Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data
This paper is devoted to the asymptotic of the solutions of the following reaction-diffusion equation with nonlinear boundary conditions, that is \[ \begin{cases} u_t-d\Delta u+f(u)=0\quad &\text{in } \Omega\\ d\frac{\partial u}{\partial n}+g(u)=0\quad &\text{on } \Gamma=\partial\Omega\\ u(0,x)=u_0(x),\end{cases}\tag{1} \] where \(\Omega\subset\mathbb{R}^N\), \(N\geq 1\), is a bounded smooth set and \(d>0\). The author deals with singular initial data, that is \(u_0\in L^p(\Omega)\), \(1<p<\infty\) or \(u_0\in W^{1,p}(\Omega)\), \(1<p<N\) and special attention is given to the case that the nonlinearities grow critically. The author's main goal is to give conditions on the nonlinear terms such that (1) is globally well-posed and moreover has a global compact attractor.
scientific article; zbMATH DE number 1764712
scientific article