## Interaction of diffusion and boundary conditions.(English)Zbl 0661.35047

The authors consider the system of parabolic partial differential equations $(1.1)\quad \partial u/\partial t=D\Delta u+f(u),\quad x\in \Omega$
$(1.2)\quad D\partial u/\partial n+\theta E(x)u=0,\quad x\in \partial \Omega$ where $$u\in {\mathbb{R}}^ N$$, $$\Omega \in {\mathbb{R}}^ n$$, $$n\geq 3$$, is a bounded open set with $$\partial \Omega$$ smooth, $$D=diag(d_ 1,d_ 2,\cdot \cdot \cdot,d_ N),$$ $$E=diag(e_ 1,e_ 2,\cdot \cdot \cdot,e_ N),$$ each $$d_ j>0$$ is constant, $$e_ j: \partial \Omega \to {\mathbb{R}}$$ is positive and continuous and $$\theta\in [0,\infty)$$ is constant. The function f: $${\mathbb{R}}^ N\to {\mathbb{R}}^ N$$ is supposed to be continuous and have a Lipschitz continuous first derivative.
The purpose of this paper is to make a modest contribution to understanding some parts of this problem. More specifically, they give some conditions on (f,E) which ensure that there is a $$d_ 0>0$$ such that, for any $$d\geq d_ 0$$, $$d=\min \{d_ j,\quad j=1,2,\cdot \cdot \cdot,N\}$$ and any $$\theta\in [0,\infty)$$, there is a compact attractor of (1.1), (1.2) which is upper semicontinuous in D, $$\theta$$ uniformly for $$d\geq d_ 0$$, $$\theta\geq 0$$. Furthermore, this attractor is a singleton for $$\theta \geq \theta_ 0$$, $$\theta_ 0$$ sufficiently large and converges to an attractor for the Dirichlet problem for (1.1). The second aspect of the paper deals with the classification of points in (D,$$\theta)$$-space as structurally stable or bifurcation points. The proof is given by reducing single equation and $$n=1$$ with $$\Omega =(0,1)$$.
Reviewer: Y.Ebihara

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B32 Bifurcations in context of PDEs 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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