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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 $\alpha v_t= \varepsilon^2\Delta v- v+Au v^2$, $u_t= \Delta u- uv+ (1- u)$ in $\bbfR^2\times \bbfR$ is considered, where $\alpha$, $\varepsilon$, and $A$ are parameters, $0< \alpha\le 1$, $0<\varepsilon\ll 1$. 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 $$\Delta w- w+ w^2= 0,\quad w>0,\quad w(0)=\max_{y\in\bbfR^2} w(y),\quad w(y)\to 0,\quad|y|\to \infty,$$ and $$L= (1/2\pi A^2)\varepsilon^2 \log(1/\varepsilon) \int_{\bbfR^2} w^2(y) dy, L_0= \lim_{\varepsilon\to 0} L.$$ Roughly speaking, the basic result can be described as follows: if $1/\log(1/\varepsilon)\ll L$ 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 $\alpha\sim \varepsilon^\gamma$, $0\le \gamma<2$ linear instability of single-spot solutions can be described in terms of the parameters $\gamma$ and $L_0$.

35K57Reaction-diffusion equations
35B25Singular perturbations (PDE)
35B35Stability of solutions of PDE
35B10Periodic solutions of PDE
35J40Higher order elliptic equations, boundary value problems
35Q80Applications of PDE in areas other than physics (MSC2000)
92E20Classical flows, reactions, etc.
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