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

We study the behavior at large time of solutions $\omega$-periodic in time to a boundary value problem for a quasilinear parabolic equation. We suppose that the problem is dissipative and has a finite number of periodic solutions. Denote by $u(x,t;u_0)$ a solution to the initial-boundary problem which takes the value $u_0$ at $t=0$. Assume that, for some natural $k$, the function $u(x,k\omega;u_0)$ does not intersect the initial data of any periodic solution. Then $u(x,t;u_0)$ converges to a unique periodic solution as $t\to+\infty$. Attractors of stable periodic solutions are studied too.