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

Abstract

This article focuses on a quasilinear wave equation of $p$-Laplacian type: \[ u_{tt} - Δ_p u -Δu_t = f(u) \] 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.