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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

$A\left(t\right)|{u}_{t}{|}^{m}{u}_{t}={\Delta }u-f\left(x,u\right),\phantom{\rule{1.em}{0ex}}u\in {ℝ}^{N},\phantom{\rule{1.em}{0ex}}\left(t,x\right)\in \left(0,\infty \right)×{\Omega },$

is studied. Here $A\left(t\right)$ is an $N×N$ 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 $k\notin {L}^{1}\left(0,\infty \right)$ such that

$\underset{t\to \infty }{lim inf}{\int }_{0}^{t}\sigma \left(s\right){k}^{m}\left(s\right)ds{\left({\int }_{0}^{t}k\left(s\right)ds\right)}^{-m}<\infty \phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{1.em}{0ex}}\sigma \left(t\right)={|A\left(t\right)|}^{m}{\left(\underset{|v|=1}{sup}\left(A\left(t\right)v,v\right)\right)}^{1-m}·$

In particular, if $A\left(t\right)={t}^{\alpha }Id$ the rest state is stable if $\alpha \le 1$ whereas in the case $\alpha >1$ solutions converging to a nontrivial steady state may exist.

##### MSC:
 35K65 Parabolic equations of degenerate type 35K45 Systems of second-order parabolic equations, initial value problems
##### Keywords:
degenerate parabolic systems; asymptotic stability