We carry out the proof of this theorem in two lemmas following the same procedures
as in [45].

Lemma 3.6 w —> (be, is continuous with values in H.

Proof. From Lemma 3.4 and 3.5, d(w) = [u(qfiw) is continuous in w and ”pen” is a
bounded function ofw. Let {wk} be a sequence tending to we. Then {(bwk} is bounded

in H. A subsequence may be chosen converging weakly in H to some v. Note v 2 0

since each 45m is positive and (,ka ——~> v a.e. on (2. Now

0 e was.) = / (Ivar + (m — cove: + «an-.91.», (3.30)

 

56

Letting w = wk —) we, by uniform boundedness of gbwk, continuity of K and d and

lower semicontinuity of weak limits, we have

Kwo(v) < 0,

and
Leo(v) 3 lim inf [wk (4)0“) = d(we).
k——+oo

By similar arguments in the proof of Theorem 3.2, we have Kwo (v) = 0, [wo(v) = d(we)
and (be, ——) v strongly in H. Then by uniqueness, v 2 (two, which completes the

proof.
Lemma 3.7 In a neighborhood in H of (two, all solutions of (3.3) lie on a C1 curve.

Proof. Write (3.3) as

—A¢+m($)¢+7¢+f($a¢) = 03 (331)
where r = —w2. Let re 2 —w3, qSe : (be, and let
£(T,v):v+(m—A+T)~lf(-,v), T> —/\1, ’06 H. (3.32)

Then £(r,v) E H since v E H C LZn/(n—Z), f(-,v) E L2n/("+2) by (f1) and (f2),
and therefore (r + m — A)‘1f(-,v) E H by elliptic theory. In fact, £(r,v) is a
C1 operator from (—)\1,00) x H into H. Note £(re,¢e) = 0. Now the operator
£0 = —A + m — wg + f’(-,(bu,o) is invertible by assumption. It follows that the
compact operator (re + m — A)‘if’(-,(be)(re + m — A)‘i' on L2 does not have —1 in
its spectrum. Hence

g—£(70a¢0) = I + (To + m _ A)_l~fl("¢0)’
v

acting from H to H, is invertible. By implicit function theorem, the solutions of
£(T,'v) = 0 in a neighborhood of (re,<be) form a 01 curve in (—)\1, 00) X H.
With these preparations, we can go on to find the derivative of d, which concludes

this section.

 

57

Lemma 3.8 Under the conditions specified in Theorem 3.3, we have

d’(w) = —w / 101.12.

Proof. From

d(w) = Jae.) = g / (1%)? + (m — w?) 141.12) + / PM.)

we have

I 6 w
d (w) = / (—A¢w + (m —w2)q§w + f(x,¢w)) 5% — w/ |¢w[2. (3.33)
The first integral in (3.33) is zero since dw is a solution of equation (3.3), and we

proved the lemma.

3.4 Stability of Standing Waves

 

We consider NLKG
u“ — Au + m(:r)u+ f(x,u) = 0 in R+ X 0,
u = 0 on 39 if 89 75 0), (3.34)
u(0,:r) : U(rc) E H, u.)(0,:c) = V($) 6 L2.

For 0 = R" it is shown in [15, 16] that strong solutions u() E C([0,T),H),
ut(-) E ([0,T), L2) exist for nonlinear interaction we are considering. For other cases
of 9, it is shown in [41, 50] that weak solutions exist and for these solutions energy
inequality holds. In this dissertation, we will only consider the weak solutions of
NLKG or NLS. The proof of stability for strong solution is relatively easier.

Let

Edam) = g / lvlz + Mu)-
Define
Rw = {(u,v) E HEB L2 l Ew(u,v) < d(w)}.
Next we introduce two invariant sets which plays a very important role in the estab—

lishment of stability.
R3,, = {(u,v) E Rw|1{w(u)> 0} U {(0,v) 6 Re} ,
R3, = {(u,v) E Rw|1{w(u)< 0}.

58

It is easy to prove that we have the following equivalent expressions

R1 {(u,v) 6 Re I Iw(u) < d(w)},

LU—

RE, 2 {(u,v) E Ru, | Iw(u) > d(w)}.

Lemma 3.9 R3, and R3, are invariant regions under the solution flow of the following

modulated equation
Utt + 2iwu, —- Au + (m(:c) — w2)u + f(x,u) = 0 in R+ X 0,
u = 0 on 30 if (99 75 (b, (3.35)
u(0,:r) = U(a:) E H, u,(0,x) = V(a:) 6 L2.
Proof. Let (U, V) E R3, and assume that there exists a 7' such that (u(r),ut(r)) ¢
R3,. Then u(r) 75 0 and Kw(u(r) S 0, i.e, u(r) 6 MJ. Let

 

s = inf{0 S t S 7' | (u(t),ut(t) ¢ Bi} , (3.36)

then Kw(u(t)) Z 0 for all 0 < t < 3. Let {5).} be the minimizing sequence for problem

(3.36), then arguing similarly as in the proof of Theorem 3.2 we have
Kw(u(s)) S limianw(u(sk)) S 0.
k—too

Note u(s) = 0 would imply that Kw(u(s)) = 0 which in turn, would imply u(sk) —)
u(s) strongly in H. Then Lemma 3.3 and u(sk) E M; would imply u(s) 76 0 which

contradicts the original assumption. Hence we have
I(w(u(3)) S 0 and u(s) 76 0. (3.37)
On the other hand
Iw(u(s))=liminf1w(u(t))
t—ts"
l
< . . ,, ,
_lt1_n_1+18nf<1w(u(t))+ -———0 +1I\,,,(u(u(t)))
S liminf Ew(u(t),ut(t)) < d(w),
t—ts—
which, in View of inequality (3.37), contradicts Lemma 3.2 and completes the proof

for the invariance of R3,.

