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

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.