. :. V , ‘1. Any! 1.4%... .m. .4 pig): . é . . .1 . tn . :3. 4 . mws .. , . .- . . 1.0... 9,. brawn.“ . w: a}? «(a ..V. 3.3. ‘ ll a. Q 14v. 1:; ‘?01)t r. .3. it! n w $4. wammfiafiliafl. wins 5! l MDU‘ a .1006 I Inn A. DV University This is to certify that the dissertation entitled Pattern Formation in Some Nonlinear Systems presented by Peng Feng has been accepted towards fulfillment of the requirements for the PhD. degree in Applied Mathematics Sit/”39% A77 Major Profeslsor’s Signature 5/2 9/05” Date MSU is an Affirmative ActiorVEquaI Opportunity Institution -._._.-.-.--.4_--—-.-,-.-.-.- -.-.—.-l_.-t-.- PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 cj'ClRC/DateDuelnddpJS Pattern Formation in Some Nonlinear Systems By Peng Feng A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2005 ABSTRACT Pattern Formation in Some Nonlinear Systems By Peng Fang We study the pattern formation arising in some nonlinear systems. The problems come from material science and biological science. We study the instability in the corresponding mathematical models and explain the pattern formation. In partic- ular, one mathematical model we studied is an elliptic equation and the other is a degenerate parabolic system. For the elliptic equation, we study the radial symmetric solutions and the bifurcation into nonradial ones. For the parabolic system, we study the existence of traveling wave solutions and the instability of the flat front. To my family. iii ACKNOWLEDGMENTS I am deeply indebted to my supervisor Prof. Zhengfang Zhou whose help, stimu- lating suggestions and encouragement helped me in all the time of research for and writing of this thesis. I would like to thank him for the uncountable number of hours spent sharing his knowledge and discussing various ideas. I would like to express my gratitude to all those who gave me the possibility to finish this thesis. Especially I am obliged to Professor P. Bates, B.S. Yan, P. Lamm, M. Fraizier, M.X. Tang. I would also like to thank those for all their help and valuable conversations: Professor Emmanuele DiBenedetto of Vanderbilt University, Professor Yi Li of University of Iowa, Professor A. Friedman of Ohio State University. Especially, I would like to give my special thanks to my wife Zhao whose patient love enabled me to complete this work. iv . - ir—uul— \,i.. ll 111111111 2«7.3).«7.3).ni.nl.nlsnl..9.9.94.9.«750... B TABLE OF CONTENTS LIST OF FIGURES Introduction 1 Pattern Formation in A Mechanical System 1.1 Introduction and Summary of result .................... 1.2 Formulation of the model .......................... 1.3 Existence ................................... 1.4 Multiplicity .................................. 1.5 Symmetry breaking .............................. 1.6 Finite Time Touchdown and Touchdown Profile .............. 2 Pattern Formation in A Biological System 2.1 Description of Biological Experiment .................... 2.2 General Mathematical Model ........................ 2.3 Existence and Large Time Behavior of Weak Solution ........... 2.4 Main result and methods ........................... 2.5 Numerical Simulation to 1D Problem .................... 2.6 Travelling Wave Solution ........................... 2.6.1 Properties of the traveling wave solutions. ................ 2.7 Existence of b for a given n ......................... 2.7.1 First-order singular boundary value problem ............... 2.7.2 Existence of a critical velocity ....................... 2.7.3 Finite traveling wave at v” ......................... 2.8 The Equivalent Problem ........................... 2.9 A fixed point ................................. 2.10 Instability of Flat Front ........................... BIBLIOGRAPHY vi 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 LIST OF FIGURES A typical pattern in bacterial colony .................... 3 A basic electrostatically actuated elastic membrane. The prime coordi- nates indicate they have not yet been scaled ............... 8 Timemap Diagram .............................. 25 Bifurcation Diagram ............................. 26 The change of sign as um crosses A axis at Ak ............... 36 A typical numerical simulation ....................... 42 Bacteria Density ............................... 50 Nutrient Density ............................... 50 Bacteria Density ............................... 51 Nutrient Density ............................... 51 Sketch of u .................................. 74 Sketch of n .................................. 84 An illustration of the dispersion relation for different D values ...... 92 vi Introduction The diversity of the natural shapes that surround us has a profound impact on the quality of our lives. For this reason alone, it is not surprising that the origins of these shapes have been the subject of serious study since antiquity. It has long been believed that a quantitative study of natural forms will help us understanding their origins and behaviors. Many physicists and mathematicians have devoted to developing general approaches toward the quantitative description of the complex patterns that are characteristic of most natural phenomena. Among these patterns, there are relatively simple ones that can arise in many simple and complex systems. For example, spiral and helix patterns are often found in sea shells. One of the most studied examples of spiral pattern-formation in reaction- diffusion systems is the Belousov-Zhabotinsky reaction. Labyrinthine patterns are also considered to be a relatively simple pattern. These maze-like patterns can be found in the well known Rayleigh-Béard convection experiments, in which a thin layer of fluid is heated from below. Another phenomenon that has been extensively studied is the viscous fingering in the well-known Hele-Shaw experiment in which a less viscous fluid is injected into a more viscous fluid in a Hele-Shaw cell. D’Archy Thompson was one of the first to attempt a mathematical description of pattern forming process in his work ”On growth and form” from 1917. Cross and Hohenberg provide an excellent technical introduction into the mathematical concepts of pattern formation. Below, we briefly describe the basic principles and Hill pizf pa lliif but mathematical tools. Pattern formation implies a change in symmetry. Spontaneous symmetry breaking is particularly evident in fluids, which are structureless in equilibrium but exhibit a surprisingly variety of patterns under nonequilibrium conditions. In this thesis, we first present a mathematical model that describes the deflection of an elastic membrane in an electromechanical system. The membrane is supported by an annular boundary. The deflection of the membrane satisfies the following parabolic equation Bu /\ . 5t—_Au+(l——u—)2 in Q, 11—0 on 69, “(Sciyiol—O- (1) Here 9 is an annulus. In this dissertation, we show that the equilibrium solution does not necessarily inherit the radial symmetry of the supporting boundary. Instead, nonradial solution may bifurcate from the radial ones. We also study the exact multiplicity of such radial solutions. We also address the finite time touch down behavior in such equation. These issues will be presented in Chapter 1. Pattern forming process can often be found in biological system. Recently, the pattern formed by a bacterial colony growing on a thin agar plate has attracted many mathematicians and biologists in an attempt to explain the intriguing patterns. A particular interesting pattern is shown in Fig. 1. Reaction-diffusion system has been often used as continuous model to describe patterns and waves arising in far from equilibrium states. In spite that the equa- tions look so simple, numerics has revealed that the equations generate complex but regulated spatio—temporal patterns. On the the typical system is the following Figure 1. A typical pattern in bacterial colony reaction-diffusion system for two components u and v: at = duAu + M)” — au — bu" (2) v, = duAv — uv'" — a(1 — v) (3) where a and b are positive constants. In particular, when m = 2 and n = 1, it is called Gray-Scott equations. For m = 2, it is known that the system generates diverse complex patterns in high dimensions depending on the values of m and n. One typical phenomenon is the occurrence of spot patterns which are generated through self-replicating process. On the other hand, when a = 0, (2)-(3) reduce to the following closed system u, = duAu + no“ — bu", (4) v, = dvAv—uv'". (5) In a bounded domain with the Neumann boundary conditions, it is proved that any solution (u,v) becomes spatially homogeneous asymptotically, that is, there occurs no pattern formation at all. Therefore, it has long been believed that such systems are not so interesting from pattern formation viewpoints. Recently, Mimura et al found that such system generate complex patterns in transient process, although it generates no pattern asymptotically. While diffusion alone tends to create uniform states, in 1952, Alan Thring sug- gested that diffusion coupled with chemical reaction may lead to spatial patterns in chemical composition. He speculated that such mechanisms would be sufficient to explain patterns such as zebra stripes. Several similar systems have been proposed by different authors to describe the growth of bacterial colony on a thin plate. The systems have the following general form ’3‘: 2 v.{Db(b,n)Vb}+g(b,n), (5) an 2 '5? = DnVn—f(b,n), (7) where D, and Dn are the diffusion coefficients of the bacteria and the nutrient, re- spectively. Here b and n represent the density of the bacteria and nutrient. The numerics showed that the system captures many patterns successfully. Another purpose of this dissertation is to present a method to understand how patterns can arise in such a system. We study the traveling wave solutionsto this system and the instability of the wave front. The results will be presented in Chapter 2. 