59

To show the invariance of R3, we just need to switch the roles of Le and Kw.
Let (U, V) E R3, and assume that there exists a r such that (u(r),ut(r)) ¢ R3,, i.e.,
Iw(u(r) S d(w). Let

s = inf {0 S t S r | (u(t),ut(t) ¢ R3,} , (3.38)

then by weak lower semicontinuity Iw(u(s)) S d(w) and Iw(u(t)) > d(w) for all
0 < t < 3. On the other hand

was» = 1331,1930 + 1) 1J.<u<t)) — we»)
5 1,131,926 + 1) 112.1140. 111(1)) — d(w))
s (0 + 1) [Ee(U, V) — d(w)] < 0,

 

which, in view of Iw(u(s)) S d(w), contradicts Lemma 3.2

Lemma 3.10 Assume d"(we) > 0. Then there exists an M(we) Z 0 such that for
every M > M(we) there exists a 6 : 6(M) such that if u(t) is a weak solution of
NLKG equation (3.34) with initial data satisfying

llU - ¢wo||H + W - in¢wo||2 < 6,

then
d(w+) S [wi(u(t)) S d(w_) V t > 0, (3.39)

and

é/Iudt) — iw:):u(t)|2 + Ji(u(t)) < d(wi) Vt > 0, (3.40)

where wi = we :1: l/M.
Proof. Set vi(t) : e‘iwi‘u(t). Then vi satisfies

'Uitt + 2iwivi¢ — AvgE + (m(:r) — wi)vi + f(x,ui) = 0 in 13+ x ft,
1).. = 0 on an if an ,1 (2), (3-41)
vi(0,$) = U(a:), vit(0,:r) = V(ir) — iwiU.

60
Note

(u)=J wi ()vzt,

/|vit|2 =/|ut(t) —iwiu(t (t)|2

The energy inequality of modulated equation (3.41) becomes

2/Ivit(t )| +Jwi(u( _ ))<2/lv— MU) +Jwi(U) (3.42)

To show (3.39) and (3.40), by invariance of Ru,i and Rat under solution flow of
modulated equation (3.41) and by energy inequality (3.42), it is sufficient to prove
that

d(w+) < [wi(U) < d(w_), (3.43)
and
EWi(U, V — tin) < d(wi). (3.44)
We first prove (3.43). Note

14401) = Ina...) + 0(5),

therefore 6 can be chosen if

d(w) < Lafitte.) < d(w)-

Set a— — 2—l(o-+1)<'i It is obvious to see that
Iw+(§bwo) : [wo(¢qu) + “(tag _w—2+-) / Wheel2
< d(we) < d(w_),
and
Ind—($0.10): [wo(¢wo) + d(wg _ WE) / l¢wol2
> d(we) > d(w+).
Note

Kw+ (¢W0)_ "‘ [{W0(¢WO)+ “’0 _w+) )W < 0

 

61

hence Lemma 3.2 implies that
d(60+) < 1w+(¢wo)-
To see d(w-) > [w_ (63%), we use d”(we) > 0 and d’(we) = —we I [deal2 to get

Ina...) = 1.44...) + d(wS — wi) / 14...)"

= (10,00).). affl‘foi(

w_ — we)d'(we)
< d(we) + (w- — we)d’(we) < d(w_).
Now we turn our attention to (3.44). It is easy to see that

Jwe(U) = Jui(¢1uo)+ 0((5 ) = on(¢h}o)+ (3-45)

                 

and

“V — iwiUll2 S “V 7‘ iw0¢woll2 + llw0¢wo _ wi¢we||2 + llwi¢WO _ wiUllz
lwo — Wil ||¢wol|2 + 0(5) (3-46)

Using (3.45), (3.46) and d”(we) > 0, and choosing 6 small enough, we obtain
Ewi(U, V — iwiU) S d(we) + (0);]: — we)d’(we) + 0(6) < d(wi).
which concludes our proof for the lemma.
Finally we can present our main result.

