Stabilization of solutions of the filtration equation with absorption and nonlinear flux (Q1895855)

\textit{N. D. Alikakos} and \textit{R. Rostamian} [Indiana Univ. Math. J. 30, 749-785 (1981; Zbl 0598.76100)] showed that solutions of the problem \[ u_t=\Delta\phi(u)\text{ in } \Omega\times(0,\infty),\;-{\partial\over \partial n} \phi(u)\in \beta(u)\text{ on } \partial\Omega\times (0,\infty) \] (with suitable initial data) converge in \(L^1\) to a constant as \(t\to\infty\) under general conditions on the function \(\phi\) and the maximal monotone graph \(\beta\). In this paper, the same result of convergence to a constant is shown if the differential equation is perturbed to \(u_t=\Delta\phi(u)-\varphi(x,u)\) for a function \(\varphi\). The hypotheses on \(\phi\) and \(\beta\) are essentially those of Alikakos and Rostamian: \(\phi\) is a continuous increasing function with \(\phi(0)=0\) and \(\beta\) is a maximal monotone graph with \(0\in \beta(0)\). The proof of convergence to a constant, which uses nonlinear semigroup theory, assumes of \(\varphi\) only that it is Carathéodory and nondecreasing in its second argument with \(\varphi(x,0)=0\) for almost all \(x\in\Omega\).