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

TL;DR

This paper studies the boundedness and long-term behavior of solutions to a one-dimensional degenerate parabolic equation with gradient nonlinearity, establishing conditions for global existence and convergence to steady states.

Contribution

It proves that solutions either have finite-time gradient blow-up or converge to a unique steady state, ruling out unbounded gradient solutions.

Findings

Solutions either blow up in finite time or converge to a steady state.

A Lyapunov functional is constructed to analyze solution behavior.

Unbounded gradient solutions do not exist globally.

Abstract

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: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad (0, +\infty)\tiles(0, 1),\qquad q > p > 2.$$ We prove that: either $u_x$ blows up in finite time, or $u$ is global and converges in $W^{1, \infty}$norm 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.