Theorem 3.4 Ifd”(we) > 0, then the ground state standing waves offrequencyw are
orbitally stable in the following sense: for every given 6 > 0 there exists a 6 = 6(6) > 0
such that
(,in (IIU— ¢|ln + “V iwotllz) S 5
implies
,ieggo (111(1) — 41., + “1110) — ween.) :6 for an 12 0.
Proof. Suppose that standing waves of frequency we are not orbitally stable. Then

there exist {(Uk, 14.)}, {tk}, and weak solutions {uk(t)} and Ge > 0 such that

,g (HUI. — 41),. + “v. — main.) —> o. (3.47)

 

62

and
152%in (Mums) — ¢HH + ||U§°(tk)||2) Z 60. (3-48)

Since Se,0 is compact in H, without loss of generality, we may assume that
(Uk,Vk) —> (v,iwev) for some v 6 5%.
From Lemma 3.10, there a subsequence of {k} such that
d(wo +1/k) S fwswkfikl) S d(Wo —1/k), (3-49)

and
IIUl‘Uk) - iw+uk(tk)lli + Jw+(uk(t1c)) < d(Mo +1/k)) (350}

where wi = we :t 1/k. (3.49) and (3.9) imply that uk(tk) is bounded in H, therefore
by continuity and (3.49) again

[wo(uk(tk)) —> d(we). (3.51)
From (3.50) it follows that there exists another subsequence of {k} such that
Je0(uk(tk)) —) d g d(we) for some d. (3.52)
Hence (3.52) and (3.51) yield
liggtxwgunte) = (o + 1) 11332) (.on(uk(tk)) — [M(ukum) S 0. (3-53)

(3-51) and (3.53) imply that {uk(tk)} is a minimizing sequence of problem (3.17),
therefore by Corollary 3.1 there exist a sequence of {It} and a (b 6 Sue SUCh that

uk(tk) —> d as k ——> 00,
which together with (3.50) implies that
k > . . 2
u,(lk) —) iweqb 1n L .

Therefore we get a contradiction to (3.48).

 

63

3.5 Nonlinear Schrédinger Equation

We recall the nonlinear Schrédinger equation
iut — Au + m(r)u + f(m,u) = 0 in R+ x Q,
u=00n89 if 89740,
u(0.x) = 11(2).
The proof of stability of standing waves of NLS will be similar to that of stability
of standing waves of NLKG which we presented in the previous sections. So we will
state the relevant lemmas and theorem without proof.
The associated nonlinear elliptic equation resulted from searching the standing

waves of the form ei“’¢(m) is

—A¢> — (ma) — as + foot) = (3.54,
ctr—0 on (90 if 80#@,
The modulated equation is
-— Au + (m($) — w)u + f(x,u) = 0 in 13+ x Q,
u = 0 on 00 if 09 # (ll, (3-55)

u(0, :13): U(:c),

The energy for modulated equation (3.55) becomes the energy for (3.54):
EwW) = Jew}-
The corresponding J, K and I functionals are
¢)=;/( IWI +< (rc—) )|¢I)+ /m we)
3):) (IWI .1 (s)— o) 13)?) + / f(x, Ill) Ill.

0 2
M4): 2(0+,)/(1IV¢| +(m —w)|¢|)

+ / (F< l¢>|)— elsl) lll)

The minimization problem is

 

d(w) = ,3), Mt),

64

where Me, 2 {ch 6 H IKw(qb) = 0, ab 74 0}.

The invariant regions R2, and R3,, in H are defined as

Rw = {u E H l Ewtu) < d(w)},
R; = {u 6 Re, I Kw(u) > 0} U {0},
R3, = {(u,v) 6 Re, I Kw(u) < 0}.

The range for frequency w is w < A1 and the derivative of d over this range is

no=—/uw
Finally the stability theorem is

Theorem 3.5 If d”(we) > 0, then the ground state standing waves offrequency w are
orbitally stable in the following sense: for every given 6 > 0 there erists a 6 = 6(6) > 0
.such that

' f — < 3
,ggwollU tile-

implies

. . _ . < ‘
$161320 IIu(t) dIIH _ e for all t_>_ 0

where Swo = {(b E Mwo I Jw0(gb) = d(we)}.

3.6 Applications

In this section, we consider several cases of nonlinearity f or domain Q where we have

orbitally stable standing waves.

Theorem 3.6 If the lower bound A1 of spectrum of operator —A + m is a positive
eigenvalue{This is certainly true if the underlying domain 9 is a bounded domain in
R“ or a compact manifold. It is also true if m($) is a potential, and the operator

—A + m has discrete spectrum to the left of a continuous spectrum), then
1. The NLKG have orbitally stable standing waves for w2 E (0, A1).

2. The NLS have orbitally stable standing waves for w E (—00, A1).

65

Proof. By Theorem 3.4 or Theorem 3.5, it suffices to show that there exists a we
such that d” (we) > 0. Again we only give the proof for NLKG. The proof for NLS is
very similar.

