Pattern formations in two-dimensional Gray–Scott model: existence of single-spot solutions and their stability

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Abstract

The Gray–Scott model has been studied extensively in recent years. However, most of the results are in one dimension. In this paper, we study the 2D Gray–Scott system. We first construct two single-spot solutions and then we establish the stability and instability of such solutions in terms of the parameters involved.

Introduction

Recently, there is a great interest in the study of self-replicating patterns observed in the irreversible Gray–Scott model [14], [15], [16] which governs the chemical reactions U+2V→3V and VP in a gel reactor, where U and V are two chemical species, V catalyzes its own reaction with U and P an inert product. Letting U=U(x,t) and V=V(x,t) denote the concentrations of U and V, respectively, the pair of coupled reaction–diffusion equations governing these reactions isVt=DVΔV−(F+k)V+UV2,Ut=DUΔU−UV2+F(1−U),x∈RN,where F denotes the rate at which U is fed from the reservoir into the reactor, the concentration of V in the reservoir is assumed to be zero, and k the rate at which V is converted to an inert product. Here the diffusivities, DU and DV, of U and V, respectively, can be any chemically relevant positive numbers.

In 1993, Pearson [37] did some numerical simulations on the Gray–Scott model in a square of size 2.5 in R2 with periodic boundary conditions. By choosing DU=2×10−5, DV=10−5 and varying the parameters F and k, several interesting patterns were discovered. One of them is that the spot may self-replicate in a self-sustaining fashion and develop into a variety of time-dependent and time-independent asymptotic states. Lin et al. [28] reported their experiments in a ferro-cyanide–iodate–sulfite reaction which showed strong qualitative agreement with the self-replication regimes in simulations of [37]. Moreover, those same experiments led to the discovery of other new patterns, such as annular patterns emerging from circular spots. See [27] for more details on the set-up. (However, no stable single-spot was observed.) In 1D, numerical simulations were done by Reynolds et al. [38], [39], independently by Petrov et al. [36], and again self-replication phenomena were observed. However, in 1D, self-replication patterns were observed when DU=1, DV=δ2=0.01. Some formal asymptotics and dynamics in 1D were contained in [36], [38]. Recent numerical simulations of [9] in 1D and [29], [33] in 2D show that the single spot may be stable in some very narrow parameter regimes.

The first rigorous result in constructing a single-spot solution was due to Doelman et al. [9] in 1997. In [9], by using Mel’nikov method, Doelman et al. rigorously constructed single- and multiple-pulse solutions for (1.1) in the case N=1, DU=1, DV=δ2≪1. In their paper [9], it is assumed that Fδ2, F+kδ2α/3, where α∈[0,32). In this case, they showed that U=O(δα), V=O(δα/3). Later, the stability of single- and multi-pulse solutions in 1D are obtained in [6], [7], [8]. Hale et al. [21], [22] studied the case DU=DV in 1D and the existence of single- and multiple-pulse solutions are established. Nishiura and Ueyama [33] proposed a skeleton structure of self-replicating dynamics. Some related results on the existence and stability of solutions to the Gray–Scott model in 1D can be found in [11], [12], [38].

In higher dimension, as far as the author knows, there are no rigorous result on the existence or stability of spotty solutions for (1.1). Since self-replicating patterns were first observed in R2 [37], it is important and interesting to have a rigorous study on spotty solutions in R2. We remark that in R2 and R3, Muratov and Osipov [29] have given some formal asymptotic analysis on the construction and stability of spot solution. In [47], the author studied (1.1) in a bounded domain for the shadow system case, namely, DU≫1, DV≪1 and F=O(1), F+k=O(1). The shadow system can be reduced to a single equation. For spot solutions for single equations, see [1], [2], [4], [13], [17], [19], [20], [24], [25], [30], [31], [32], [34], [35], [40], [41], [42], [43], [44], [45], [46], [48], [49], and the references therein.

