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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 \(\mathbb{R}^2\times \mathbb{R}\) is considered, where \(\alpha\), \(\varepsilon\), and \(A\) are parameters, \(0< \alpha\leq 1\), \(0<\varepsilon\ll 1\). 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 \[ \Delta w- w+ w^2= 0,\quad w>0,\quad w(0)=\max_{y\in\mathbb{R}^2} w(y),\quad w(y)\to 0,\quad|y|\to \infty, \] and \[ L= (1/2\pi A^2)\varepsilon^2 \log(1/\varepsilon) \int_{\mathbb{R}^2} w^2(y) dy, L_0= \lim_{\varepsilon\to 0} L. \] 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 \(\alpha\sim \varepsilon^\gamma\), \(0\leq \gamma<2\) 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 in context of PDEs
35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
35J40 Boundary value problems for higher-order elliptic equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
92E20 Classical flows, reactions, etc. in chemistry
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