Suppose not, then d”(w) S 0 for all w E (0, m) which implies that
d'(w) = —w/ IquI2 is decreasing for w E (0, A1),
which yields that there exists a positive constant C independent of w such that

/I<,/>WI2 2 C for w E (6, VT?)

for any constant 0 < e < m. By definition of d(w), we have

 

d(w) = to.) > 6 “1

—u0+n/XW%J+“n—wflalremwwmn_on,an»

where A1 = A1(0, C) is a positive constant independent of w. Next we estimate an
upper bound for d(w). Let v(:z:) be the first eigenvalue of operator —A + m, and
v5(:c) = 6v(:v). We can find 6 = 6(w) so that

Kw(v5) = 62 / (IVvI2 + (m —w2) IvI2) + 6/vf(:t,6v) = 0,

(). —w2) / v2 = —§ / view),

which implies from Remark 3.1 that

01'

6 = 6(w) ——> 0 as w2 —> A1. (3.57)

Using (f3) and alternative expression for d(w) we get

2

d(w) 3.1.4621): 5; / (IVvI2 + (m — w?) Ivlz)

/\1—-UJ2

2

 

.r/pV243amrornfi) (3%)

where A2(0, v) is a positive constant independent of w. Combining (3.56) and (3.58)
gives

A1
0 —<(52
<A2_,

66

a contradiction to (3.57), and the theorem is proved.
The second application we consider is for the case 9: IR" with n > 2. We

investigate the stability of standing waves for NLKG 1n the following form.

— Au + u — Iqu_1 u = 0 in IR" >< IR, (3.59)
which corresponds to m($) E 1 and f(cc,u) = — Iqu_1u, and for NLS 1n the form
to. — Au — Iqu—1 u = 0 in 13” x R, (3.60)
which corresponds to m(:z:) E 0 and f($,u) : — Iu|p_1 u

Theorem 3.7 The NLKG (3.59) and NLS (3.60) have orbitally standing waves for
1 < p < 1 + 4/n.

Proof. We consider the NLKG first. Due to the scaling property of the solutions in
R", we can find an explicit expression for d(w). Let gbw be the positive radial symmetric

solution satisfying

—Aqbw + (l — w2)gbw — 5,: 0.
Put v(a:) = (1/6)q$w(ac/fl), then

—652Av + (1 — w2)6v — 6%” = 0.

which is transformed to

—Av+v—v”=0

if we select 52 = 67"1 2 1 — w2. Therefore we have

 

(1(0) 2 JOfv) : é/(IV’UI2 + v2) — IJ-l—l— v7)+1
6—2: 2
= (fin 2|V¢o(x/fi)l +453, (ac/m) 2:le p+1( (')/fl

   

: [611— 26— 2Jw(¢w):fln_ 26— 2d(£0),

which implies that
d(w) = 6262—"d(0) = (1 — w2)ad(0),

67

where
4+(p—1)(2—n)
2(1) - 1) '

 

a:

Taking the second derivative we find
d"(w) : 2a [—1 + (2a — 1)w2] (1 — w2)°’-2d(0),

which shows that if (.122 < 1, then

 

{w|d"(w)>0}={w|0< <w2<1}.

2a—1

The set in right hand side is nonempty if 1 < p < 1 + 4/n.
As for NLS

and

d"(w) = d(a — 1)(—<.o)c'"2

which is positive for all w < 0 and 1 < p < 1+ 4/n.

Remark 3.3 o The same stability result for standing waves of NLKG {3.59) was
obtained by Shatah in [43], but the approach in [43] can not handle the case

n = 1 or n = 2 due to the usage of a diflerent functional Kw which is not well

defined for n = 1 and n 2 2.

0 Similar orbital stability result for standing waves of NLS (3.60) was obtained by
Cazenave and Lions in [.9], their method is very different from the one developed
here. In [9], the frequency w can not be prescribed to find corresponding ground

state. Instead, they solve minimization problem

1
inf(/ |Vu|2 — 73—; / Iulr‘ | Hull. = u u e H‘(R")}

where ,a > O to find a ground state, then use Lagrange multiplier to find corre-

sponding frequency.

1 II-

Chapter 4

Finite Time Blow Up for Nonlinear

Klein-Gordon Equation

4. 1 Introduction

In this chapter, 9 is R" or a bounded domain in R" or n—dimensional compact man—

ifold. We will only consider the following nonlinear Klein-Gordon equation:

utt — Au + m(:1:)u + f(x,u) : 0 in R+ x Q,
u=0 on 30 if 80750, (4-1)
u(0,:z:) = U(:z:), ut(0,:z:) = V(.r),

where condition on m(:c) is the same as defined in last chapter, i.e., m(.r) is a real
bounded function and if Q 2 IR”, m(.r) is assumed to be radially symmetric, i.e.,
m(:r) = m(|.r|). We also assume that the lower bound A1 of spectrum of operator
—A + m is positive. We still assume f(x,u) = g(x, |u|)u. We prove that for a class
of nonlinearity f the steady states of the least energy, i.e., ground states are unstable
in a very strong sense: there is a region on boundary of which ground states lie,
such that that every solution of NLKG (4.1) starting from this region will blow up
in finite time. This type of instability basically means the nonexistence of global
solutions for some initial data and some nonlinearities. Keller’s work [27] represents

one of the earliest results in this direction. Since then, a number of authors (e.g.,

68

69

Berestyski and Cazenave [2], John [11], Glassey [12], [13], Levein [29], Payne and
Sattinger [36], Sternberg [48] and Tsutsumi [54]) have investigated the conditions on
which the solutions of (4.1) will blow-up in finite time. The second kind of instability
is that the solution of (4.1) starting near ground state may exist globally but will
approach infinity in L2 norm as time t approaches infinity. Shatah’s work [43] and
Keller’s work [25] represent the work in this direction. The third type of instability
is that the solution of (4.1) may exist globally and may not approach infinity as time
t approaches infinity, nevertheless the ground state is unstable. Shatah and Strauss’
paper [45] and Keller’s paper [25] represent the work in this direction.

