Asymptotic behaviour of the doubly nonlinear equation $u_t={\Delta }_p{u}^m$ on bounded domains

Abstract

We study the homogeneous Dirichlet problem for the doubly nonlinear equation $u_t={\Delta }_p{u}^m$, where $p>1,m>0$ posed in a bounded domain in ${\mathbb{R}}^N$ with homogeneous boundary conditions and with non-negative and integrable data. In this paper we consider the degenerate case $m(p-1)>1$ and the quasilinear case $m(p-1)=1$. We establish the large-time behaviour by proving the uniform convergence to a unique asymptotic profile and we also give rates for this convergence.

47 pages, 4 figures