Decay estimate for the solution of the evolutionary damped $p$-Laplace equation

Abstract

In this note, we study the asymptotic behavior, as $t$ tends to infinity, of the solution $u$ to the evolutionary damped $p$-Laplace equation \begin{equation*} u_{tt}+a\, u_t =Δ_p u \end{equation*} with Dirichlet boundary values. Let $u^{\ast }$ denote the stationary solution with same boundary values, then the $W^{1,p}$-norm of $u(t)-u^{\ast }$ decays for large $t$ like $t^{-\frac{1}{(p-1)p}}$, in the degenerate case $p>2$.