In this chapter, we shall deal with first type of instability. We first prove the
existence of the ground state, then establish more properties of weak solutions. Finally
we prove that every solution of N LKG (4.1) starting from some region with ground

state on its boundary will blow up in finite time.

4.2 Steady State and Weak Solution

In this section, we prove the existence of steady state of the least energy, i.e, ground
state, and establish more properties of weak solutions of N LKG (4.1). For the ex—
istence of ground state, we mainly state the relevant results since the conditions
imposed on nonlinearity f in this chapter will be the same as in those in last chapter
except that we do not assume that fu < 0 which was used to prove the continuity
of d(w), therefore the proofs for some related lemmas and theorems(for instance, the
theorem of the existence of ground states) in this chapter are also similar and will be
omitted. Now let us go to the existence of ground state. The steady state of NLKG

(4.1) satisfies the nonlinear elliptic equation

-A<t+ m(-7?)</> + f(rvat) = 0,
¢=0 on an if 852M.

(4.2)

We know that every solution u E H of (4.2) is a critical point of potential energy

functional

J(v)=-1,-/ (|er + mu) Ivlg) + / Fm),

 

 

70

where F(x, v) = F(x, |v]) = 0M f(x, s)ds. It is also easy to check that every solution
of (4.2) satisfies

I((v) s f (|er + m(:v) W) + / |v|f(:c, lvl) = 0.
Therefore every solution of (4.2) lies on hypersurface
M:{v€H|K(v)=0andv#0}.

To find the steady state of the least energy, it is natural to turn to minimization

problem

d = 3355 Jon). (4.3)

We make following assumptions on f to ensure J to attain its minimum.

(H1) f E C1(fl X R) and f,’,(:v,s), f(cc,s) —> 0 uniformly in Q as s —> 0.

(H2) There exist constants 2 < l < 2n/(n — 1) and C such that If,’,(:1:,s)| _<_ 03”2

for large s > O and for all a: in 9.

(H3) There exists a constant 0 > 1 such that sf,’,(:c,s) g 0f(:c,s) for all cc 6 fl and

all s > 0.

Note that the following functions satisfy the above assumptions (H1)—(H3).

f(rc,s) 2 —sp with 1 < p <1+4/(n — 2),

f(x,s) = —sp+sq with 1 < q<p<1+4/(n—2),

Introduce the I functional and the region M— in H

I(v) = ~2—(06——;-11—)/(|Vv]2+m(33)]v]2) +/(F(a:,v)—— 0:1 |v|f(:v,|v|)) ,
M— ={v€H|K(v)SOandv7$0}.

 

Lemma 4.1 M is a Cl-hypersurface in H, and both M and [W— are bounded away

from zero.

71

Lemma 4.2

d: ' fl = ' f I .
vlélM (’U) vélli/I- ('0)

Theorem 4.1 Let f satisfy conditions (H1), (H2) and (H3), then
1. d is positive;

2. Every minimizing sequence of problem {4.3) posses a convergent subsequence.

In particular, d is attained at some 45;
3. This minimizer qt can be chosen positive;

4. Every minimizer of problem (4.3) is a solution of Equation (4.2) and is called
the ground state.

We define the set of ground states

Next we present some properties about the weak solutions of N LKG (4.1) (see [36],
[49] and [54] for details). Suppose that u is a weak solution of (4.1) on Q X [0,T),
then the following statements hold

(1) u(t) is weakly continuous from [0, T) to H, ut(t) is weakly continuous from [0, T)

to L2. So ||u(t)||H and ||u,(t)||2 are bounded on compact subsets of [0,T).

(2) There exists a weakly continuous mapping from [0, T) to L2 denoted by u,, such

that
(W) :3 = / (utudt,

for 0 3 t1 3 t2 and (b E L2 where (u,v) = I

Q u'vda: is standard inner product on

complex valued L2(Q).

(3) For any 2b : [0, T) —> H satisfying (1) and (2), we have

(Unit) if =/2[(ui,¢i)-(Vu,V¢)—(m(')ua¢)—(f(HULI/uldt (4-4)

11

72

It follows from (2) that (u(t), qt) is Lipschitz continuous for any gt in L2, i.e., u(t) is

weakly Lipschitz continuous in t. In (4.4), if we put it = u, we have

(utvu) [if : [2 [llutllg _ “Valli _ (m()uvu) _ (f('7u)7u)] dt' (4'5)

t1

Let u(t) be weak solution of NLKG (4.1), define
N(t) = (u(t),u(t)).
We follow the approaches in [36] to get some smoothness of N (t)

Lemma 4.3 N”(t) exists a.e. in [0, T), and N'(t) is Lipschitz continuous there.

