Convergence in a quasilinear parabolic equation with Neumann boundary conditions

Consider the problem u(t) = a(u(x))u(xx) + f(u(x)) (vertical bar x vertical bar < 1, t > 0), u(x)(+/- 1, t) = k(+/-)(t, u(+/- 1, t)) (t > 0), where k(+/-) are smooth functions which are periodic in both t and u. Brunovsky et al. proved in their paper (Brunovsky 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 V-t + c are also studied. (C) 2010 Elsevier Ltd. All rights reserved.