Convergence in a quasilinear parabolic equation with time almost periodic boundary conditions

Consider the problem ut=a(ux)uxx+f(ux)(|x|<1,t>0),ux(±1,t)=±g(t)(t>0), where g(t)g(t) is an almost periodic function. We prove the following convergence results. If a time-global solution uu is bounded, then it converges to an almost periodic solution which is unique up to space shift, stable and asymptotically stable; if uu goes to infinity with positive lower average speed, then it converges to an almost periodic traveling wave which is unique up to space shift, stable, asymptotically stable and has positive average speed.