Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term

We investigate the boundedness and large time behavior of solutions of the Cauchy–Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity:

u_t=(|u_x|^p−2u_x)_x+|u_x|^q in (0,∞)×(0,1),q>p>2.

We prove that: either u_x blows up in finite time, or u is global and converges in W^1,∞(0,1) to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.