Title: 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.