Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type

TL;DR

This paper investigates the existence and extension of solutions to a strongly damped wave equation involving the p-Laplacian, with boundary damping and nonlinear source, in a bounded domain, addressing supercritical source challenges.

Contribution

It establishes local and global existence results for solutions to a p-Laplacian wave equation with boundary damping and supercritical source terms, extending previous work to more general nonlinearities.

Findings

Proved local existence of weak solutions.

Extended solutions globally under growth conditions.

Addressed supercritical nonlinear source challenges.

Abstract

This article focuses on a quasilinear wave equation of $p$-Laplacian type: $$ u_{tt} - \Delta_p u - \Delta u_t=0$$ in a bounded domain $\Omega\subset\mathbb{R}^3$ with a sufficiently smooth boundary $\Gamma=\partial\Omega$ subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator $\Delta_p$, $2 < p < 3$, denotes the classical $p$-Laplacian. The nonlinear boundary term $f (u)$ is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from $W^{1,p}(\Omega)$ into $L^2(\Gamma)$. Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the source term satisfies an appropriate growth condition.