Convergence in a quasilinear parabolic equation with Neumann boundary conditions

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