In this paper, we rigorously establish the existence and stability of single-spot solution in 2D. Let us first transform the system (1.1). We follow the notations in [29]. Setϵ2=DVFDU(F+k),A=FF+k,α=FF+k,x=DUFx̄,t=1Ft̄,V(x,t)=Fv(x̄,t̄),U(x,t)=u(x̄,t̄).Let us drop the bar from now on. It is easy to see that (1.1) is equivalent to the following:αvt2Δv−v+Auv2,ut=Δu−uv2+(1−u),x∈R2.Note that there are three parameters (α,ϵ,A) in Eq. (1.2). Throughout this paper, we always assume that0<ϵ≪1.By definition, we must haveα≤1.To study (1.2), we first consider the stationary equation of (1.2):ϵ2Δv−v+Auv2=0,Δu−uv2+(1−u)=0,x∈R2.From (1.1), it is also required that u and v are positive.

Before we state our results, let us first gain some insights into Eq. (1.5) and therefore introduce an important quantity. We are interested in radially symmetric solutions to (1.5). By (1.3), u is a slow variable and v a fast variable. Hence near 0, formally one can assume that u(x)∼u(0). Substituting that into equation for v, we have that v satisfiesϵ2Δv−v+Au(0)v2=0,v>0,v∈R2.By the following scaling:x=ϵy,v(x)=1Au(0)v̂(y),we see that satisfiesΔv̂v̂+v̂2=0,v̂>0,v∈R2.Problem (1.7) has been studied in [18], [23]. It was proved that there exists a unique radially symmetric solution to the following problem:Δw−w+w2=0,w>0,v∈R2,w(0)=maxy∈R2w(y),w(y)→0as∣y∣→+∞.

Remark 1.1

Unlike 1D case, the solution to (1.8) cannot be written explicitly.

Let us denote the unique solution of (1.8) by w(y)=w(∣y∣). Going back to the equation for v, we see thatv(x)∼1Au(0)w∣x∣ϵ.From the equation for u, we have that1−u(0)=R2K(∣z∣)u(z)v2(z)dz=1A2u2(0)R2K(∣z∣)u(z)w2∣z∣ϵdz,where K(∣z∣) is the fundamental solution of −Δ+1 in R2.

It is well known thatK(∣z∣)=−1log∣z∣+O(1)+O(∣z∣)for∣z∣<1.By , , we obtain1−u(0)∼ϵ22πA2u(0)log1ϵR2w2(y)dy,which suggests that the following quantity play a very important role:L≔12πA2ϵ2log1ϵR2w2(y)dy.Thus formally, we have by the definition of L in (1.13)1−u(0)=(1+o(1))Lu(0).Eq. (1.14) admits a positive solution for u(0) if and only if L<14. It is easy to see that for each L<14, (1.14) admits two solutionsu(0)=ξϵ112(1−1−4(1+o(1))L)<12,u(0)=ξϵ212(1+1−4(1+o(1))L)>12.Thus formally, we should have at least two solutions — one is small and another is large. This is exactly what we will prove in this paper.

Section snippets

Main results: existence and stability

We now state our main results in this paper. The existence result will be stated in terms of the quantity L introduced in (1.13) and the stability result will involve both L and α. Note that both L and α may depend on ϵ.

LetL0limϵ→0L.Our first result is on the existence of single-spot solution to (1.5).

Theorem 2.1

Assume that1log(1/ϵ)≪L,L0<14.Then problem (1.5) admits two solutions (vϵi,uϵi), i=1,2 with the following properties:

  • 1.

    vϵi,uϵi are radially symmetric functions.

  • 2.

    vϵi=(1/Auϵi(0))(1+o(1))w(∣x∣/ϵ),i=1,2,

Outline of proofs and two technical lemmas

We prove Theorem 2.1 by the following steps.

Step 3.1

We rewrite the system (1.5) as a single equation with a nonlocal term (rescaled):S[v]≔ϵ2Δv−v+1ξϵAT[v]v2=0,∈R2,where T[v] solves the following equation:ΔT[v]+1−T[v]−1A2ξϵ2T[v]v2=0,∈R2,where ξϵ=ξϵ1 or ξϵ2. We will show that T is a well-defined operator.

