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Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability. (English) Zbl 0981.35026

In this paper the pair of coupled reaction-diffusion equations $\alpha {v}_{t}={\epsilon }^{2}{\Delta }v-v+Au{v}^{2}$, ${u}_{t}={\Delta }u-uv+\left(1-u\right)$ in ${ℝ}^{2}×ℝ$ is considered, where $\alpha$, $\epsilon$, and $A$ are parameters, $0<\alpha \le 1$, $0<\epsilon \ll 1$. The above system represents a model of chemical reaction $U+2V\to 3V$, $V\to P$ in gel reactor, where $U$ and $V$ are two chemical species, $V$ catalyzes its own reaction with $U$ and $P$ an inert product. The author first constructs two single-spot solutions and then investigates their stability and instability in terms of the parameters involved. The characteristic parameters $L$ and ${L}_{0}$ are defined in the following way: let $w$ be (unique) radially symmetric solution to the problem

${\Delta }w-w+{w}^{2}=0,\phantom{\rule{1.em}{0ex}}w>0,\phantom{\rule{1.em}{0ex}}w\left(0\right)=\underset{y\in {ℝ}^{2}}{max}w\left(y\right),\phantom{\rule{1.em}{0ex}}w\left(y\right)\to 0,\phantom{\rule{1.em}{0ex}}|y|\to \infty ,$

and

$L=\left(1/2\pi {A}^{2}\right){\epsilon }^{2}log\left(1/\epsilon \right){\int }_{{ℝ}^{2}}{w}^{2}\left(y\right)dy,{L}_{0}=\underset{\epsilon \to 0}{lim}L·$

Roughly speaking, the basic result can be described as follows: if $1/log\left(1/\epsilon \right)\ll L$ and ${L}_{0}<1/4$, then the system has two single-spot solutions; if ${L}_{0}>1/4$, then there are no single-spot solutions.

In the case $\alpha \sim {\epsilon }^{\gamma }$, $0\le \gamma <2$ linear instability of single-spot solutions can be described in terms of the parameters $\gamma$ and ${L}_{0}$.

##### MSC:
 35K57 Reaction-diffusion equations 35B25 Singular perturbations (PDE) 35B35 Stability of solutions of PDE 35B10 Periodic solutions of PDE 35J40 Higher order elliptic equations, boundary value problems 35Q80 Appl. of PDE in areas other than physics (MSC2000) 92E20 Classical flows, reactions, etc.
self-replication