Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions
This paper is concerned with the asymptotic dynamics of a reaction-diffusion equation in a bounded domain \(\Omega\subset\mathbb{R}\) with nonlinear Neumann boundary conditions: \[ \begin{cases} u^\varepsilon_t- \text{div}(p_\varepsilon(x)\nabla u^\varepsilon)+\lambda u^\varepsilon= f(u^\varepsilon) &\text{in }\Omega,\\ {\partial u^\varepsilon\over\partial\vec n}= g(u^\varepsilon) &\text{on }\partial\Omega,\\ u^\varepsilon(0)= u^\varepsilon_0.\end{cases}\tag{1} \] There is considered the case for which the diffusion coefficient \(p_\varepsilon\) blows up in \(\Omega_0\subset\Omega\) which is an open domain such that \(\overline\Omega_0\subset\Omega\) and \(\Omega_0= \bigcup^m_{i=1}\Omega_{0,i}\), where \(\Omega_{0,i}\) are smooth disjoint, connected subdomains of \(\Omega_0\) and \(\Omega_1= \Omega\setminus\overline\Omega_0\). More precisely, the coefficient \(p_\varepsilon\in C^1\) is assumed to satisfy \[ p_\varepsilon(x)\to \begin{cases} p(x)\text{ uniformly on }\Omega_1,\\ \infty\text{ uniformly on compact subsets of }\Omega_0,\end{cases} \] when \(\varepsilon\to 0\), where \(p\in C^1(\overline\Omega_1,[m_0,\infty))\) for some \(m_0> 0\). The continuity at \(\varepsilon= 0\) of the family of global attractors \(\{A_\varepsilon: a\in[0,\varepsilon_0]\}\) associated to the family problems (1) is proved. There is studied the upper semicontinuity and the lower semicontinuity of the attractors.
scientific article; zbMATH DE number 5563848