We study the Gray-Scott model in a  bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives
together with two nonlocal eigenvalue problems
both play a major role in the analysis.
For symmetric spots, we establish a threshold behavior for stability:
If a certain inequality for the parameters holds
then we get stability, otherwise we get instability of multiple spot solutions.
For asymmetric spots, we show that they can be stable within a narrow parameter range

Wei, J

Winter, M

English

Brunel University Research Archive

EXISTENCE AND STABILITY OF MULTIPLE SPOT SOLUTIONS FOR THEGRAY-SCOTT MODEL IN R2JUNCHENG WEI AND MATTHIAS WINTERAbstract. We study the Gray-Scott model in a bounded two dimensional domain and estab-lish the existence and stability of symmetric and asymmetric multiple spotty patterns. TheGreen’s function and its derivatives together with two nonlocal eigenvalue problems both play amajor role in the analysis. For symmetric spots, we establish a threshold behavior for stability: Ifa certain inequality for the parameters holds then we get stability, otherwise we get instability ofmultiple spot solutions. For asymmetric spots, we show that they can be stable within a narrowparameter range.1. Introduction: Self-replicating spotsWe study the existence and stability of multiple spotty patterns in the two-dimensional Gray-Scott model. The Gray-Scott system, introduced in [9], [10], models an irreversible reactioninvolving two reactants in a gel reactor, where the reactor is maintained in contact with a reservoirof one of the two chemicals in the recation. In nondimensional variables, it can be written as(GS)Vt = DV∆V − (F + k)V + UV 2, x ∈ Ω, t > 0Ut = DU∆U − UV 2 + F (1− U), x ∈ Ω, t > 0∂V∂t =∂U∂t = 0 on ∂Ω,where DU > 0, DU > 0 are the two diffusivities, F denotes the feed rate, k > 0 is a reaction-timeconstant, and Ω ⊂ RN , N ≤ 3 is the container. For various ranges of these parameters, (GS) areknown to admit a rich solution structure involving pulse, spots,, rings, stripes, traveling waves,pulse-replication pattern, and spatio-temporal chaos. See [21], [22], [23], [14], [15] for numericalsimulations and experimental observations.Some important analytic work is the following, first for the case of 1-D: single and multiple pulsesolutions [7], stability [5], [6], stability index [4], slowly modulated two-pulse solutions [2], [3],dynamics of pulses (formal) [22], [23], skeleton structure, spatiotemporal chaos [18], dynamics [8],case of equal diffusivities [11], [12], scattering and separators [19], [20], symmetric and asymmetricpatterns, Hopf bifurcation, pulse-splitting in a bounded interval [24].In higher dimensions there are the following results: formal approach in 2-D and 3-D [16],shadow system in higher dimensions [26], ground state in 2-D [27], bounded domain case in 2-D,symmetric and asymmetric multiple spots, which is the basis for this paper [28], [29].Let us first rescale the system (GS). Set²2 =DVF + k, D =DUF, A =√FF + k, τ =F + kF,x =√DUFx¯, t =1F + kt¯, V (x, t) =√Fv(x¯, t¯), U(x, t) = u(x¯, t¯).1991 Mathematics Subject Classification. Primary 35B40, 35B45; Secondary 35J40.Key words and phrases. Pattern formation, Self-replication, Spotty solutions, Reaction-diffusion systems.12 JUNCHENG WEI AND MATTHIAS WINTERLet us drop the bar from now on. Then (GS) is equivalent to(GS1)vt = ²2∆v − v +Auv2, x ∈ Ω, t > 0τut = D∆u− uv2 + (1− u), x ∈ Ω, t > 0∂u∂ν =∂v∂ν = 0 on ∂Ω.Throughout this paper, we assume that ² << 1, D = D(²)→∞ as ²→ 0, A > 0 may dependon ², τ > 1 does not depend on ², Ω ⊂ R2 is a bounded and smooth domain.We define two important parameters:η² =|Ω|2piDlog√|Ω|², L² =²2∫R2 w2 dyA2|Ω| .Let us assume thatlim²→0L² = L0 ∈ [0,+∞], lim²→0 η² = η0 ∈ [0,+∞].All our results will be stated in terms of (the real constants) L0, η0, and τ .Let w be the unique solution of the following problem (ground state):(G) ∆w − w + w2 = 0, w > 0 in R2, w(0) = maxy∈R2w(y), w(y)→ 0 as |y| → +∞.The uniqueness of the solution to (G) was proved [13]. We shall prove the existence and stabilityof steady-state solutions to (GS1) of the following shape:v² ∼K∑j=11Aξ²jw(x− P ²j²), u²(P ²j ) ∼ ξ²j ,where K is the number of spots, P ²j ∈ Ω, j = 1, ...,K is the location of the spots, 1Aξ²j , j = 1, ...,Kis the amplitude of the spots, w is the shape of the spots.We call the steady state “K− symmetric spots” if the amplitudes of the spots are asymp-totically the same in the leading order, i.e.,lim²→0ξ²iξ²1= 1, for all i = 2, ..., N.Otherwise, we call it “K−asymmetric spots”, i.e., iflim²→0ξ²iξ²16= 1, for some i = 2, ..., N.2. Existence and Stability of Multiple Symmetric SpotsThe result on the existence of multiple symmetric spots is the followingTheorem 2.1. Suppose that(T1) 4(η0 +K)L0 < 1and(T2)(2η0 +K)2η0L0 6= 1.Then, for ² sufficiently small and D not too large, problem (GS1) has two steady-state solutions(v±² , u±² ) with the following properties:(1) v±² (x) =∑Kj=11Aξ±²(w(x−P ²j² ) + o(1)) uniformly for x ∈ Ω¯. Hereξ±² =1±√1− 4(η0 +K)L02.GRAY-SCOTT SYSTEM 3(2) u±² (x) = ξ±² (1 + o(1) uniformly for x ∈ Ω¯.(3) P ²j → P 0j as ²→ 0 for j = 1, ...,K.The locations of the spots are determined by using a Green’s function and its derivatives asfollows: Define G0(x, y) by∆G0(x, y)− 1|Ω| + δ(x− y) = 0, x ∈ Ω,∂G0(x, y)∂νx= 0, x ∈ ∂Ω,where y ∈ Ω. Set H0(x, y) = 12pi log 1|x−y| −G0(x, y).For K ∈ N and P = (P1, ..., PK) ∈ ΩK with Pj 6= Pl for j 6= l we defineF0(P1, ..., PK) =K∑i=1H0(Pi, Pi)−∑j 6=lG0(Pj , Pl).Then, if P0 = (P 01 , ..., P0K) is a nondegenerate critical point of F0, the solutions exist.We call (v−² , u−² ) the small solution and (v+² , u+² ) the large solution.Next we consider the stability of such solutionsTheorem 2.2. Assume that (T1) and (T2) hold. Let P0 = (P 01 , ..., P0K) ∈ ΩK be a nondegeneratelocal maximum point of F0. Let (v±² , u±² ) be the K−spot solutions constructed in Theorem 2.1.Then, for ² sufficiently small and D not too large, the solution (v+² , u+² ) is linearly unstablefor all τ ≥ 0. For the small solutions the following holds:Case 1. η² → 0.If K = 1, there exists a unique τ1 > 0 such that for τ < τ1, (v−² , u−² ) is linearly stable, whilefor τ > τ1, (v−² , u−² ) is linearly unstable.If K > 1, (u−² , v−² ) is linearly unstable for any τ ≥ 0.Case 2. η² → +∞.Then (v−² , u−² ) is linearly stable for any τ ≥ 0.Case 3. η² → η0 ∈ (0,+∞), (i.e., D ∼ log 1² ).If L0 < η0(2η0+K)2 , then (u−² , v−² ) is linearly stable for τ small enough or τ large enough.If K = 1, L0 > η0(2η0+1)2 , there exist τ2 > 0, τ3 > 0 such that (v−² , v−² ) is linearly stable forτ < τ2 and linearly unstable for τ > τ3.If K > 1 and L0 > η0(2η0+K)2 , then (v−² , u−² ) is linearly unstable for any τ ≥ 0 .3. Existence and Stability of K-Asymmetric SpotsOur first theorem shows that the asymmetric patterns exist only in a narrow parameterrange.Theorem 3.1 Asymmetric patterns can