Convergence in time-periodic quasilinear parabolic equations in one space dimension

We consider time-periodic quasilinear parabolic equations in the domain {(t,x)∈R2|00}, where the right boundary r(t) of the spatial interval is a positive function which might be periodic, or asymptotically periodic, or a function tending to infinity, or infinity. We show that, in the first case (that is, r(t) is a periodic function), any bounded solution of the equations converges as t→∞ to a periodic one; in the other three cases, any positive bounded solution converges as t→∞ to a nonnegative periodic one. Using such a result, we study the long time dynamics of the initial-boundary value problem on the half line, as well as the Stefan free boundary problem, of a general heterogeneous reaction-diffusion equation. Also, we use the convergence result to study the long time dynamics of the initial-boundary value problem for a time-periodic (mean) curvature flow equation.