Attractors for the strongly damped wave equation with $p$-Laplacian

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional strongly damped wave equation with p-Laplacian, establishing existence and properties of attractors in specific function spaces for different p ranges.

Contribution

It proves the existence of local and global attractors for the strongly damped wave equation with p-Laplacian, extending understanding of its asymptotic dynamics.

Findings

Existence of weak local attractors for p>2.

Existence of strong global attractors for 2<p<4.

Global attractors are bounded in W^{1,∞} spaces.

Abstract

This paper is concerned with the initial boundary value problem for one dimensional strongly damped wave equation involving $p$-Laplacian. For $p>2$, we establish the existence of weak local attractors for this problem in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$. Under restriction $2<p<4$, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$ and this attractor is a bounded subset of $W^{1,\infty }(0,1)\times W^{1,\infty }(0,1)$.