Asymptotic behaviour of the doubly nonlinear equation $u_t=\Delta_p u^m$ on bounded domains

TL;DR

This paper analyzes the long-term behavior of solutions to a doubly nonlinear PDE on bounded domains, establishing convergence to a unique profile and providing convergence rates for different parameter regimes.

Contribution

It characterizes the asymptotic behavior of solutions to the doubly nonlinear equation in both degenerate and quasilinear cases, including convergence rates.

Findings

Solutions converge uniformly to a unique asymptotic profile

Established convergence rates for large-time behavior

Analyzed both degenerate and quasilinear cases

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.