51in; CHAPTER 1 Pattern Formation in A Mechanical System 1.1 Introduction and Summary of result In 1968, at the age of 82, in the context of investigating fundamental questions in electrohydrodynamics, Geoffrey Ingram Taylor 3‘ one of the great physicists of the twentieth century, studied the electrostatic deflection of elastic membranes. Tay- lor’s device is a soap film as the membrane material. He then applied a fixed high voltage potential difference between two supported circular membrane. He showed experimentally that the two membranes snap together and touch at a critical volt- age. At smaller voltage, even though the membranes remained separate, they either became unstable or failed to exist. This instability is now known to researchers in microelectromechanical system (MEMS) and nanoelectromechanical systems (NEMS) Although many researchers are familiar with one or another of Taylors contributions, few seem to be aware of the incredible breadth of his scientific publications and their relevance to important research questions today. The same person who is commonly remembered as the namesake for several basic fluid flow instabilities (TaylorCouette, RayleighTaylor, and SaffmanTaylor) also was the first to show experimentally that a diffraction pattern produced by shining light on a needle does not change when the intensity of light is decreased. And these topics are only the beginning. Taylor made fundamental contributions to turbulence, championing the need for developing a statistical theory, and performing the first measurements of the effective diffusivity and viscosity of the atmosphere. «i (it- fields as the pull-in instability. The use of electrostatic forces to provide actuation is now a method of central importance in MEMS and is of growing importance in NEMS. Here, we study the electrostatic deflection of an annular elastic membrane. We investigate whether elec- trostatically deflected membranes always inherit the radial symmetry of the mem- brane’s domain. In the recent work of Pelesko, Bernstein and McCuan [19], they derived a mathe- matical model for the deflection of the membrane at the equilibrium state. The model is essentially a semilinear elliptic equation with the corresponding boundary condi— tions. This model incorporated the voltage as a bifurcation parameter. They showed that asymmetric solutions exist through numerical investigation. A bifurcation dia- gram was obtained. They conjectured that there are an infinite number of branches intersecting the upper radially symmetric solution branch. However, they were un- able to obtain the complete bifurcation diagram numerically where the equation is very close to become singular. Motivated by their work, we shall study this problem theoretically and we will obtain the complete bifurcation diagram and prove their conjecture that there are indeed infinitely many symmetry breaking point at the upper branch of the radial solutions. The problem we will study is the following semilinear elliptic equation with Dirich- let boundary condition: /\ —A’U, = W in 9, (1.1) u = O on 80 (1.2) where {I = {2: E R2 : 51 < |:1:| < 1} is an annulus in R2, u is the displacement of the membrane. A serves as a bifurcation parameter and its exact meaning will be shown in the next section. 1.2' u;- l iifd' I an} We investigate the exact number of positive radial solutions and non-radially symmetric bifurcation of the above problem. The exact number of positive radial solutions may be 2, 1, or 0 depending on the value of A. The upper branch of radial solutions has non-radially symmetric bifurcation at infinitely many A N E (O, A‘). The proof of multiplicity result relies on the characterization of the shape of the time- map and shooting method. The proof of bifurcation result relies on a well known bifurcation theorem by Kielhbfer [11]. In fact, we can prove the following results: Theorem 1.1 There exists a A“ such that the problem has no positive radial solution for A > A', one radial solution for A = A“ and exactly two radial solutions for 0 1 — u. The result is the following semi-linear elliptic equation for the displacement u: in Q, (1.16) u=0 on 69. (1.17) 1 .3 Existence In this section, we shall study the following semilinear elliptic equation with Dirichlet boundary condition. A —Au=(I——T)2 on Q, (1.18) u=0 on 80. (1.19) First we show that there is no solution to our problem when A is sufficiently large. That is, the membrane fails to exist when the voltage is sufficiently large. We prove Theorem 1.3 There exists a A" such that when A > A“ there is no solution to equa- 11 tion (1.1) and equation (1.2). Proof. Let A1 be the lowest eigenvalue of —Au 2 Au on Q, (1.20) u=0 on on, (1.21) with ul is the corresponding eigenfunction which can be chosen strictly positive on Q. Multiplying equation (1.1) by ul and integrating gives fn—uAul — AL ———(1 _ ”)2. AIAUUI—A12—(l—uy. Since ——1— > 2—7u, we have (l—u)2—4 11.1 27 A/uuzA/——>—/uu. 1n 1 0(1—U)2_4 01 Hence, A 3 ii“ Remark. Any smooth solution u must be nonnegative by the maximum principle. Next we shall obtain the existence result for small A. We have the following theorem. Theorem 1.4 There exists a solution to equation (1.1) and equation (1.2) for some Small A. 12 7.0 (hr Pr To prove this theorem, we should apply the well-known method of upper and lower solutions [18]. Let us recall the following definition: Definition 1.1 A function a E 02(0) is called an upper solution of equation (1.1) and equation {1.2) if it satisfies the inequalities —A’L—l 2 m OD Q, (1.22) 1120 on 89. (1.23) Similarly, y is called a lower solution if it satisfies all the reversed inequalities. The following two lemmas provide us with a proper choice of lower and upper solu- tions. Lemma 1.1 Any constant c < 0 is a lower solution. Lemma 1.2 y = i—vl is an upper solution when A S fialm. Here 01 and v1 is the first eigenvalue and eigenfunction for the following problem: —Av 2 av on (2', (1.24) v=0 on 00', (1.25) where Q’ is a proper domain with smooth boundary which contains (2 and has been chosen such that m 3 v1 3 1 on 9. Proof. It is sufficient to show that 13 In fact, §> A _ A 4 — (1 - 1/3v1)2 ‘ (1 —n)2' 1 1 1 A —Aa = —§Av1 = 5011711 _>_ 301mg 3- This completes the proof. Using Lemma 1.1 and 1.2 and the well-known theorem that a solution exists between an ordered pair of upper solution and lower solution, we obtain the existence result. 1 .4 Multiplicity In this section we are concerned with the multiplicity of positive radial solutions. Consider the following equations: Au+f(u) = 0 in Q, (1.26) it = 0 on 09, (1.27) where Q is either a radial or annular domain in R" and f is a strictly convex C2 function on [0, 00). According to the well-known result of Gidas, Ni and Nirenberg [7] every positive solution of the above equations is radially symmetric if the domain is a ball, i.e., it’s only a function of r 2 |x|. The number of positive solutions has been widely studied for different types of f on a general bounded domain using variational and topological methods, see e. g., [1, 2]. The problem is both fundamental and often difficult. During the last decade, there has been tremendous progress in studying these problems when (I is a ball or entire R", see, e. g., [16, 17]. If the domain is a ball, then ODE techniques can be applied to get more information on the number of solutions [23]. Similarly, if the domain is a ball, we can also derive the exact multiplicity of 14 radial solutions [8, 12, 13, 14, 25]. A radial solution has the form it = u(r) where r = [x] = (x? + . . . + 27%)”? First notefori=1,...,n, 87‘ 1 r 8113 : §($i + + mill/22$“ - _— We thus have If; n $12 $12 ux. : ”’02—: Utah U. (T)? + u,( ) (T _ :5) Consequently, I! — 1 Au = u (r) + n T WT) Hence the radial solution of equation (1.1) and equation (1.2) satisfies the following equations ”(r) + lu’(r) + A — 0 in (e 1) (128) u r (1 — u)2 — 1’ ’ ' u(el) = u(1) = 0. (1.29) Let s = — 1n r, w(s) = u(r), then w(s) satisfies w” + Ae’23——1—— = 0 in (0 — In 61) (1.30) (1 _ w)2 ’ 1 10(0) = w(—ln 61) = 0. ‘ (1.31) Henceforth, we shall consider the following initial value problem ’U,”(T) + Ae—2rm : O in (0,-ln61), (1.32) u(0) = 0 and u'(0) = p. (1.33) 15 Definition 1.2 Let u(-) = u(-,p,A) be the solution of equations (1.32) and (1.33) and we define the time-map associated to the above initial value problem to be the following function R: R(p, A) = min{R > 0 : u(R, p, A) = 0}. We shall prove in the next lemma that R(p, A) is well defined for all p. By the boundary condition, it has exactly one critical point, at which it takes the maximum value. We shall denote this critical point by r(p, A). Hence u'(r) > 0 for r E (0,r(p,A)) and u'(r) < 0 for r E (r(p,A),R(p,A)). Also note that u satisfies the following integral equation u(r) 2 pr + AA (3 — r)e"23Z1—_;(8—))2ds. To prove our multiplicity result, we need to establish several useful lemmas. Lemma 1.3 R(p, A) is well defined. Proof. First we claim that it is indeed well defined for p sufficiently small or large. Suppose otherwise that limHJr00 u’(r) = 0. Multiplying equation (1.32) by u’ and integrating gives Hence 2 _ 1—u( 16 lo I] Let h(r) = or lf—des, then we have 1 1 . Ah'(r) + 2Ah(r) — 5p2 + 5'u'(r)‘2 — A = 0. When r is sufficiently large, h(r) = [lilhfls = —§/;u"(s)(1—u(s))ds = l [—(1 — u)u' +p —/ u'2(s)ds] A o P < —. 1.34 _ , < ) Hence for sufficiently large r, , 1 2 1 , 2 Ah (r) = -2Ah(r) + fip — 2U (r) + A 1 1 > _ _ 2 _ _ I 2 _ 2p + 2p 2u (r) + A 2 c > 0 (1.35) for some constant c and p either sufficiently large or small. Therefore, e—2r 1— u(r) c >- 0 "A> for sufficiently large r. It follows that saw = I 17 for p sufficiently large or small. Applying L’Hopital’s rule we have —‘2r 2 i—2r _4 _—2r lim h'(r) = lim e = lim e = lim e r—>+oo r—>+oo 1 -— u(r) r—>+oo u’(7‘) r—>+oo u”('r) , . 1 — u(r) , : l 4 —2T___ : — -’, ‘ z . _ r—iE-noo e 6"” r—liil-nooéu1 (1(7)) 0 (1 36) This is a contradiction to the previous conclusion that h’ (r) 2 c > 0. Hence R(p, A) is well defined for p sufficiently large and small. By continuous dependence on para- meters, R(p, A) is well defined for all p. This completes the proof. Lemma 1.4 lim R(p, A) = lim r(p, A) = 0. p—>0+ p—>0+ Proof. Suppose otherwise, there exists a A > 0, e > 0 and a sequence pk —-> 0+ such that RI: 5 RU?!