Proof. Let Q(t, s) = (u(t), u(s)). Since u(t) is weakly absolutely continuous(by (2)),

and ut(t) is weakly continuous

N’(t) = (gt-Q(t,s) + %Q(t,s)) [gt 2 2Re(ut,u). (4.6)

From special form of our nonlinearity f(rc, u) = g(x, |u|)u and (4.5) we conclude that

(ut,u) is real, so (4.6) and (4.5) yield

N’(t2) — Nu) = 2 f '2 [llutlli — “Wu: — (m(')uau) — (f<-,u>.u)] dt (4.7)

1!1
for all 0 g t1 < t2 < T. Since each term in the integrand is bounded on compact
subsets of [0, T), we see that N’ is Lipschitz continuous on such sets. Therefore, N”

exists a.e. in [0, T), and
N"(“) = 2 [Hutlli — ”Vii“: — (mf'fitau) — (f(nulaull- (4-8)

This concludes the proof for the lemma.

4.3 Finite Time Blow Up

In this section, with preparation made in last section, we shall prove the finite time

blow up of solutions starting from a region that has ground states on its boundary.

73

First we define total energy(E) as the sum of potential energy(.](u)) and kinetic

2
energfltllvllz)

l
E(u,v) = 1(a) + ,Ilvni.
For weak solutions u(t) of NLKG (4.1), we have energy inequality
E(u(t1),ut(t1)) Z E(u(t2),ut(t2)) V 0 S t1 S t2.
Next define the NLKG solution—invariant sets

R1 = {(u,v)E R I I((u) > 0}U{(0,v) E R},
R2 = {(u,v) E R I I((u) <0},

where
R: {(u,v) E HEEL2 I E(u,v) < d}.
By definition of d, K and I, R1 and R2 have the following alternative definitions

R1={(u,v)E R I I(u) < d}U{(0,v) E R},
.82 = {(u,v) E R I I(u) > d},

Lemma 4.4 R1 and R2 are invariant regions under the solution flow of NLKG (4.1).

Proof. The proof is similar to the proof of similar invariance lemma in last chapter,
so we skip it.

Now we are ready to present our main result.

Theorem 4.2 Let u be a weak solution of Equation (4.1) with initial data (U, V) E
R2, let [0,T) be the existence interval. Then T must be finite.

Proof. Suppose that T 2 00. Note I((u) = (0 + 1) (J(u) — I(u)). From invariance
of R2 and energy inequality, we have for all 0 < t < oo
I((u) g (0 +1)(E(u, ut) — d) (4.9)
3' (6+1)(E(U,V)—d) :—: —€<0. (4.10)

74

Let N (t) = (u(t), u(t)), recall from last section that
N"(t) = 2 [Ilutlli — MW“: -— (m(-)u,u) — (f(-.u),u)] (4.11)
Therefore by (4.10) and (4.11) we have
NW) = 2lll’utll2 — KM] Z 26 > 0

which implies that N' is strictly increasing for t E [0, 00), and there exists some t1 2 0
such that
N’(t) 2 N’(t1) > 0 V t 2 t1 (4.12)

From (H1) and (H3) we get
(6 +1)/F(x.u) 2 (Jr-w)u).
Energy inequality implies that
/ F(w) 3 EU]. V) — g1,- (Wan: + (mum) + Halli)-
Hence we arrive at

—(f(~.u),u) 2 —(0 +1)/F(:c,u)

> 9L1
— 2
which yields by (4.11)

(”Willi + (mum) + llutlli) — (9 +1)E(U,V),

Mi) 2 (0 + 3)llut||§ + (0 ~1))1I)uni— 2(6 +1)E(U.V).

Note that N(t) is strictly increasing for t > t1 from (4.12), therefore there exists a
t2 2 t1 such that
N"(t) > (0 + 3))Imll3.

Hence for t > t2, we have that

NN”—- fig
4

(W > (o + 3)||u|l§||ut||§ — (0 + 3)(u,..,)2 2 0,

which leads to

II CY

(N‘O’) Z—NC'” NN”—(oz—I—1)(N’)2 <0fort>t2,

 

75

where a = («9 — 1)/ 4 > 0, and so M ‘0‘ is concave and decreasing for t 2 t2. Therefore
there exists a To such that N"" ——> 0 as t —> To which implies N —> 00 as t ——> To.
Thus we get contradiction to T : 00.

Finally we need to show that R2 is not empty. Choose initial data U (:c) = rqb(:z:)
with (t E S, and choose V(a') E 0. Then E(U,V) = J(U) = J(rgb) and I((U) =
I((rqS). It suffices to find To so that

E(U, V) < d and I((U) < O.

The following arguments complete our proof. Let h(r) = J (rd) and g(r) 2 I((rqfi).
Then

12(1) = M) = d. 9(1) = 1(a) = o.
and by (H3) and 1<(¢) = 0 we have
9(1) = / (IV¢12+ mu) W + W m. |¢|))
: (a — 1) f lemma)
=(1— «9) / (IWV + m(rv) W) < 0,

and

h'(r) = %g(r) and h"(1) = g'(1) < 0.

