##
**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 \(\Omega =(0,1)\).

\[ (1.2)\quad D\partial u/\partial n+\theta E(x)u=0,\quad x\in \partial \Omega \] 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 |

### Keywords:

Lipschitz continuous; compact attractor; upper semicontinuous; singleton; Dirichlet problem; bifurcation
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\textit{J. K. Hale} and \textit{C. Rocha}, Nonlinear Anal., Theory Methods Appl. 11, 633--649 (1987; Zbl 0661.35047)

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