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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 conditionfeaturing boundary damping and a nonlinear source term. The operator$\Delta_p$, $2 < p < 3$, denotes the classical $p$-Laplacian. The nonlinearboundary term $f (u)$ is a source feedback that is allowed to have asupercritical exponent, in the sense that the associated Nemytskii operator isnot locally Lipschitz from $W^{1,p}(\Omega)$ into $L^2(\Gamma)$. Under suitableassumptions on the parameters we provide a rigorous proof of existence of alocal weak solution which can be extended globally in time provided the sourceterm satisfies an appropriate growth condition.

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