On stabilization of solutions to boundary value problems for quasilinear parabolic equations periodic in time (Q1335929)

We study the behavior of large time solutions to a boundary value problem for a quasilinear parabolic equation, \(\omega\)-periodic in time. It is shown that if the problem has an \(\widetilde{\omega}\)-periodic solution \(\widetilde {\varphi}(x,t)\) with period \(\widetilde {\omega}\) incommensurable with \(\omega\), then \(\widetilde{\varphi} (x,t+\gamma)\) is an \(\widetilde {\omega}\)-periodic solution, too \((\gamma\in \mathbb{R})\). Assume the problem to be dissipative and to have only finitely many periodic solutions. Then the period of a periodic solution is commensurable with \(\omega\). Replacing, if necessary, \(\omega\) by \(k\omega\), we can assume that the period of every periodic solution equals \(\omega\). It is also shown: 1) The problem has maximal and minimal periodic solutions. 2) If the solution \(u(x,t)\) does not intersect for \(t= k\omega\) the initial data of any periodic solution then it stabilizes to a periodic solution.