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

This article focuses on a quasilinear wave equation of $p$-Laplacian type: $$u_{tt}-{\mathrm{\Delta }}_{p}u-\mathrm{\Delta }{u}_t=f(u)$$ in a bounded domain $\Omega \subset $\mathbb{R}$^3$withasufficientlysmoothboundary$\Gamma=\partial \Omega$ subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator ${\mathrm{\Delta }}_p$, $2<p<3$, 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 $W^{1,p}(\mathrm{\Omega })$ into $L^2(\mathrm{\Omega })$ or $L^2(\mathrm{\Gamma })$. 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.