Remark 4.1 Interesting enough, if a weak solution u starts from invariance region

R1, then the solution exists globally. To see this, energy inequality implies that

 

Ewe). mm) = 311ml); + I(u) +

2 I((u) S E(U, V),

l
0 + 1
and since R1 is invariant, then I((u(t)) > 0 for all t. Therefore

1
EHalli + I(u) s E(U,V).

So IIuIII2 and IIuIIH are uniformly bounded, and this implies the global existence of the

solution.

List of References

List of References

1. V. Benci and D. Fortunato, The Dual Method in Critical Point Theory, Multiplic-
ity Results for Indefinite Functionals, Ann. Mat. Pura Appl. 132, 215-242(1982).

2. H. Berestycki and T. Cazenave, Instabilité des états stationaries dans les

e’quations de Schrodinger et des Klein-Cordon nonline’aries C. R. Acad. Sc.

293, 489-492( 1981).

3. H. Brezis, J. Coron and L. Nirenberg, Free Vibrations for a Nonlinear Wave
Equation and a Theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33, 667-
698(1980).

4. H. Brezis and E. Lieb, A Relation between Pointwise Convergence of Functions

and Convergence of Functionals, Proceedings of American Mathematical Society

88(3), 486—490(1983).

5. H. Brezis and L. Nirenberg, Positive Solutions of Nonlinear Elliptic Equations
Involving Critical Sobolev Exponent Comm. Pure Appl. Math. 36, 437-477(1983).

6. H. Brezis and L. Nirenberg, Forced Vibrations for a Nonlinear Wave Equation,

Comm. Pure. Appl. Math. 31, 1—30(1978).

7. H. Brezis and L. Nirenberg, Characterizations of the Ranges of Some Nonlinear
Operators and Applications to Boundary Value Problems, Ann. Sc. Norm. Pisa
5, 255—326(1978).

76

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

77

T. Cazenave, Stable Solutions of the Logarithmic Schrodinger Equation, Nonlin-
ear Analysis, Theory, & Application 7, 1127—1140(1983).

. T. Cazenave, and P. L. Lions, Orbital Stability of Standing Waves for Some

Nonlinear Schrodinger Equations Comm. Math. Phys. 85, 549—561(1982).

T. Cazenave and P. L. Lions, Existence d’one Solitaires dans les Problems Non-

line’aires du Type Klein—Gordon, Arch. Rational Mech. Anal. 82, 316—338(1983).

F. John, Blow—up of Solutions of Nonlinear Wave Equations in Three Dimension

Manuscripta Math. 28, 235-268(1979).

R. T. Glassey, Blow-up Theorems for Nonlinear Wave Equations Math. Z. 132,
182-203(1973).

R. T. Glassey, Finite-time Blow-up for Solutions of Nonlinear Wave Equations,
Math. Z. 177, 323-340(1981).

M. Grillakis, Linearized Instability for Nonlinear Schrodinger and Klein-Cordon
Equations, Comm. Pure Appl. Math. XLI 747—774(1988).

J. Ginibre and G. Velo, The Global Cauchy Problem for the Nonlinear Klein-
Cordon Equation, Math. Z. 189, 487—505(1985).

J. Ginibre and G. Velo, The Global Cauchy Problem for the Nonlinear Klein-
Cordon Equation - II, Ann. Inst. Henri Poincaré —Analyse non linéarie 6(1),

1535(1939).

J. Ginibre and G. Velo, The Global Cauchy Problem for The Nonlinear
Schrodinger Equation, Ann. Inst. Henri Poincaré —Ana1yse non linéarie 2(4),

309-327(1935).

M. Grillakis, J. Shatah and W. Strauss, Stability Theory of Solitary lVaves in
the Presence Of Symmetry, I, J. Funct. Analysis 74(1), 160—197(1987).

M. Grillakis, J. Shatah, and W. Strauss, Stability Theory of Solitary Waves in
the Presence of Symmetri, II, J. Funct. Analysis 94(2), 308-348(l990).

20

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

78

D. Hartree, The Wave Mechanics of an Atom with a Non-Coulomb Central Field,
Part 1, Theory and Methods, Proc. Camb. Philos. Soc.24, 89-132(1968).

D. Jerison, C. D. Sogge and Z. Zhou, Sobolev Estimates for the Wave Operator
on Compact Manifolds, Comm. P.D.E. 17, 1867—1887(1992).

C. K. R. T. Jones, An Instability Mechanism for Radially Symmetric Waves of
a Nonlinear Schrodinger Equations, J. of Diff. Eq. 71, 34-62(1988).

L. V. Kapitanskii, Cauchy Problem for a Semilinear Wave Equation, II, J. of
Soviet Math. 623, 2746—2777(1992).

P. L. Kelly, Self-focusing of Optical Beams, Phys. Rev. Lett. 15, 1005-1008(1965).

C. Keller, Stable and Unstable Manifolds for the Nonlinear Wave Equations with
Dissipation, J. of Diff. Eq. 50, 330—347(1983).