“ A) 2 6- Since u(r,pk) = pkr + A/ (s — r)e'2’———1————ds o < pkr - —, (1.37) Thus RIC < 2% 18 Hence, 3,, 2 1 = —A — R 7 3—————d ”R" [0 (s ”e (1— u(sW S 2 A/ (e — s)e_25ds 0 > 0. (1.38) This is a contradiction. Hence limp_,0+ R(p, A) = 0. It follows that limp_,0+ r(p, A) = 0. This completes the proof. Lemma 1.5 lim R(p,A)= lim r(p, A) =0. p—) 00 p—i +00 Proof. Suppose limpnoo7-(p, A) 74 0, then there exists a To > 0 and a sequence Pk —> +00 With uk(r) E u(r,pk, A) > 0 and u;(r) > 0 in (0, To). Let 7" = To / 2, we claim lim supuk(7") = 1. k—i +00 Otherwise, there exists 6 > 0 such that 0 < uk(?) 3 1 — c. It follows that - — ”T fr—fe"2r———1 r Uk(7') — pk +A/(;( ) (I—u(r))2d 2 pk? + 32/ (r — 1")e‘2’dr (1.39) 0 which is impossible since pk -—> +00. Hence choosing a subsequence if necessary, we may assume lim uk('f) = 1. k-H-oo Note that uk satisfies u(r) = in (firo). Let 1 Mkzinf{/Uk :rE(f.T0)}, (1— Uk)2 then lim M), = 00. k—>+oo Note that Ae‘zr 2 A6470 in (fire). Let vk solves v"(r) + Ae‘2TOMkv(r) = 0 in (fire). It follows that vk has at least two zeros in (7", To) when k is sufficiently large. By Sturm Comparison Principle, u), has at least one zero in (f, To). But this is impossible. Hence lim r(p, A) = 0. p—H-oo Finally, we prove limp_,+oo R(p, A) = 0. Otherwise, there exists a point r0 > 0 and a sequence pk —> +00 with uk(r) > 0 and u],(r) g 0 in (rk,r0) where u,c E u(r,pk, A) and r), :- r(pk, A). Let 7" = 52“, in view of previous lemma that limp_,+oor(p, A) =: O, we may assume 1“ > Tk for any It. We claim that lim sup uk(f) < 1. k—>+oo Otherwise, by Sturm Comparison Principle again, u,c has zeros in (nu?) when k is sufficiently large which is impossible since 7",, ——> 0 as k —-> +00. 20 Note that ' Ae“2"‘ u'(r) = — / —,—ds, Tk (1 — “(8))2 and 1 , Ate-2r ’ 2Ae‘2' — ' = - . 1.40 (2" +1—u(r)) 1—u(r) ( ) Integrate equation (1.40) on (rt, r), we have 1 2 Ae"2f e42c /' 2Ae‘2s — ' ’ - — A—— — ——d . 2” (T) 1— u(I“) + 1— u(r).) n. 1— u(s) 8 On the other hand, we have _ < _— 2u(r) +/T,,1—u(3)ds _ u(r) +/ (1—u(s))2ds 5 ., 1 I — 2 I — S 2“ (r) +2|u(r)|. (1.41) Hence Ae—2f 8—27';‘ 1 —— A—<—"2 2". 1.42 1—u(f) + 1—u(T,.) - 2““) + Mr“ ( ) Integrate equation ( 1.40) on (0, rk), we have Ae‘m T" 2Ae’2’ 1 —— ——d = - 2 A. 1—u(rk) +/0 l—u(s) S 2pk+ Therefore, Ae"2"’= 1 1 > — — 2 A . 1.43 Combining equation (1.42) and (1.43), we have u'(7‘“) —> —oo. 21 Thus for r > r, we have uk(r0) < uk(f) + u;(f)(r0 — 7‘) —> —00, a contradiction to uk(r0) > 0. This completes the proof. Lemma 1.6 Define R = R(A) = sup{R(p, A),p > 0}. Then R(A) is strictly decreas- ing. Proof. Let 0 < A1 < A2 and U2 is a solution at A2 on (0, R(A2)). Let v(s) = cu2(r) with r = s/c where c is some constant greater but close to 1. It’s easy to see v(O) = 0 and v(R(A2) + e) = 0 for e = (c — 1)R(A2). Note that 6—23 1 e—2s II A = _ II A v + 1(1—v(s>)2 8‘2”“ ‘(l—cu2(r))2 1 e-2r 6—23 2 —— — < 1. coin-um)? A‘s—cushy)” ( 44) When c is sufficient close to 1. Hence v is a lower solution for e-2r v”(7") + Aim = 0, 12(0) 2 0, v(R(A2) + e) = 0. Hence R(Al) 2 R(Ag) + 6. Hence R(A) is strictly decreasing. This completes the proof. Lemma 1.7 lim,\_,0+ R(A) = +00, 11m“... R(A) = 0 Proof. Suppose lim,\_,o+ R(A) 79 +00, then there exists a number R“ > 0 such that ~ ~ a sequence Ak —+ 0+ with limk_,+00 R(Ak) = limk_,+oo R(Ak,pk) = R‘. Let us write 22 uk(r) = u(r, Ak,pk), then R. I z * 2 * /\ — * ~23 d.' 0 uk(R) ka + k/o (s R)e (1—uk(s))2 s A R‘ 2 ml?“ + :2’3/ (3 — R*)e_23ds. (1.45) 0 Hence pk —> 0+ or R" = 0. But this contradicts the facts that limp_,0+ R(p, A) = 0 and R(A) is strictly decreasing. This completes the proof. Finally for any given A, we study the shape of R(p). Note that R(p) is determined by the implicit equation v(R(p),p) = 0. (1.46) Differentiating equation (1.46) we get the following equations for the derivatives of R: ur(R(p),p)R’(p) + up(R(p),p) = 0, (1-47) tin-(Mp),1012(2))2 + 2urp(R(p), p)R'(p) + ur(R(p), p)R”(p) + upp(R(p), p) = 0- (1-48) We write h(r, p) = up(r, p), z(r, p) = upp(r, p) and v(r, p) = u,.(r, p), then equation (1.47) can be written as v(R(p),p)R'(p) + h(1301),!» = 0- (149) If R’(p) = 0, we conclude from equation (1.48) that v(R(p),p)R"(p) + 2(R(p),p) = 0- (1-50) Applying equation (1.50), we may prove the following important Lemma. Lemma 1.8 For a given A, if R’(p) = 0, then R”(p) < 0. 23 Proof. Note that h(r, p) satisfies the following initial problem 2Ae"2" h” + (1— ur h(r,p) = 0, (1.51) h(O, p) = 0, h’(0, p) = 1. (1.52) If R'(p) = 0, then equation (1.47) gives us h(R(p),p) = 0. We claim that h(r, p) > 0 on (0, R(p)). Otherwise let h(§(p),p) = 0 and h > 0 on (0, €(p)). Note that v satisfies the following ,, 2Ae‘2’" 2Ae‘2’ 0—103” (1 —u>2 = 0, (1.53) v(0,p) = p, v'(0,p) = -A. (1.54) Recall that v(r(p),p) = 0. If £(p) Z r(p), then v < 0 on (£(p),R(p)). By Sturm Comparison Theorem, v should have a zero on (6 (p), R(p)) since h(R(p), p) = 0. This is impossible. If {(13) < T09), then v < 0 0n (7(a),R(p))- Since 0 = v(r(p),p) > h(TlpLP), by Sturm Second Comparison Theorem, v > h on (r(p), R(p)) which is impossible since h has to cross over v and reaches zero at R(p). Next we claim z(R(p), p) < 0. Note that 2Ae"2' 6Ae‘2’" " ——h2 = 0 1.55 z +(1—u)3z+(1—u)4 , ( ) z(0,p) = 0, z'(0,p) = 0. (1.56) We claim z is negative in some neighborhood of 0. Otherwise by observing equation (1.55), we have 2” < 0. It follows that z’ < 0 in the neighborhood of0 since z’(0,p) = 0. This contradicts the assumption. 24 Figure 1.2. Timemap Diagram Next we claim 2 < 0 in (0, R(p)]. Otherwise, let z(r1, p) = 0 with z < 0 in (0, r1). Comparing equation (1.51) and equation (1.55), it follows that h must have a zero in (0, r1) which contradicts our previous statement. Hence z(R(p), p) < 0 and it follows from equation (1.50) that R” (p) < 0. We are now in position to prove Theorem 1.1. Proof. In view of the above lemmas, we may obtain the timemap diagram as shown in Figure 1.2. Horn which we can easily conclude the theorem. In fact, for any given cl > 0, 3 A“ such that R(A”) = —ln £1 and there is a unique q such that R(A‘, p) = — In 61, thus there exists a unique radial solution at A = A‘. For A < A‘, we can find p1, p2 such that R(A, p1) = R(A, p2) = — In 61. The problem has two radial solutions in this case. For A > A‘, since R(A) < — In 61, there is no radial solution. This result is shown in Figure 1.3. 25 //U//m Figure 1.3. Bifurcation Diagram 1 .5 Symmetry breaking In previous section, we studied the multiplicity of radial solutions. Our purpose in this section is to study how radial symmetry can be broken, that is, to describe the bifurcations of these radial solutions to non-radial solutions. The problem of non-radial bifurcation from radial solutions on balls were studied by Dancer [4] and Smoller and Wasserman [22, 24], on an annulus by Lin [15] and others. Let us first introduce a few basic concepts and the celebrated result by Crandall and Rabinowitz [3]. We will also illustrate the theorem by an example. Then we shall introduce a very useful bifurcation theorem due to Kielhbfer [11] which is what we applied in this thesis. For a summary on bifurcation theory, see for example [9]. We shall consider two real Banach Spaces, U and V, as well as a nonlinear abstract operator F: RxU—>V of the form F(A, u) = L(A)u + R(A, u) 26 and the associated nonlinear abstract equation F (A, u) = 0 where the following assumptions are assumed to be satisfied: 0 There exists A0 6 R and a,b E R, a < A0 < b, such that L(A) is a linear operator from U to V for all A E (a, b). Moreover, 3 r 2 2 such that the map A —-) L(A) is of class C" and L(Ao) is a Fredholm operator of index zero.(R[L(A0)] is a closed subspace of V and dimN[L(A0)] = codimR[L(A0)] < 00. o R is an operator of class C' such that R(A, 0) = 0 and DuR(A, 0) = 0 for each A E (a, b). Definition 1.3 (A0, 0) is a bifurcation point from the curve of (A, 0) if there exists a sequence (Amun) E (a, b) x (U\{0}) such that lim(A,,,u,,) = (A0, 0) and F(Amun) = 0. Definition 1.4 A0 is a nonlinear eigenvalue of L(A) if (Ao,0) is a bifurcation point from the curve (A, 0) and R(A, it) satisfies the second assumption. In other word, A0 is a nonlinear eigenvalue of L(A) if the fact that bifurcation occurs is exclusively based on the linear part. Definition 1.5 zero as a simple eigenvalue of L(Ao) if N[L(A0)] EB R[L(A0)] = V. For this, we need to assume U C V. Definition 1.6 A0 is an eigenvalue of the pair (L0,L1) if zero is an eigenvalue of L0 — AOL] where L0 := L(AO) and L1 :2 %(AO). Crandall and Rabinowitz found that if F (A, u) is of class C2 and dimN[L0] = 1, 27 L1(N[Lo]) EB R[L0] = V, then (A0,0) is a bifurcation point from (A,0). h'Ioreover, under these assumptions, the set of solutions bifurcating from (A, 0) at A = A0 consists of a curve of class C1. The second condition is usually referred to as the Crandall-Rabinowitz transversality condition, or nondegeneracy condition. Due to the huge number of applications of the result, it has become one of the most celebrated in nonlinear functional analysis. We shall state this theorem more rigorously as following Theorem 1.5 (Crandall-Rabinowitz[3]) Suppose F (A, u) is of class C' for some r 2 2 and zero is a simple eigenvalue of (L0, L1). Let Y C U be a subspace such that Nuqev=a Then, there exists 6 > 0 and two mappings of class C"1 A:(—e,e) —)R, y:(—e,e) —)Y such that and for each s E (—e, e) Fouynn)=o u(s) := s(¢o + 31(8))- Moreover, there exists p > 0 such that if F(A,u) = 0 and (A,u) 6 Bp(A0,0), then either it = 0 or (A,u) = (A(s),u(s)) for some 3 E (—e,e). Here Bp(A0,0) is a ball centered at (A0,0) with radius p. Furthermore, if F is assumed to be real analytic, then so are A(s) and u(s). 28 Note that (A'(0),q§0) is the tangent vector to the curve (A(s),u(s)) at (A0,0); it is usually called the bifurcation direction. We shall illustrate the application of Crandall-Rabinowitz theorem by the follow- ing example. Example 1.1 Consider the nonlinear boundary value problem I! —u (x) = Au(x)[1+ h(u(x))], x E (0,1), (1.57) u(0) = u(1) = 0, (1.58) where A is a real parameter and h is a function of class C3 such that h(u) = h2u2 + 0(u2) asu—>0 withh2750. Note that u E 0 is a solution of eq. (1.57) for any A E R. Our goal is to apply Crandall-Rabinowitz theorem to show the existence of nonzero solution of eq. (1.57) with small amplitude and arbitrary nodal behaviour. To put it under the abstract framework introduced earlier, we consider the Banach spaces U = 0300,11). v = C(io. 1)) then the operator F defined by F(A,u) = u" + Au[1+ h(u)], (A,u) E R x U makes sense and is of class C3 if [luiloo is sufficiently small. We also have L(A)-u :2 u" + Au, R(A, u) := Auh(u). 