On wave equations of the $p$-Laplacian type with supercritical nonlinearities

TL;DR

This paper studies a quasilinear wave equation with supercritical nonlinearities of p-Laplacian type, establishing local and global existence of solutions and conditions for blow-up, advancing understanding of nonlinear wave dynamics with boundary damping.

Contribution

It provides the first rigorous proof of local and global solutions for a p-Laplacian wave equation with supercritical sources and boundary damping, including blow-up results.

Findings

Existence of local weak solutions under certain conditions.

Global solutions are obtained when damping dominates sources.

Blow-up occurs for solutions with negative initial energy.

Abstract

This article focuses on a quasilinear wave equation of $p$-Laplacian type: \[ u_{tt} - \Delta_p u -\Delta u_t = f(u) \] 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 interior and boundary terms $f(u)$, $h(u)$ are sources that are allowed to have a supercritical exponent, in the sense that their associated Nemytskii operators are not locally Lipschitz from $W^{1,p}(\Omega)$ into $L^2(\Omega)$ or $L^2(\Gamma)$. 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 damping terms dominates the corresponding sources in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.