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

## TL;DR

This paper investigates the long-term decay behavior of solutions to the damped p-Laplace equation, showing that the difference from the stationary solution diminishes at a specific polynomial rate in the degenerate case p > 2.

## Contribution

It provides a decay estimate for the solution's convergence to the stationary state in the degenerate p > 2 case, extending understanding of asymptotic behavior for this class of equations.

## Key findings

- The W^{1,p}-norm of the difference decays like t^{-1/((p-1)p)} as t approaches infinity.
- Decay rate is explicitly characterized for the degenerate case p > 2.
- Results contribute to the theory of long-term behavior of nonlinear damped evolution equations.

## 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 =\Delta_p u \end{equation*} with Dirichlet boundary values. Let $u^*$ denote the stationary solution with same boundary values, then the $W^{1,p}$-norm of $u(t) - u^{*}$ decays for large $t$ like $t^{-\frac{1}{(p-1)p}}$, in the degenerate case $ p > 2$.