C. Keller, Large-Time Asymptotetic Behavior of Solutions of Nonlinear Wave
Equations Perturbed from a Stationary Ground State, Comm. P. D. E. 8, 1073-
1099(1983).

J. B. Keller, On Solutions of Nonlinear Wave Equations, Comm. Pure Appl.
Math. 10, 523—530(1957).

T. D. Lee, Particle Physics and Introduction to Field Theory, New York, Har—
wood Academic Publishers 1981.

H. Levine, Instability and Nonexistence of Global Solutions to Nonlinear 'Wave
Equations of the Form Putt 2 —Au +F(u), Trans. Amer. Mathematical Society,
192, 1—21(1974).

E. H. Lieb, Existence and Uniqueness of the Minimizing Solution of Choquard’s
Nonlinear Equation, Stud. Appl. Math. 57, 93—105(1977).

P. L. Lions, Principe de Concentration-Compacité en Calcul des Variatons C.

R. Acad. Sc. Paris, 294, 261-264(1982).

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

79

J. L. Lopes, Gauge Field Theori, An Introduction, Pergamon Press, Oxford,
1981.

H. Lovicarova, Periodic Solutions of a Weakly Nonlinear Wave Equation in One

Dimension, Czech. Math. J. 19, 324—342(1969).

B. Marshal, W. Strauss and S. Wainger, LP - L" Estimates for the Klein-Gordon
Equations, J. Math. Pures Appl. 59, 417—440(1980).

C. Miiller, Spherical Harmonics, Lecture Notes in Mathematics No. 17, Springer—
Verlag, 1966.

L. E. Payne, and D. H. Sattinger, Saddle Points and Instability of Nonlinear
Hyperbolic Equations, Isreal J. Math. 22, 273—303(1975).

P. Rabinowitz, Free Vibrations for a Semilinear Wave Equation, Comm. Pure

Appl. Math. 31, 31-68(1978).

P. Rabinowitz, Periodic Solutions of Nonlinear Hyperbolic Partial Difierential
Equations, Comm. Pure Appl. Math. 20, 145-205(1967).

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to
Difierential Equations, AMS, Providence, Rhode Island, 1983.

P. Rabinowitz, Variational Methods for Nonlinear Elliptic Eigenvalue Problems,
Indiana Univ. Math. J.23, 729-754(1974).

D. H. Sattinger, On Global Solutions of Nonlinear Hyperbolic Equations, Arch.
Rat. Mech. Anal., 30, 148—172(1968).

D. Sattinger, Stability of Nonlinear Hyperbolic Equations, Arch. Rational Mech.
Anal. 28, 226-244(1968).

J. Shatah, Unstable Ground State of Klein-Gordon Equations, Trans. Of Ameri-
can Mathematical Soc., 290(2), Aug. 1985.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

80

J. Shatah, Stable Standing Waves of Nonlinear Klein-Gordon Equations Comm.
Math. Phys. 91, 313-327(1983).

J. Shatah and W. Strauss, Instability of Nonlinear Bound States, Comm. Math.
Phys. 100, 173-190(1985).

C. D. Sogge, Oscillatory Integrals and Spherical Harmonics, Duke Math. J.
53(1), 43—65(1986).

P. E. Souganidis, and W. A. Strauss, Instability ofa Class of Dissipersive Solitary
Waves, Proceedings Of the Royal Society Of Edinbergh, 114A, 195—212( 1990).

N. Sternberg, Blow Up near Higher Modes of Nonlinear Wave Equations, Trans.
of Amer. Mathematical Soc., 296(1), (315-325)1986.

W. Strauss, On Weak Solutions of Semi-linear Hyperbolic Equations An. Acad.
Brasil. Ciénc., 42, 645-651(1970).

W. Strauss, Existence of Solitary Waves in Higher Dimensions, Comm Math
Phys. 55, 149—162(1977).

W. Strauss, Stability of Solitary Waves, Contemporary Math. 107, 123-
129(1990).

S. R. Strichartz, Restrictions of Fourier Transforms to Quadratic Surfaces and

Decay of Solutions of Wave Equation, Duke Math. J. 44, 705-714(1977).

main B. R. Suydam, Self-focusing of Very Powerful Laser Beams, US. Dept. of
Commerce N. B. S. Special Publication 387.

M. Tsutsumi, On Solutions of Semilinear Diflerenlial Equations in a Hilbert
Space, Math. Japan 17, 173—193(1972).

M. Weinstein, Lyapunov Stability of Ground States of Nonlinear Dispersive Evo-
lution Equations, Comm Pure Appl. Math. 39, 51—68(l986).

 

81

56. Z. Zhou, The Existence of Periodic Solutions of Nonlinear Wave Equations on
S”, Comm. P. D. E. 12, 829-882(1987).

57. Z. Zhou, The Wave Operator on S", Estimates and Applications, J. Funct. Anal.
80, 332—348(1988).

 

 

 

LIBRQRIES

HICHIGQN sraTE UNIV

I
I
1 I
I

IIIIIIIIII III
5553088

IIII

IIIIII IIIII
3129301

IIIIII

II

[III

II