Stabilization for degenerate diffusion with absorption (Q1811789)

The present article is a continuation of the author's study [Commun. Partial Differ. Equations 26, 1385-1408 (2001; Zbl 1056.35096)], wherein the author was interested in the asymptotic behavior of solutions of initial-boundary value problems for the so-called filtration equation. The purpose of this article is to study the limit in \(L^1(\Omega)\) of solutions of general initial-boundary value problems of the form \[ u_t = \Delta w - g(x,u),\quad u\in \beta(w) \] in a bounded domain \(\Omega\) with general boundary conditions of the form \[ \partial_{\eta}w + \gamma(w) \ni 0, \] where \(\beta\) and \(\gamma\) are maximal monotone graphs and \(g:\Omega\times \mathbb{R}\to R\) is a nonincreasing continuous function in \(r\in \mathbb{R}\). The author proves that a solution stabilizes in \(L^1(\Omega)\) as \(t\to\infty\) to a function \(\underline u\in L^1(\Omega)\) which satisfies \(\underline u(x) \in \varphi^{-1}(c)\cap g(x,\cdot)^{-1}(0)\) a.e. \(x\in\Omega\), with \(c\in\gamma^{-1}(0)\).