29 It is easy to see that the eigenvalues of L(A) are An = n27r2 where n E N, n 2 1. Moreover, for each n 2 1, N[L(An)] = span[sin(n7rx)] and due to Fredholm alternative, 1 R[L(An)] = {v E V :/ v(x) sin(n7rx)dx = 0}. 0 Thus, for any integer n 2 1, sin(n7r - x) ¢ R[(L)\n)] and hence zero is a simple eigenvalue of L(An). Therefore, applying Crandall- Rabinowitz theorem, we have the following result Corollary 1.1 Let n _>_ 1 be an integer and consider 1 Yn 2: {u E U :/ u(x)sin(n7rx)dx = 0}. 0 Then there exist 6 > 0 and two mappings: An : (—e,e) —) R yn : (—c,e) —> Yn of class 02 such that Au(O) = An, yn(0) = 0 30 and for each s E (—e, e) F(A,,(s), s[sin(nrr-) + y,,(s)]) = 0. Moreover, if (A,u) is sufficiently close to (An,0) and F(A, u) = 0, then either 21 = 0 or there exists 3 E (—c, e) for which (A, u) = (Ans), slsin(mr-) + yawn). Therefore, for any integer n 2 1, eq. (1.57) has a curve of solutions (Au(s),u,,(s)) emanating from (A, 0) at A = An where un(s) :2 s[sin(n7r~) + yn(x)]. Let a(A) denote the classical eigenvalue of the family L(A) perturbed from the zero eigenvalue of L(AO): 0&0) = 0, (MAO) = (20, where (to spans N (L(A0)). Then the following result holds Proposition 1.1 If zero is a simple eigenvalue of L(Ao), then a'(Ao) 75 0 if and only if zero is a simple eigenvalue of the pair (L0, L1),i.e., L1(N[L0]) EB R[L0] = V. As we recall Crandall’s theorem which states that if zero is a simple eigenvalue of the pair (L0, L1), then (A0, 0) is a bifurcation point. In other word, (A0, 0) is a bifurcation point if a'(A0) sé 0. However, the meaning of nondegeneracy condition is not so clear, in [3] it is a technical condition which was just needed to apply the implicit function theorem to a modified equation where the trivial solution was eliminated. The following theorem deals with degenerate eigenvalue and is due to Kielhiifer [11]. 31 Theorem 1.6 Assume U C V and zero is a simple eigenvalue of L(AO). Then A0 is a nonlinear eigenvalue of L(A) if and only if a(A) changes sign as A crosses A0. With the aid of this result, we now study the symmetry breaking problem. We shall consider the linearized problem about a given radial solution 21: 2A A ————: = . w+(1_u)3w 0 We may write w in the spherical harmonic decomposition form: 00 w = Z 0N(T)¢N(9), N=0 and aN satisfies the equation: a"+1a'+ 2A —Ea —0 N r N (1 — u)3 r2 N _ together with the boundary conditions aN(1) = 0 = aN(el). If the above equation admits a nonzero solution aN 79 0 for some N 2 1, then radial symmetry breaks. We consider the following eigenvalue problem: 2A N2] 0N = —#N,Ica~- all-+10, + —__ N r” (1—u)3 r2 Let U = C02(el,1) and V = C(61,1). We have the following lemma. Lemma 1.9 If u is a radial solution on the upper branch, then for arbitrary positive integer N, pN,1(A) < 0 for A sufi‘iciently close to zero. 32 Proof. It is well known that the eigenvalue pNv,1(A) can be characterized by firm = inf {fi%, (15 E C02([61,1])} , l where If u is a positive radial solution, then l'vu'2=/..(1—u)2 Since u is a solution on the upper branch, HUIIoo —> 1 as A —> 0+. Note that for arbitrary p > 0, there exists a > 0 such that 2u l—u 2p for uZl—a. Letfl1={xefl:u21—a},92={xEQ:u<1—a}. Hence _ 1 2u 2 u2 2M“) “ Ah((l—uv‘(1—u)3)““V 2a 1 u2 2_ 2 _ [QIVUI A/nU—U) (l—ul2u+N 9T2 N2 2u A < _ 2 _ 2_ _ . - 1 “(Q'WHe 9'“ (tips —u)l (l-uVu S 1-p+——/ V142- ( 621,1) al | (1.59) for some constant M > 0 which is independent of A. Hence for any given N > 0, pit/,1 < 0 since p > 0 can always be chosen to be 33 sufficiently large. Remark. It’s easy to see that if u is an upper branch solution, then (/,,|vul2)i 2 2”“ 1—61 as A —> 0. In fact, u(r) = fru’(s)dsS(1—el)i(/ (u'(s>)2ds)% (fn Ivul )2 (1.60) S (1— 60% 1 \/ 271'61 Lemma 1.10 p0,1(A*) = 0. Proof. Let p1(A) be the principal eigenvalue of 2 l/Jfl‘l' _¢_ — _#(——_1_)wa {CE (61,1) Clearly, u1(/\ )=inf[/ rim/[:03 M) ii 2603mm}. On the other hand, note that 1 it, = inf [Q(¢)/ / new, d) e 03([61.1])} . where QM) = / 11 r (v2 — (1 303%) dr. Thus #a1 2 inf [lad/0 — A] - at}; f: (132),,ivr}. 34 By a more general result by Keller [10] which states that pl(A‘) 2 A‘, it follows that [10,1 (AI) = 0. Definition 1.7 A0 is called a non-radial bifurcation point with mode It ifnk,1(A0) = 0 where k is a positive integer. We now apply the above result to prove the symmetry breaking result. Proof of Theorem 1.2. Since u0,1(A*) = 0, it follows that pN,1(A*) > 0 for N Z 1. By Lemma 1.9, for any N 2 1 there exists AN 6 (0, A“) such that pN,1(AN) = 0 and u(A) changes sign as A crosses AN. See Figure 1.4. Hence by Theorem 1.6, there is a bifurcation at A N where the radial symmetry breaks. The proof is completed. Figure 1.4. The change of sign as pm crosses A axis at Ak 1.6 Finite Time Touchdown and Touchdown Pro- file In this section, we modify a recent result by Flores et al. [6] and show that above a critical voltage, the membrane will touch the plate in finite time. These are apparently more important issue in the actual design of a MEMS device. For example, how to increase the stable operating range by increasing the critical voltage (pull-in voltage) V... This increase in the stable Operating range may be important for the design of microresonators. Another example is how to decrease the time for touchdown, thereby increasing the speed of switch. Now we consider the following time-dependent deflection of the membrane 36 u(x, y, t) which satisfies Bu ‘a—t —AU+ (1_u)2 in (I; u=0 on Oil; u(x,y,0)=0. (1.61) The solution u of (1.61) is said to touchdown at finite time if the maximum value of u reaches 1 at some finite time t = T. < 00. At such time, the membrane touches the fixed ground plate. We prove the following theorem. Theorem 1.7 Let A1 > 0 and u] be the smallest eigenvalue and the corresponding eigenfunction of the Dirichlet eigenvalue problem —Au = Au on O; u = 0 on 09. (1.62) IfA > A E 52171, then the solution u of (1.61) reachese -1 at finite time. Proof. This proof is based on a key result in [6]. Without loss of generality we assume that ul > 0 in Q, and we normalize ul so that f” u1 = 1. Multiplying (1.61) by ul and integrating over O, we have d U1 _ = A . (”Lulu Lu] u+A/n(l—u)2 Using Green’s Theorem, we obtain d U1 — = —A A ————. dt nulu l/Qulu+ A(l—Ul2 Introducing an energy-type variable E (t) by E (t) = fQ ulu, we have dE 'll1 — A E: A —. 1.6 dt + ‘ [fin—u)? ( 3) 37 Applying Jensen’s inequality on the right-hand side of (1.63), we obtain dE A —- AE>——————. dt +1 -(1—E)'2 On the other hand, we have E(O) = 0, E(t) g supu/u1 = supu. n n 11 Now we let F (t) solves dF A 35“” " (—1—F)2’ ”0) “0' “'64) It follows by a standard comparison principle that E (T) 2 F (t) on the domains of existence. Therefore, we conclude F(t) g E(t) S sup u. a On the other hand, we may determine t in terms of F by separating variables in (1.64). And we have that when F = 1. T is finite if the integral is finite. A simple calculation shows that the integrand is finite if A > A E 3.371. Thus ift = T. where T. < T, 11 reaches 1. 38 CHAPTER 2 Pattern Formation in A Biological System 2.1 Description of Biological Experiment Bacteria grown on thin agar plate may develop colonies of various spatial patterns, depending on both bacterial species and environmental conditions. We briefly de- scribe an experimental study done by Ohgiwari (1992). They used a bacterial strain of Bacillus subtilis. The bacteria was point-inoculated at the center of an agar plate containing peptone as a nutrient in a plastic petri dish with a diameter of 88mm. The swimming of the bacteria is a random walk type of movement and can be done only in a fluid with low viscosity. To produce such fluid the bacteria cooperatively secrete lubricant in which they can swim. In order to move, reproduce and perform other metabolic activities, the bacteria consume nutrients which are given in limited supply. The growth of a colony is thus limited by the diffusion of nutrient toward the colony—-the bacterial reproduction rate is limited by the level of nutrient available for the cells. If the nutrient is deficient for a long enough time, the bacteria begin the process of sporulation. They stOp normal activities and change into a spore. The sporulating bacteria may emit a wide 39 range of materials, some of which are unique to sporulating bacteria. These emitted chemicals might be used to signal other bacteria about the condition at the location of spores. Even though the initial seed of bacteria is radially symmetric, the colonies may develop various morphological patterns in response to environmental conditions. Ob- servations of such patterns occurred as early as 1938. It was first reported by Fujikawa and Matsushita [32] that the colony may exhibit branching patterns similar to the type known from the study of fractal formation in the process of diffusion-limited aggregation (DLA). When the agar medium becomes softer, the colony turns to Show a dense-branching morphology (DBM) with a smooth circular envelope. When both the nutrient concentration and the softness of the agar are high, the colony grows al- most homogeneously into a radial shape. Other types of patterns such as concentric rings and Eden-like pattern may also appear. 2.2 General Mathematical Model To explain each characteristic colony pattern, various models have been developed. For example, a diffusion limited aggregation model for DLA pattern proposed by Fujikawa and Matsushita [33], a communicating walkers model for DLA and DBM patterns by Ben-Jacob et a1. [26, 27]. In a recent paper, Kawasaki et al. [39] devel- oped a simple reaction diffusion model which closely captured all five different colony patterns. Other reaction diffusion models can be found in, for example, Kitsunezaki et al. [40], Matsushita et al. [43], Lacasta et al. [41]. All of the above models contains the population density b(x, t) of the bacterial cells at time t and spatial location x and the concentration of the nutrient n(x, t). 40 The models can be generalized into the form I g; = v-{D.