exist only if lim²→0 η² = η0 ∈ (0,+∞). In other words,D ∼ C log 1² .Our next theorem shows that the asymmetric patterns are generated by exactly two typesof spots.Theorem 3.2 The asymmetric solutions is generated by exactly two kinds of spots, called typeA and type B, respectively, which differ by their amplitudes.The following theorem gives the existence of K−asymmetric spots.4 JUNCHENG WEI AND MATTHIAS WINTERTheorem 3.3. Fix any two integers k1 ≥ 1, k2 ≥ 1 such that k1 + k2 = K ≥ 2. IfL0 ≤ η04(η0 + k1)(η0 + k2) ,there are asymmetric K−spotty solutions with k1 type A spots and k2 type B spots. The locationsof these K spots are determined by a Green’s function which depends on the number k1, k2 of thespots.The last theorem classifies the stability of asymmetric patternsTheorem 3.4.(1) (stability) The K-asymmetric spots are stable ifη0(2η0 +K)2< L0 ≤ η04(η0 + k1)(η0 + k2)and τ small.(2) (Instability) Assume thatL0 <η0(2η0 +K)2orτ is large enough.Then the K-asymmetric spots are unstable.4. Main Steps in the Existence ProofThe existence of K symmetric and asymmetric spots is obtained by the Liapunov-Schmidtreduction method.We calculate the equations for the amplitudes by solving the second equation of (GS2) witha suitable Green’s function and expanding this Green’s function (here G0 is needed). Assumingasymptotically thatlim²→0 ξ²,j = ξj , j = 1, ...,K,we obtain the following system of algebraic equations1− ξi − η0L0ξi=K∑j=1L0ξj, i = 1, ...,K.In the symmetric case, i.e., ξ1 = ... = ξK , we haveξ1 = ... = ξK = ξ,where ξ satisfies the quadratic equation1− ξ − η0L0ξ=KL0ξ.This gives two branches of solutions.In the asymmetric case, it follows by an elementary argument that (assuming w.l.o.g. thatξ2 6= ξ1) for ξj , j = 3, . . . ,K, we have either ξj = ξ1 or ξj = ξ2. This shows that asymmetricpatterns are generated by exactly two types of spots.Let k1 be the number of ξ1’s in {ξ1, . . . , ξK} and k2 the number of ξ2’s in {ξ1, . . . , ξK}. Thenξ1 must satisfy1− ξ1 = (k1 + η0)L0ξ1+k2η0ξ1and therefore(k2 + η0)ξ21 − η0ξ1 + (k1 + η0)η0L0 = 0,GRAY-SCOTT SYSTEM 5which has a solution if and only ifη0 ≥ 4(k1 + η0)(k2 + η0)L0.Thus we have determined the amplitudes of the spots. Now we have to glue the spots togetherin Ω. This is be done by the Liapunov-Schmidt reduction process and a fixed point theoremargument.5. Stability Proof: Systems of NLEPsTo study the stability of K-spots, we consider two cases: Large eigenvalues λ² → λ0 andsmall eigenvalues λ² → 0. The small eigenvalues are related to the domain geometry via theGreen’s function G0.The analysis of the large eigenvalues gives us the critical threshold.By some lengthy asymptotic analysis, we arrive at the following system of nonlocal eigen-value problems (NLEP-system):∆Φ− Φ+ 2wΦ− 2B∫R2 wΦ∫R2 w2w2 = λ0Φ, Φ =φ1φ2...φK ∈ (H2(R2))K ,whereB = L0(F + L01 + τλ0E)−1 (η0I + 11 + τλ0E),F = ξ21 + L0η0. . .ξ2K + L0η0 , E = 1 · · · 1... ... ...1 · · · 1 ,and I is the identity matrix.Diagonalizing the matrix B, we are lead to the study of the following single NLEPs :(NLEP) ∆φ− φ+ 2wφ− 2µi(τλ0)∫R2 wφ∫R2 w2w2 = λ0φ, i = 1, 2, φ ∈ H2(R2),where µ(z) is an analytic function of z = τλ0.We need a key result about (NLEP) from [25]: if τ = 0, then (NLEP) is stable if 2µi(0) > 1. 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Existence and stability of multiple spot solutions for the gray-scott model in R^2

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