Pointwise decay for the solutions of degenerate and singular parabolic equations

Abstract

We study the asymptotic behavior, as $t\to\infty$, of the solutions to the evolutionary $p$-Laplace equation \[ v_t={\operatorname{div}}( | {\nabla v} | ^{p-2}\nabla v), \] with time-independent lateral boundary values. We obtain the sharp decay rate of $\max_{x\in{\Omega}} | {v(x,t)-u(x)}| $, where $u$ is the stationary solution, both in the degenerate case $p > 2$ and in the singular case $1 < p > 2$. A key tool in the proofs is the Moser iteration, which is applied to the difference $v(x,t)-u(x)$. In the singular case, we construct an example proving that the celebrated phenomenon of finite extinction time, valid for $v(x,t)$ when $u\equiv 0$, does not have a counterpart for $v(x,t)-u(x)$.