(b,n)Vb}+g(b,n), (2.1) g; = awn—f(b,n), (2-2) where D, and D" are the diffusion coefficients of the bacteria and the nutrient, re- spectively. It is usually assumed that the diffusion coefficient Dn is a constant, while the diffusion coefficient of the bacteria Db depends on both bacterial density and the nutrient concentration. In [39], they chose Db(b, n) = onb, g(b,n) = nb, f(b, n) = nnb. In [40], Db(b,n) = ob", g(b, n) = nb - pb, f (b, n) = bn. Note that the first term in g(b, n) represents the rate at which the nutrient is consumed, the second term represents bacteria becoming stationary. The initial concentration of bacteria and nutrient is set to be b((li, 0) = b0($), n(x, 0) = no, where no is a positive constant since the nutrient is initially uniformly distributed, and b0(x) is the initial bacteria density which is a function with compact support. A typical numerical simulation is shown in the next figure. 41 Figure 2.1. A typical numerical simulation 2.3 Existence and Large Time Behavior of Weak Solution We consider the following nonlinear diffusion system: 2—1: :: A(uk) + um'l) — a(u, ’U)'U.n in QT 22 Q X (01 T), (2'3) (69—: = dA(vl) — umv in QT, (2.4) where k,l,m and n are positive integers, d is a positive constant and a(u,v) is a strictly positive function of u and v. Q is a smooth domain of RN and T > 0. The following boundary and initial conditions are assumed g;— : g}: = 0 on 89 x (0, T), (2.5) (u(x, 0), v(x,0)) = (u0(x), vo(x)) for all x 6 0. (2.6) 42 In the context of bacteria growth, u is the bacteria density and v is the nutrient density. When m = n = 1 and k = l = 1 and a(u,v) is a constant, the model is the well-known Gray-Scott model in chemistry. A special case is the scalar equation ut = A(u"), (2.7) which is called the porous medium equation for k > 1. To understand some basic behavior of the solution, we shall first present this equation as an example. Specifi- cally, we look for a ”similarity” solution of the porous medium equation and observe its property. Example 2.1 Let us look for a solution it with the form u(x,t) = —w( x a: E R" (2.8) t0 t—fi)’ where the constants a, fl and the function w : R" ——> R are to be determined. We insert (2.8) into (2. 7) and obtain at"(°+”w(y) + flit-(“"31 - Dw(y) + t'(°k+2‘3)A(w")(y) = 0 (2-9) for y = t‘Bar. To put (2.9) into an expression involving only variable y, we require a + 1 = alt + 25. Then (2.9) reduces to aw + By - Dw + A(wk) = 0. (2.10) Let us further assume that w is in fact radial, that is, w(y) = w(|y|). Then (2.10) 43 becomes H n- — 1 + aw + lirw' + (wk) (wk)' 2 O. (2.11) where r = Iyl, ’ 2 dir. Now if we set a = nfi, we have (r"_1(wk)')' + fi(r"w)' = 0. Thus r"‘1(wk)' + firnw = a for some constant a. Assuming limrnoow = lim,_+00 w’ = 0, we conclude a is in fact 0. Hence Consequently, where we took the positive part of the right hand side to ensure w 2 O. Recalling w(y) = w(r) and the rescaling, we obtain 1 k—1Ix2“”rl at :_ b_— f- a where n [3 __ 1 mk—n+2""mk—n+2 a: This solution is called Barenblatt’s solution to the porous medium equation. Based upon this special solution, one can easily observe that it has compact support 44 for each time t > 0. This is a general feature for nonnegative weak solutions of the porous medium equation with compactly supported initial data. The porous medium equation becomes degenerate wherever u = 0, and this type of parabolic equation is generally called degenerate parabolic equation. So the set u > 0 moves with a finite propagation speed which is a general feature of degenerate parabolic equation. Thus it is often regarded as a better model of diffusive spreading than the linear heat equation ( which predicts infinite propagation speed). One may also observe that for k = 2, the first derivative of the solution is discon- tinuous but finite at the boundary of the support, and for k > 2, the first derivative diverges at the boundary of the support. This is also a general feature of degenerate parabolic equation, that is, the solution may not be classical. Hence one shall seek solutions in a weaker sense. Definition 2.1 We say (u, v) is a weak solution on [0, T] if it satisfies: (i) u, v 6 C(QT) and u, v _>_ 0, (ii) For all 45 E C2’1(QT) such that 2:3 = 0 on 69 x [0,T], we have for allt E [0, T]: Lu(t)¢(t) = Luo¢(0)+/Ot/n(ukA¢+u¢t+ (umv—a(u,v)u")¢), (2.12) Lump“) = [gum/5(0) +AtL(dle¢+u¢t —umv¢). (2.13) In a very recent study by Mimura et al. [30], they studied the existence and large time behavior of such weak solutions. The proof is very technical and tedious, we refer interested readers to their paper. Here we shall state their main results: Suppose that 1) U0, v0 6 0(9) and 0 S u0,vo g M for some constant M > 0. and 2) a(u, v) is strictly positive and locally Lipschitz or a = O, and k+2/N, ifN23, 13 m < (2.14) k+1, ifN=1,2. In the case a 7t 0, the problem admits a unique weak solution (a, v) satisfying 0 S u(r,t) _<_ C0, and 0 S v(x,t) g M for all (x,t) E Q X (0, T), for some constant C0 > 0. There exists a constant v°° such that lim (u(t), v(t)) = (0, v°°) uniformly in Q. t—>oo Furthermore, if 1 S m < n, v°° = 0, while if 1 g n g m, then v°° > 0. Especially if m = n, then v°° _<_ a(O, v°°). For the case a = 0, they obtained similar result and lim (u(t),v(t)) = (< uo + v0 >,O) uniformly in Q. t—>oo 2.4 Main result and methods As we indicated earlier, bacteria grown on the surface of thin agar plates often develop colonies of various spatial patterns, such as fractal morphogenesis, dense-branching pattern. Recently Kawasaki et al. [39] proposed a degenerate parabolic system with cross diffusion that captures the qualitative features of the growth patterns. The 46 model is g; = DbV - {nbe} + nb, (2-15) 92—: = DnVZn — nb (2'16) with initial conditions b($,y,0) = b0, n(x, y,0) = no. Here b is the bacteria density and n is the nutrient density. Recently, Maini et al. [45] considered the one-dimensional version of the model and a special case 0,, = 0. They studied the existence and uniqueness of traveling wave solutions. The found that such solutions exists only for speeds greater than some threshold speed and the wave with the minimum speed has a sharp profile. For speeds greater than this minimum speed the waves are smooth. By considering the special case Dn = 0, the authors were able to reduce the problem to a phase-space analysis in R2. In this thesis, we consider a more general model and we will not assume D" = O. The model we will study takes the form 8b E = va.{npkab}+n‘Ib’, (2.17) g" = DnV2n—nqb’. (218) To our knowledge, there is little theory on traveling wave solutions on such coupled degenerate diffusion system with cross-diffusion. Our method is based on Schauder-fixed point theorem. By fixing n in a properly chosen S ace V we investi ate the existence of travelin wave solution b. For such , 47 a traveling wave b, we can find a traveling wave solution a which lies in a compact subset of V. In this way, we may define a continuous mapping T : V —> V with Tn = n. Invoking Schauder-fixed point theorem, we conclude the existence of a traveling wave solution pair (b, n). A key part of the thesis is to investigate the existence of finite traveling wave solution for a degenerate cross-diffusion equation. Our method is inspired by a very recent study of a similar problem by Malaguti [42]. We will transform the existence of traveling wave solution b to the solvability of a first-order singular boundary value problem which can be done by typical shooting and comparison argument. For more information on singular ODE, see [28, 34, 38, 42, 44]. Our main goal is to prove the following theorem. Theorem 2.1 For k = 1, q 2 1, p = 0, l > 1, there exists a constant velocity U. such that the system admits a traveling wave solution (b({),n(§)) where b is a monotone finite traveling wave solution and n is a monotone classical traveling wave solution. Here 5 = :2: — vt is the usual wave coordinate and by finite traveling wave we mean 5" = SUp{€: b(é) > 0} < 00. Remark. Even though our result is very restricted on the values of k and p, most part of our analysis in this thesis works through without such restrictions. We shall keep them until the occasion arises when we need to impose such restrictions. It is our future concern to extend our result to more general cases. In section 2.5, we simulate the 1D problem and oberseve the long time behavior of the solutions. In section 2.6 we will derive some properties of traveling wave solutions. In section 2.7 we investigate the existence of finite traveling wave b for a given 72. For this part, the result is general and we place no restriction on k. In section 2.8, we 48 derive the equivalent problem and study the existence of a traveling wave n for the determined finite traveling wave solution b. In this part we shall require that k = 1 in order to derive the equivalent problem. In section 2.8 we apply Schauder fixed point theorem to deduce the existence of traveling wave solutions (b, n) where b is of finite type. In section 2.10, we study the instability of the flat front and explain the fingering pattern in the bacterial colony. 2.5 Numerical Simulation to 1D Problem In this section we consider the one dimensional problem on [0, 1] 6b 6 6b 3n 02n with Neumann boundary condition and initial data is chosen to be b(x,0) = —(:1: — 0.25)(.v — 0.75), n(x,0) = 1. We numerically simulate the problem on time interval [0, 0.2] and we have the results shown in Figure 2.2 and 2.3. The numerical results show b stays compact supported. The long time behavior can be seen when we calculate the density functions up to t = 9, the results are shown in Figure 2.4 and 2.5. We can see that b tends to a uniform state and n tends to 0 asymptotically. 