Convergence in a quasilinear parabolic equation with Neumann boundary conditions

Consider the problem ut = a(ux)uxx + f(ux) (Ixl < 1, t > 0), ux(±1, t) = k±(t, u(±1, t)) (t > 0), where k± are smooth functions which are periodic in both t and u. Brunovský et al. proved in their paper (Brunovský et al., 1992 [8]) that if a time-global solution u is bounded then it converges to a periodic solution. We prove that if u is unbounded from above, then it converges to a periodic traveling wave V(x, t) + ct in case k± = k±(t) (or k± = k±(u)), where V is a time periodic function and c > 0. In addition, the periodic traveling wave is unique up to space shifts (or time shifts), it is stable and asymptotically stable. The average traveling speed c and the instantaneous speed Vt + c are also studied.