Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain
Let \(\Omega\) be a two-dimensional domain. This paper deals with the simplified system \[ v_t=\varepsilon^2\Delta v-v+uv^2,\quad \varepsilon^2u_t=D\Delta u+a-\varepsilon^{-2}uv^2, \qquad x\in\Omega, \tag{1} \] associated with the homogeneous Neumann boundary conditions on the boundary \(\partial\Omega\), where \(0<\varepsilon\ll1\), \(D>0\), \(a>0\), are parameters. The aim of the authors is to study the behavior of spot-patterns for (1), as \(\varepsilon\) tends to zero. The one-spot solutions and the multi-spot-solutions are studied separately. In each case, a quasi-equilibrium solution is constructed, by means of matched asymptotic expansions , and its stability is investigated numerically. Some properties of the corresponding Green's functions, for the unit disk and for the rectangle, are given. Several experiments, starting with one-spot or multi-spot quasi-equilibrium solutions, in a square or in the unit disk, are presented. Some open problems are listed at the end of the paper.
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