49 b(x,!) Distance x Figure 2.2. Bacteria Density n(x,!) Distance x Time I 012 Figure 2.3. Nutrient Density 50 b(x,t) Distance x Figure 2.4. Bacteria Density n(x,!) Distance x 1 0 Time t Figure 2.5. Nutrient Density 51 2.6 Travelling Wave Solution We consider the system (9b (9 8b — : —-- P k— ‘ql 8t D8$(n b (92:) + n b, (2.21) an 8271 a : 5-1:; — nqbl, (2.22) where (:13, t) 6 R x R+ and D is the rescaled non-dimensionalised diffusion coefficient of b. Let us denote the spatially uniform steady states by (b, n) = (b,, 0), (0, n,), where n, and b, are some constants. We may assume that initially the nutrient is uniformly distributed in the plate and there is no bacteria seed. Hence a prOper initial steady state is given by b=0, n=1, for all —ooOasx—):l:oo, for all t>0 ( 3) Let 6 = :1: — vt be the wave coordinates in which b and n solve D(n”b"b’)’ + vb’ + nqb’ = 0 (2.24) n" + vn’ — nqbl = 0 (2.25) where ’ denotes the derivative with respect to wave coordinate 5. Equations (2.24)- (2.25) are to be solved subject to the following boundary conditions ahead of the 52 wave b—+0,n—>1as£—>+oo, (2.26) and behind the wave b —-> b,,n —> n, as 5 —> —oo. (2.27) 2.6.1 Properties of the traveling wave solutions. Property 1. n, = 0, b3 =1. Proof. Integrating on both sides of equation (2.24) from —00 to +00 gives +00 —vb, + / nqb' = 0 (2.28) 00 from which we conclude that b, > 0 since u > 0. But nabs = 0, hence n, = 0. To conclude that b8 2 1 we integrate Equation (2.24) from —00 to f, we have 6 npbkb’ + v(b - b,) +/ nqb’ = 0. (2.29) —00 On the other hand, integrating equation (2.25) from —00 to g, we have E / nqbl = n’(§) + vn(£). (2.30) Substituting expression (2.30) into equation (2.29) and passing 5 to +00, we have —vbs + v = 0. (2.31) Therefore b, E 1. Pr0perty 2. If n, b are the traveling wave solutions, then n is monotone increasing and b is monotone decreasing if 0 < b < 1. 53 Proof. The monotonicity of n follows directly from equation (2.25). In fact, 77.” + vn' = n‘lbl 2 0, (2.32) or equivalently (n'e”£)’ 2 0. (2.33) Integrating this inequality from —00 to 5, we obtain n’e”é > 0. Hence n’ > 0 for all 5. For a traveling wave solution b({), we prove that b’(§) < 0 for 0 < b({) < 1. We first suppose that b’ ((30) == 0 for some {0, then from equation (2.24) we have anbkb" + D(n”b")’b’ + vb’ + nqb’ = 0. (2.34) Hence ,, 4b ‘ b:—;.::::.sfzi 0 for all s E (€,{0).}. (2.36) Therefore 5,, > —oo => b’(€.) 2: 0 since 0 < b < 1 = b(-—oo) = 1. Furthermore, b’ (6..) = 0. On the other hand, from equation (2.24) we conclude that npbkb’ is positive and strictly decreasing in (5., {0], in contradiction with b’(.£,) = 0 = b’ (50). Thus if b’ (E) > 0 in some interval (61,52) with b’ (£1) = 0, we necessarily have b(él) = 0 which again in contradiction since npbkb’ is positive and decreasing in 54 (61,52). Therefore we conclude that b’(€) < 0 when 0 < b < 1. 2.7 Existence of b for a given ii Let £1=inf{{:b({)<1}6 RU {—00}; £2 = sup{£ : b({) > 0} E RU {+00}. Since b is strictly decreasing on (61,52), it follows that the inverse of b({), denoted by 5 = 6 (b) is well defined on (0, 1) and takes value in (51,52). Therefore we may define n(b) = n(6(0) (2-37) and u(b) = an(b)b"b’(§(b)) < 0 for all b 6 (0,1). (2.38) In fact, {1 = —00. We state this in the following lemma. Lemma 2.1 {1 = —00. Proof. The proof is similar to the proof of property 2 in previous section. Let n(b) E V where V is the closed convex set of the Banach space C°([0,1]) defined by n(b) V :2 {n(b) E CO[0,1], 0 S n g 1, limsup S L, n(b) Z 1/L(1— b)} an- -b where L is a sufficiently large constant and will be chosen later. We shall see later in the thesis the reason why we define such a space. 55 Differentiating both sides of equation (2.38) with respect to f we have u’(b)b’(£) = anbkb" + (anbkyb' = —vb' — nu}. (2.39) Therefore a as a function of b satisfies p+q b bk+l u'(b) = —v — Dn u( ) , b 6 (0,1). (2.40) Our next step is to derive the boundary condition at 0 and 1. Lemma 2.2 u(0+) = u(1‘) = 0. Proof. If 52 = +00, i.e., b is a classical traveling wave, it follows that u(0+) = lim an(b)bkb'(§) = 0. {—2+oo If 62 < +00, i. e., b is a finite traveling wave solution, we may assume that it vanishes at 5 = 0. We expect that as E —) O b(6) ~ A(—§)°. (2-41) Substitute this expression into anbkb” + D(n”b")’b’ + vb’ + nqb’ = o where the derivative is with respect to 6, we obtain anAk+l[a(a _1)+ ka2](_€)(k+l)a~2 + Dpnp—l7l’(—Ak+10)(—€)(k+l)O-l —vAa(-—g)"-‘ + n"A’(—§)"’ = 0. (2.42) 56 Equating the dominated terms at g = 0 we have (k+l)(r—2=a—1, Dn‘pAk+1[a(a -1)+ ka2] — v.40 = 0 where n" = limb_,0 n(b) > 0. Hence we have a =1/k, kv A = . Dn*” From the definition of u, we have u(0+) = lim Dn”(b)b’°b’(§) b—>0+ = lim anAk(—§) . (—Aa)(—§)1/k_1 {40+ =0. u(1‘) = 0 follows from the definition. Therefore u solves the following singular boundary value problem: an+q (b) bk+l u u'(b) = —v — for b 6 (0,1), subject to u(0+) : u(1") = 0. We now consider the solvability of this singular problem. 57 (2.43) (2.44) (2.45) (2.46) (2.47) 2.7 .1 First-order singular boundary value problem Given n(b) E V, we consider the following existence problem: Problem P1. Find a pair (v, u) with v > 0 such that .u'(b) = —v — anJflwakH for b 6 (0,1), (2.48) u(0+) = u(l‘) = 0, (2.49) u < 0 in (0,1). (2.50) 2.7.2 Existence of a critical velocity In this section we show that P1 is solvable with a unique negative solution it = u(b) if and only if v 2 v" for some positive v“ which depends on the choice of n(b). We first prove the following lemma Lemma 2.3 If there exists qt 6 01(0, 1) such that an-t—q (b) bk+l ¢(b) for b 6 (0,1), (2.51) ¢’(b) > -v - such that ¢(0+) = 0 and ¢(b) < 0 for b 6 (0,1). Then P1 is solvable and the solution 0 > u(b) > (b(b). Proof. The proof is based on shooting argument and some comparison techniques. First, for a fixed constant b0 6 (0, 1), note q§(bo) < 0, let a E [¢(b0), 0), we consider the following initial value problem: k+lp+q u'=—-v-Db n (b), 0b‘+ Since tlfitmtwu—Derruvr=—Dav“%rur)<2 hence we can find a constant c > 0 such that u' : —vu — Dbk+lnp+q(b) > ;c_ > 0 u u(b) for all b* < b < b" + c, which would imply that u > 0 for b* < b < b” + c, clearly a contradiction. Next we assert if a > ¢(bo), then (b(b) < u(b) < 0 for 0 0, then u(b) = (b(b) and I Dbk+lnp+q(b) n(b) — - u(lz) : _V_Dr“wva) $02) < 49(2), 59 which is clearly absurd. In view of the arguments above, we have proved that u is well defined in (0%). Now we may define (0, b0) to be the maximal existence interval for solution u. The aim is to show that b0 2 1 for some a E [0(b0), 0). Let u 1 and u; be two distinct solutions corresponding to initial value 011 and 02 respectively. Suppose for definiteness that C21 < (12, then u1(bo,0) there exists a sufficiently small constant M > 0 and A < 2M such that Db"+’np+q(b) M _v _ _— ’U. ’U. 60 for all —A < u < 0 and b0 _<_ b< b0+/\. Leta > —)t and define w = —\/o"2 — 2M(b — b0) for be g b 3 be + 2% which solves the following initial problem: M W = _—a ll) "(b(bo) =0. We have 02 (”+274- 0, hence u(b) > u(bo) = a > —2\ in a right neighborhood of b0. Put 02 I = [b0,min{bo + 2M’baH’ we deduce that u(b)>——A for all beI. Apply a similar comparison argument as before to conclude u(b) > w(b) for all b E (be, be) 0 (b0, b0 + %). Since (b(bo + 2115;): 0, we have 2 a ba(bo),0) : b0, < 1}, then b0. = 1, therefore the corresponding solution u is defined and negative on (0,1) and u > (15 in (0,1) and u(0+) = 0. This completes the proof. We are now in position to prove the following solvability result for Problem P1. Theorem 2.2 There exists v“ > 0, such that for all v > v“, Problem P1 has a unique 61 negative solution. Proof. We first show P1 is solvable for v sufficiently large. To this aim, we let an+q(3)8k+l c = supsem,” s , (2.52) which is well defined. Let W?) = -\/Eb- Then, for v > 2\/E, we have Dbk+lnp+q (b) Dbk+lnp+q (b) —v — < —2 c + cab) f m s —2\/E + W: = ¢’(b) (2-53) for all b 6 (0,1). Hence ¢(b) satisfies condition of Lemma 2.3. Therefore P1 is solvable for every v > 2\/E. We now show that P1 is not solvable for v = 0. Otherwise, let u solves I Dbk+lnp+q(b) u:— u defined on some interval (a, 1) with 0 < a < 1 and u(b) < 0 for all b E (a, 1). Integrate the equation above in [b, b] with a < b < b < 1 we obtain 5 u2(b) = u2(b) — 2/ Dsk+lnp+q(s)ds. (2.54) b Therefore, if u(l‘) = 0, we have u(b) = —\/2 [bl Dsk+lnp+q(s)ds (2.55) which implies u(0+) < 0, a contradiction. 62 We now let v" = inf{v : P1 is solvable} which is well defined and v* > 0 based on the observation above. We prove that for every v > v“, P1 is solvable. Given v > v", take 17 such that P1 is solvable with v < v and the unique solution a for 27. Since _, _ Dbk+lnp+q(b) Dbk+lnp+q(b) u = —v — _ > -—v — _ , ”U. U hence it satisfies condition of Lemma 2.3. Therefore, we conclude the solvability for v. Finally we prove that P1 admits at most one solution. Suppose for contradiction that ul and u2 are two distinct solutions of P1. For definiteness, we assume u1(bo) > U2(bo) for b0 6 (0,1), it follows that , , Dbk+'n"+q b Db"+'n”+q b U1(bO) - U2(bO) z .— 0u1(bo)( 0) 0u2(b0)( 0) > 0. (2.56) Hence if u1(b0) > u2(b0), then u’1(b0) > u§(bo). Therefore, it is impossible that This completes the proof. 63 2.7.3 Finite traveling wave at v* The main goal of this section is to show that b is a finite traveling wave at the minimum wave speed. We will also show that b is a classical traveling wave if v > v”. The idea is to characterize the type of b by the value of u'(0+). Lemma 2.4 u'(0+) = 0 or u’(0+) = —v. Proof. We first show that if u’ (b‘) = 0 for some sufficiently small b“ then u"(b“) > 0. To this aim, we write g(b) = Dbk+lnp+0(b), it follows that Note that g'(b) = Dbk+l‘lnp+"—1[(k + 1)n(b) + (p + q)bn'(b)] > 0 for b sufficiently small. Hence if there exists b‘ sufficiently small such that u’(b‘) = 0, then u”(b*) —_- Jig-l > 0. Therefore we conclude that there exists 0 < b < b‘ such that u"(b) > 0 or u”(b) < 0 on (0, b). Case 1: u"(b) > 0 on (0, b). In this case u’(0+) < 0 exists and it follows from (u' + v) - 3;- : —Db"+"‘np+q(b) that u'(0+) = —v. Case 2: u"(b) < 0 on (0,b). In this case, —v < u’(0+) S 0, similarly we obtain u'(0+) = 0. 64 Lemma 2.5 b is a finite traveling wave if and only if u’(0+) = —v. Proof. If b is finite traveling wave, then from Lemma 2.2, at g = 0, we have kv b = — V". 2.’ (6) WA 5) ( o7) Therefore by equation (2.39) we have , + _ , _ _ n"bl u(0 ) _ bl—llbu v b’ ) kv : _ 1- q __ l—lk _ l/k—l/k+l "+5351” (an) ( E) = _v, (2.58) On the other hand, if u’(0+) = —v, we have . I _ . n(b) b (SENS) _ til—131+ b BMW 1' ” 1m —, b—+O+ Dbk—lnp (2.59) which is — if k = 1 and —00 if k > 1. This implies that b is a finite traveling Dn‘P wave solution. Lemma 2.6 For v sufficiently large, b({) > 0 for allé E R. In other word, b(é) is a classical traveling wave solution. Proof. Note that for v sufficiently large, there exists A > 0 such that >“It: —v + g —(k + in. (2.60) We claim that u(b) 2 —Ab‘““, (2.61) 65 for all b 6 [0,1]. Suppose u(b) < —)\bk+’ for some b 6 (0,1), then - Dbk+lnp+q(b) ’ b = — — _ U( ) u(b) D5k+lnp+q(5) < -—v + A5k+l < —v + P- /\ g —A(k +1) 3 —/\(k +1)B’c+‘—1. (2.62) Hence u(b) + Ab“l < u(b) + Abk‘” < 0 for all b > b (2.63) which contradicts u(1‘) : 0. Now for any {0 E (—00,§2), let b({0) 2 b0, we have 50 50 - £2 = 64de 0 _ b°an(b)b" ‘ / u(b) db 1 b°an(b) _ g 7/0 b, db——00. (2-64) Hence {2 = +00 which implies that b is a classical traveling wave solution. Note here we applied the fact that l 2 1. Now our goal is to show that when v = v“, b is a finite traveling wave solution. In view of Lemma 2.5, we shall show that when v = v“, the solution of satisfies u’(0+) = ——v‘. 66 The trick is to construct a converging sequence of solutions which satisfy this property. To Show this, we define a continuous function g,,(b) on [0, 1] as: 1 ’n+2 ] , 0 S gn S Dbk+l'rip+q(b) for b E ] 1 1 ] nb=0 0 , g() on ] n+2n+1 l n b =Dbk+l p+qb 1 . andg() n () on [n+1’] It holds that %S%% We consider the following problem: To find a function un : (0,1) -) R‘ and vn > 0 such that un(0+) = un(1‘)= 0. Let us emphasize here that vn is also an unknown of the problem. We shall prove the following theorem. Theorem 2.3 There exists a unique solution un : (0, 1) —+ R‘ and vn of the problem. u is of class C1. Proof. The problem is equivalent to 1 9n 1 = — n — — ,1 , 2.6" un v Un on [n + 2 ] ( o) 1 n = — nb 0, , 2.66 u v on I: n + 2] ( ) 1 I Z _ ln~ 2.67 u,,(1—) = 0. (2.68) 67 Note that the solution of this equivalent problem is of C1 since u’($) = —v,,. We apply the shooting argument and comparison techniques on [$, 1] similar as in Section 3 to the problem: , 9n 1 = — n — — ——,1, 2.69 un U an 0n [n+2 l ( ) 1 vn ,, = — , 2.70 u (n+2) n+2 ( ) 1 ' : — n. 2.71 un(n+2) 'U ( ) We can show that there exists a unique vn such that un(1') = 0. We now examine several properties of an. Some of them will be applied in the proof of the next lemma. The proof is all the similar shooting and comparison argument. We shall briefly show the proof of the second one. Lemma 2.7 an satisfies: p1 un(b) = —vnb on [0, :35]. p2 un(b) Z —v“b for all b 6 [0,1]. p3 u" 2 un+1 for all b 6 [0,1]. Proof. p2 Since un(b) = —v,,(b) > —v*b on (0, #2], hence if 3 bsuch that un(b) < 1]. Furthermore, 3 b E ( 1 b) such that ”n(b) = —v*b for some n, b must be in ( 31—2, _1_ n+2’ —v*b and u;,(b) < —v*. However, which is a contradiction. Now we will prove that v,, determined in the theorem above has the following properties. Lemma 2.8 {vn} is monotone increasing and lim,,_,.r00 2),, = v’. 68 Proof. First we show that {vn} is monotone increasing. Suppose Bun > vn+1 for some n. Then, 1 — v" — v,, + 1 1 n : <' n - “(n+2) n+2 n+2 u +1(n+2) We also have 1 gn+1( ) 1 up 2 _Un < _vn _ n+2 I "(714-2) +1 un+l(n+2) n+l(n+2) since ( 1 )> 0 gn+l 72. + 2 - Hence un(b) < un+1(b) in a right neighborhood of 5—1—2, therefore in this neighborhood I 911 9n+1 I b = — n - — — n — — = b . un( ) ’U u" < ’t) +1 “n+1 un+l( ) Hence 1 “n(b) _ un+l(b) < un( )_ un+l( ) < 0 n+2 n+2 for alle[1 1]. n+2’ This is a contradiction with un(1‘) = u,,+1(1‘) = 0. Hence ”n S vn+l- We now prove that vn S v“. Suppose 3 vn > v“, then 1 —vn -—v“ 1 = < < n+2) n+2 n+2 ”(n+2 un( ), where we have applied the fact that u’ > —v‘ for all b 6 (0,1). We also have 1 ) < *< '( 1 :_’U‘n —’U U n+2 n+2 Ui.( ). 69 1 Hence un(b) < u(b) < 0 is a right neighborhood of "+2. For everv b in this nei rhbor- . E» hood, we have n n l)lk+l p+q l 1111(1)) : ”'Un —' L < —"U‘ — g— < —"Ut — ) n ()) : U,(b) un u u which contradicts with un(1") = u(l‘) = 0. Finally, we show limn_,+oo v1, = v“. Let 27 = limsupvn S v“. n—H-oo Note that we may define u(b) = limn_,+oo un(b), we have 1 . 1 u(b)—u(nH) — "143100048 ‘“"(n+1)) b = lim u n—++oo _;_ n+1 _ 1' b g" — 1m —v,, — — n—H-oo n_l_l un b k+l p+q = / —t—Db " (b). (2.72) 0 u Hence u solves k+l p+q u' = —v — Db Z (b) on (0,1), Moreover, we have u(0+) = 0 and u(1‘) = 0. 70 By the definition of v“ we conclude that 27 2 v*. Hence 1‘; E v“. Note that u(b) = lim un(b) 2 —v*b for b 6 [0,1). n—++00 Applying monotone convergence theorem to {an} we can show that it solves k+lp+qb u=—v—Db Z () in (0,1), Hence Therefore This shows that the traveling wave at minimum wave speed v‘ is of finite type. We conclude this section with a few properties of the negative solution u. Lemma 2.9 Let u = u(b) be a negative solution of Problem P1. Then, there exists u'(1") = 0. Proof. Let u(b) be a solution of P1. Let M :2 limsupb_,1_ 35—91) and m 2: liminf),_,1— :4}? 2 0. There are three possible cases: either m = M > 0 or M > m 2 0 and M = m = 0. We show the first two are impossible. In fact, if m = M > 0, then an+qbk+l b — 1 . I __ _ . :— bgm_u(b)— v bl—l+rl1_ b—l u(b) v<0, a contradiction. Here we applied the fact that lim supbal- 7:9,] S L. 71 If M > m 2 0. Then for any given m0 6 (m, M) there exists a sequence {b,,} which converges to 1 such that u(bn) : m0 and (1‘92), 3 0. b=bn Hence, 1 D p+q bn bk+l lim u'(bn)= lim —v—— n ()n =—v<0, n—)+00 n—++00 m0 bn — 1 therefore we may assume u’ (bn) < 0 for every n. On the other hand, (:Ebl);=bn 2 b1, 1—1 law") _ 1:021] = bn1—1lu'(b")_ mo] > 0. I A contradiction to previous statement that ($3), b S 0. Therefore, M = m = 0, i.e., there exists u’(1') = 0. Using this lemma, we can further show that Lemma 2.10 For the solution u of Problem P1, there exists C1 < 0, C2 < 0 such that C2(1 — b)?” S u(b) S C1(1 — b)p+q for b sufficiently close to 1. Proof. We only need to show that it is impossible to find negative constants C1, C2 such that u(b) Z C1(1 - b)p+q+7 or u(b) S C2(1 —- b)”+"‘7 for any 7 > 0 as b -—> 1‘. Otherwise, we either have an+qbk+l n(b) an+qbk+l > —v - .. — 01(1 _ b)p+q+1 D(1/L)P+‘1(1 — new“! (31(1 — b)P+‘I+7 u'(b) = —v—- IV 72 for b sufficiently close to 1. A contradiction to Lemma 2.9. Or we have DLP+q(1 — b)P+qbk+’ U — .. — 01(1 - b)P+‘I—’7 as b —> 1’. A contradiction to Lemma 2.9 again. In fact, we can improve this result in the next lemma. Lemma 2.11 There exists C2 < 0 such that u(b) Z C2(1 — b)p+q for all b E [0,1]. Proof. Suppose that there exists b 6 [0,1] such that u(b) < C2(1 — (3)9”. Then we have — an+q(b)bk+‘ ’ b = — — _ u( ) v u(b) DLP+ b. A contradiction to u(1’) = 0. This ends the proof. This result is illustrated in Figure 2.6. 73 u(b) <— 02(1-p)p+q Figure 2.6. Sketch of u 2.8 The Equivalent Problem Now that we have shown that for any n E V, there exists a v" which depends on the choice of n such that b is a finite traveling wave, we may assume that b({) = 0 for E Z 0. (2.73) We shall simplify the problem by reducing it to a system in the interval 6 < 0 only. Note that n satisfies n" + vn' = 0 for 6 Z 0. (2.74) We have the following lemma: 74 Lemma 2.12 The problem n" + vn' = 0 in (0, +00), (2.75) 72(0) 2 n" with 0 < n" < 1, (2.76) n'(0) = v(l - n*), (2.77) has a unique bounded solution that satisfies lim€_,+00 n({) = 1. Proof. We consider the following ODE system: A phase plane analysis shows that every trajectory can intersect the n — axis at most once. Hence p’ changes sign at most once, and consequently n(+00) exists. Let n(+00) = c, a direct integration shows that +00 +00 / n =-/ 1m, 0 0 hence —v(1 — n") = —vc+ vn“. Therefore lim n({) = c =1. £—>+00 In view of this lemma, we may reformulate the problem into: Problem P2. Find (v, b, n) with v > 0 such that D(n”bk)" + vb' + nqb’ = 0, g < 0, (2.78) 75 n" + vn' — n"bl = 0, g < 0, (2.79) n'(0) = v(l — 72(0)), n(—00) = 0, (2.80) b(—00) = 1, b(0) = 0. (2.81) We have seen that as { varies from —00 to 0, b = b({) decreases monotonicity from 1 to 0. We can therefore define g:§(b) as the inverse function of b({) where b varies from 0 to 1 and 5 takes value in (—00, 0). As before we define u(b)=Dn”b"b’(§(b))<0 for all be(0,1) (2.82) and M0=nk®h 980 Since d _ , d _ u(b) d d—g-b(€)'a-ana‘a’ we can transform problem P2 into the following equivalent problem: Problem P3. Find (v,u, n) solves Dbk+lnp+q(b) u' = —v — , (2.84) it u ( u ,),+ u I .— Dbknp Dbknp' vaknP n' — nqb’ = 0, (2.85) u,(()+) = u(1’) = O, (2.86) 76 kn” n'(0+) 28(1—n(0)).b1351+l:b(b) = —(1—n(0))Dn(0)p(k =1), n(1_)= 0, (2.87) u<0 in (0,1) (2.88) we shall also need 1 k p / Db ” db = —00. (2.89) 0 Remark. The first equality in equation (2.87) is equivalent to the first equality of equation (2.80) only for k = 1. Therefore from here we shall take k = 1. 2.9 A fixed point Give n(b) E V, let (v, u) be the unique solution of Problem P1 such that u’ (0+) = —v, i.e., the corresponding b is a finite traveling wave. Consider the following: Problem P4. Find n(b) such that u ,, , ~, _ Dn“’ii"b’+1 _ , (_Dbnpn) + vn ——u — 0 in (0,1), (2.90) a’ + D(1 — ii)n” = 0 at b = 0, (2.91) 71(1) 2 0. (2.92) We shall show that Problem P4 admits a unique solution that is also in V. To prove this fact, we begin with the following local existence result. Lemma 2.13 it ~ , ~, annquk , (Dbknpn') + vn — -—u—— = 0 in (0,1), (2.93) 72(0) 2 no, 0 < 710 < 1, (2.94) 77 n'(0) = D(n0 — 1)n*”, (2.95) n(b) > 0 for b > 0 and close to O. (2.96) has a local solution. Here n* : limb_,0 n(b). Proof. Take E = {fl(b)lfl(b) E C([O,b0]),0 S ’fl S no for 0 S b S ()0, and some small b0 > O to be determined later}. Then E is a closed convex set in C ([0, M). Before proceeding it is helpful to rewrite equation (2.93) as an equivalent integral equation. Assuming that n is a smooth solution on the interval (0, b0] with b0 3 1, we integrate once to get u ~ _ u(s) ~ ~ b annqsl“ '— l ' b — = —. 