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