Pucci, Patrizia; Serrin, James Asymptotic stability for nonlinear parabolic systems. (English) Zbl 0874.35056 Antontsev, S. N. (ed.) et al., Energy methods in continuum mechanics. Proceedings of the workshop on energy methods for free boundary problems in continuum mechanics, Oviedo, Spain, March 21–23, 1994. Dordrecht: Kluwer Academic Publishers. 66-74 (1996). The problem of asymptotic stability of the rest state \(u=0\) for the parabolic system \[ A(t)|u_t|^m u_t=\Delta u -f(x,u), \quad u\in\mathbb{R}^N, \quad (t,x)\in(0,\infty)\times\Omega, \] is studied. Here \(A(t)\) is an \(N\times N\) positive matrix, \(m>1\) and \(f\) is a nonlinearity satisfying certain growth conditions. The main result is that the rest state is asymptotically stable provided that there exists a nonnegative function \(k\not\in L^1(0,\infty)\) such that \[ \liminf_{t\to\infty} \int_0^t \sigma(s)k^m(s) ds \left(\int_0^tk(s) ds\right)^{-m} <\infty \quad\text{where}\quad \sigma(t)=|A(t)|^m \left(\sup_{|v|=1} (A(t) v,v)\right)^{1-m} . \] In particular, if \(A(t)=t^\alpha Id\) the rest state is stable if \(\alpha\leq 1\) whereas in the case \(\alpha>1\) solutions converging to a nontrivial steady state may exist.For the entire collection see [Zbl 0855.00021]. Reviewer: D.Ševčovič (Bratislava) Cited in 16 Documents MSC: 35K65 Degenerate parabolic equations 35K45 Initial value problems for second-order parabolic systems Keywords:degenerate parabolic systems; asymptotic stability PDFBibTeX XMLCite \textit{P. Pucci} and \textit{J. Serrin}, in: Energy methods in continuum mechanics. Proceedings of the workshop on energy methods for free boundary problems in continuum mechanics, Oviedo, Spain, March 21--23, 1994. Dordrecht: Kluwer Academic Publishers. 66--74 (1996; Zbl 0874.35056)