2.97 ow" .355. DsnP(s)n +””( ) ”"0 f0 u(s) ( ) Since - u(s) ~I _ —'U _ tp 31—155) DsnP(s)n — Dn"‘l"D(n0 1)” ’ we have u b annqsl“ Dbflf’fi, + ’Ufl(b) — ’U 2/0‘ ——u—(s—)——. (2.98) A second integration yields 5 n” b p 8 npfiqTHl For small b0 and 0 g b 3 b0, we have Ogngn0<+oo, 78 (17$ C< +00. Dbn”(b) - 1)an fbonpnqrw u(b) 0 11(7) This shows the operator defined above is a compact operator. Therefore it has a fixed point in E, i.e., it has a local solution. To establish the existence and uniqueness of solution to P4, we shall also need to following lemmas: Lemma 2.14 If 72(0) 2 no is sufi‘iciently close to 1, then 21(1) 2 0. If 21(0) 2 no is sufficiently close to 0, then n(b) = 0 for some 0 < b < 1. Proof. We prove this Lemma by contradictions. We start with the following equa- tion: va bow '+1 Db? m” "fi=[v+/ ""8 ] ". (2.100) n 0 u(s) 11 Let h(b)= ——which 18 clearly nonpositive. The above equation can be rewritten as b (efé’thmy = ef:”h(s)dsh(b) [11+ / h(sws’ds] . (2.101) 0 Let H (b) = cf: ”WW3, and J (b) = v + fob h(s)nqs’ds. Integrating both sides, we have Ham—71(0) = fl): —(3H' b = —H(b)J(b)——H( 0)J(O)— )/(8): -1-H(s)h(s)fi"(s)s'd8 : 111(0)] (b)—1—/0b:—sH( (33.)‘ds (2.102) ’U Suppose that no is sufficiently close to 1 but n(b) = 0 for some 0 < b < 1. Then the left side of the above equation is sufficiently close to -1. But since H (b) > O, J(b) = Dgnm' + 1)n(b) > 0 and fob %H(s )h(s )fz"(s)s’ds < 0. Thus the equality above is impossible. Hence we conclude that n( 1) 2 0 for no sufficiently close to 1. 79 To prove the second statement, we note that u ’ Dbl p+q u(0)=11mu+v=lim- n =0 b—>0 b—>0 u for l > 1. Since u fin _ -( u )I + ’0 fl, + anflqbl+l Dan _ Dan u u’ u upnp‘ln’ , annqbl“ : _ _ _ _ , 2.103 _Dan Db2nP Don?!” + v u ( l and lim u’ _ u __u"(0) b—+0 Dan 00%? _ 2Dn“? Thus ifp = 0, l > 1, we have a”(0) < 0. Note that there exists 6 > 0, such that I (22+L — i) >3 and 3)- < LEI <21) on [0,6]. Db D02 2 We claim that n" < 0 on{b E [0, 6] : n > 0}. If not, El 61 6 (0,6) such that a (61) = 0, H(b) < 0 on (0, 6,). Thus if we let We have [(61) = 0. 80 On the other hand, u~, an(0) I> —§n(0)— ”/2 6’ > 0 for 73(0) << 1. This ends the proof of the lemma. Remark. Note in the proof of this lemma, we added the restriction on p and I such that p = 0, l > 1. We expect the result to be true for more general p and I. For the rest of this thesis, keep in mind that p = 0 even though our argument as follows does not necessarily need this condition and thus we shall still keep this notation. Lemma 2.15 Let n1, in be the solutions corresponding two different initial value 7110 and noo with ”10 > n2o, then n1 > no on [0,min{T(n1o),T(n2o)}] where T(n,o) represent the corresponding maximal existence interval in the sense that n(T) = 0. Proof. Suppose otherwise there is a bo E [0, min{T(n1o), T(n2o)}] such that n1(bo) = n2(bo) and a, > no on [0, bo), then n((bo) < n’2(bo). We should have H(bo) ~' " u(bO) ~I ~ DbonP(bo)m(b0) + vn1(b0) v > DbonP(bo)n2(b0) + vn2(bo) v, (2.104) while 50 17“? 1+l b0 p~q 1+1 / _Dn M < f 27212:, (2.105) 0 u(s) 0 21(8) which is clearly a contradiction to (2.98). In view of the above Lemmas, we have established the following theorem Theorem 2.4 Problem P2 admits a unique solution. The solution it satisfies the fol- lowing property: ’Il lim sup 5 L, B—H‘ 1_ b —M g 71’ S 0. Proof. Since equation (2.90) is only degenerate at b = 1. The boundedness of n’ on [0, 1) follows from standard ODE theory. We shall study the boundedness at b = 1. To this purpose, we let n(b) ~ (1 — b)"t with a 2 1 as b —> 1. Let n(b) ~ (1 — b)5 as b —-> 1. In view of equation (2.90), we have an+q it ~ — as b -—> 1. 21 Thus as b —> 1, it satisfies q ~q (1W + m’ + 21-73- = 0. (2.106) _v ”Q Suppose that B < 1, matching the leading singular terms in the equation above, we have fl—1=Q(B—a)7 or _aq—l fi— q__1. This is clearly impossible since a 2 1. Thus we conclude fl 2 1, or equivalently, 77(1) exits. In fact, by matching the coefficients, one can show that ~ gL. lim sup b—>l' " b 82 Let W : 0;”,71’ and differentiate (2.85), we get 'll Db pW’)’ + on” — quHii/b’ — H100“ = 0. (2.107) ‘71 ( Also W(0) > 0, W(1) = 0. The maximum principle yields W > 0 in (0,1), i.e., n’ < 0 in (0,1). Note that to deduce W(1) = 0, we have applied the fact that 731(1) is bounded. By Lemma 2.14, we have the bound on 71(0). The uniqueness follows from Lemma 2.15. This ends the proof of this theorem. We shall now combine Lemma 2.15 with Theorem 2.4. For every n E V, we define (u, u) to be the solution of P1 and n by Theorem 2.4, and introduce the mapping T by Tnzn. Clearly, T maps V into itself, and its image lies in a compact subset of V (since -—M S 11' g 0). By the uniqueness part in Lemma 2.15, it also follows that T is continuous. Invoking the Schauder fixed point theorem, we conclude that there exists at least one fixed point for T. We shall denote it by (v‘, u, n). To show that the corresponding (22*, b, n) is a solution to problem P2, we shall only need to show equality (2.89). In fact, we do have the following conclusion. 1 k p Lemma 2.16 / mdb 2 —oo. 0 u(b) Proof. We shall start with the following facts. There exists a constant L > 0 such that n(b) 2 1/L(1 - b) and a constant C2 < 0 such that u(b) Z C2(1 — b)”+q where (u, n) is the pair in the fixed point stated above. The first. fact is trivial. The second 83 Figure 2.7. Sketch of n fact follows from Lemma 2.11. Thus lobknv 1 Db"(1/L)P(1— b)? ___ l. mom’s/o Cam—b)?” " Remark. Note that we kept k and p here and it is true for general k and p as long aqul. 2.10 Instability of Flat Front In this section, we carry out the linear stability analysis of the traveling wave front. A similar analysis on a nondegenerate reaction diffusion system can be found in the papers by Horvath et al.[36, 37]. They introduced a small spatial perturbation of the 84 traveling wave solution. Upon linearization of the system, they obtained a dispersion relation. Using numerical method, they obtained a detailed diagram of the dispersion relation and explained the instability phenomenon. Because of degeneracy at the flat wave front, we not only perturb the wave front but also the bacteria and nutrient function at the front. The system we will analyze takes the form 8b a = DAbk+1+nb, (2.108) 987% = An—nb. (2.109) For the convenience of analysis, we introduce the perturbed coordinate: n = € + 66332900 c08(99) in terms of which the bacteria and nutrient density can be expressed as b(n, y, t) = bo(77) + 6b1(n)e$p(x\t)COS(91/), "(0. y, t) = n0(0) + en1(n)exp(lt) 608(99)- Here 5 is the traveling wave coordinate, bo and no are the traveling wave solutions. n, = (no),,(—vo + eAexp(/\t) cos(qy)) + 6(n1),,(—uo + 6Ae$p()\t) cos(qy))ea:p()\t) cos(qy) + enlAexp(/\t) cos(qy), ”xx 2 (”ohm + 6(71'1)nne$p()‘t) COS(qy), 85 " (710),,(-69)BIPW)8in(q;I/) + 6(n1)n(—6q)mp(/\t) sin(qz/)6IP(M) 608019) + €7116$I)(’\t)(-q)3in(qy)v (um—momma cos + (no.4—«1)2m:p<2A0(sin (1y)? +6nlexp(/\t)(—q2) cos(qy) + 0(62). Substituting the above expressions and Equalizing the 6 terms in equation (2.109), we obtain (no),,/\ — 0007.1)" + n1/\ 2 (711),],7 — Q2010)" — Q2721 — Tlobl — nlbo. (2.110) Similarly we calculate b. = (b0)n(—Uo + dam/V) c08((19) + 6(b1)n (——vo + cAexp(At) cos(qy))e:rp()\t) cos(qy) + eblAe$p(At) cos(qy) (W1). = (k + 1)b"((bo)n + e(b1)nexp(x\t) cos(qy)) (bk-Him = (k + 1)kbk—l((b0)n + 6(b1)nexp()\t) C05(‘ly))2 +(k + 1)bl‘((bo),m + 6(b1),,,,e:z:p()\t) cos(qy) (bk+l)y = (k + 1)b'°[(bo)n(-é<1)e$p(/\t) Sin(qy) + 6(b1)n(-6q)e$p(/\t) 8in(qy)el‘p(/\t) C08(99) + 6b1(-q)ervp(/\t) sin(qy)l (bk+l)yy = (k + 1)b’°[(bo),,(—£q)e2:p(/\t) sin(qy) + 6(b1)n(—cq)e:rp()\t)sin(qy)exp(/\t) 008(99) + 61)i(-q)e:1310(/\t)sin(€19)l2 + (k + llb"[(bo)nn(€q)2 eXemt) si1120131) +(bo).,(-692) expW) C08(99) + ebl exp(At)(-q2) C08(99) + 0(8)] (2-111) Substituting the above expressions into equation (2.108) and equalizing the e 86 terms, we have (bo),,A—vo(bl).+bv\ = (k+ aka—1112:; 212100): +2kb§ 10050). +(k+ +(k +1)(b0)kl-qz(b0)n “ q2bll + nibo + nobl- 1)kb§‘1b1(bo),,,, + (k +1)(b0)k(b1)mz (2.112) Using equations (2.110) and (2.112) we can linearize the original system (2.108)— (2.109) into L11 L12 b1 L21 L22 n1 where Ln = L12 = L21 = L22 = 82 Df’— b0) —n03 62 an _2 +110— /\+Df'q2 O 0 /\+q2 , 6 Db?(f)+vo— +710 30 02 .. Bbo an 8 +(2Df D7 +2120)??? Bbo fl" ”H2 — 2+ 710, f(— 3 —b. 8n 0 Here f 2: b3“ and the differentiation is with respect to bo. Similarly, if we have the following system: bt: D V (n v bk+1)+ nb n¢=An—nb 87 ”I + (”(07) (2.113) (2.114) (2.115) (2.116) (2.117) (2.118) After some tedious calculation, we arrive at the following linearized system: I:11 L12 £21 £22 Now if we consider earized system: L11 L21 This implies that b A+D ’ 2 0 b + b 1 = 71on 1 ( Ola 2119) n1 0 A + q2 n1 + (no),7 02 , B = Dnoa— n2 0 we have _ 9 3 02711 877.1 2 2 Ono 7977441077) -(/\+q )n1_(’\+q )5; which can be solved. In fact, n1 = (1 — Co)voe$P(_’Uo'7) + d0€$P(-’W77)- . -‘/12 /\ 2 . . . . . w1th w = ”0 10:“ +q ) and do 18 to be determmed usmg the boundary condltlon at 77 = 0. Since n1 and its derivative are continuous at n = 0, we have n1(0) = (1 — co)vo + do, and 0,,n1 = —(1 — co)u§ — dow. Since we require that bl/bo is bounded, we may assume that at 77 = 0 b1 vanishes as bl = B ( ‘77)3- By a straightforward calculation we obtain B = A (Eloy/k _k_voD and 89 Hence b A b—;—>-k—U for 7)—>0 which is finite. The boundary conditions at n —> —00 are given by b1 —) 0, anbl —) 0, n1 —) 0, 6,,111 ——> 0. Note A is small when q is small. Also n1 2 O and bl = 0 for q = 0 and so for small q, [)1 and m are both of order q2 too. This implies that in equation (2.113) for q small, the terms on the right hand side involving b1 and n1 are of order q“. Hence to order q2, we have L11 L12 b1 2 /\ + ID‘f’q2 0 (bo),, (2125) L21 L22 n1 0 /\ + (12 (n0)n Let L“ be the adjoint Operator of L which is obtained from °° (151 °° .. zPl / (la/114%”! =/ (¢1:¢2)L (2126) *°° (b2 “°° $2 Let z/q and 1,122 be the eigenvector of I 62 6 L“ = Df 7’77" _ ”00—" +120 1”" (2.127) b0 567,3 U061" — b0 90 Hence by the definition of L‘, we have °° b1 0° 7491 f (21,222» =/ (b1,b2)L* 00 711 00 $2 Hence oo A + D f’q2 0 in / (101402” an = 0 ”°° O /\ + q2 679'? from which we obtain the following relation +00 abo 8710 _ 2 +00 , abo 6710 A/wo ($18—77 +1/12—g) — ‘4 [_00 (lef 5-1; “228—77) So when q is small, we have dA II agile—0 ‘T with +°° , 0b 8n 11=‘/—00 (Di/hf .5; +KZ’28—no) and +°° 8b 6 I2 =/ (#2119779 + $201170). 00 (2.128) (2.129) (2.130) (2.131) (2.132) (2.133) From this dispersion relation, one may be able to conclude that for fix k, there exists a threshold Dk such that the flat interface is unstable for small q if D < Dk and it is stable for all q if D > Dk. 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