Step 3.2

We construct solutions of (3.1) of the following form:x=ϵy,v(x)=w(y)+φϵ(y),where φϵ is suitably small in some Sobolev spaces.

To this end, it is important to study the linearized operatorLϵφ=S′[w](φ).As ϵ

Proof of Theorem 2.1

In this section, we prove Theorem 2.1. We construct two single-spot solutions in R2 for the following Gray–Scott system:ϵ2Δv−v+Auv2=0,Δu+(1−u)−uv2=0,x∈R2,u=u(∣x∣),v=v(∣x∣),v>0,0<u<1,v(x)→0,u(x)→1as∣x∣→+∞.First we rescale Eq. (4.1). Recall that there are two solutions ξϵ1<ξϵ2 to Eq. (1.14). Moreover, it is important to note that by (2.2),limϵ→0(1−ξϵi)≠12,i=1,2.From now on, we take ξϵξϵ1. The other case can be treated similarly. We rescalev≔1ϵv̄.Then (v̄,u) will satisfyϵ2Δv̄v̄+uξϵ(v̄)2=0,

Spectrum analysis I: reduction to NLEP and large eigenvalue estimates

Let (vϵi,uϵi),i=1,2 be the two solutions constructed in Section 2. Without loss of generality, we use (vϵ,uϵ) to denote any one of the solutions. We now study the eigenvalue problem associated with (vϵ,uϵ). We assume that α=α0ϵγ for some 0≤γ<2 and positive number α0.

We need to analyze the following eigenvalue problem (letting x=ϵy):ΔyΦϵ−Φϵ+AΨϵvϵ2+2AuϵvϵΦϵ=αλϵΦϵ,ΔxΨϵ−Ψϵ−Ψϵvϵ2−2uϵvϵΦϵϵΨϵ,x,y∈R2,λϵC.Let vϵ=(1/Aξϵ)v̄ϵ, λ̄ϵ=αλϵ. Problem (5.1) becomesΔyΦϵ−Φϵ+1ϵ2Ψϵv̄ϵ2+21ξϵuv̄ϵΦϵ=λ̄ϵΦϵ,ΔxΨϵ−Ψϵ1A2

Spectrum analysis II: study of small eigenvalues

In this section, we discuss Case II: λ̄ϵ→0 and finish Proof of Theorem 2.2(3). In this case, we need to show that λ̄ϵ=0 and the multiplicity is exactly 2. We only need to consider (vϵ,uϵ)=(vϵ1,uϵ1).

It is easy to see that Φi≔Aξϵ(∂vϵ/∂xi,∂uϵ/∂xi)=(∂v̄ϵ/∂xi,Aξϵ(∂uϵ/∂xi)),i=1,2 are solutions of (5.1) with λϵ=0. (Here we use x=(x1,x2)∈R2, y=x/ϵ=(y1,y2)∈R2.) Furthermore, since (vϵ,uϵ) are radially symmetric functions, we have that Φ1Φ2 in L2(R2)⊕L2(R2). Here, we equip L2(R2)⊕L2(R2) with the

Some discussions

There are several interesting problems left.

  • 1.

    When ϵ2α≪O(1) andL0<141−γ4−γ2from Proof of Theorem 2.2, we can actually prove that all unstable eigenvalues for (vϵ1,uϵ1) must satisfy that λ̄ϵ=αλϵ=O(τ) (see (6.24)). That is, there are only small eigenvalues. (For large eigenvalues, by , , μ0=limϵ→0μϵ>1. So they are all stable by Theorem 5.1(1).) We can also decompose (Φϵ,Ψϵ) as in , . However, we cannot show that (Φϵ,Ψϵ)=(0,0) (one can see this from (6.24)). This means that the unstable modes, if

Acknowledgements

This research is supported by an earmarked research grant from RGC of Hong Kong. The author would like to thank Prof. S.-I. Ei, Prof. Y. Nishiura, and Prof. M.J. Ward for useful discussions. I thank the referees for their valuable remarks and suggestions which greatly improved the presentation of the paper.

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