Article Contents
Article Contents

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

• * Corresponding author: Nicholas J. Kass has been partially supported by NSF grant DMS-1211232
• This article focuses on a quasilinear wave equation of $p$-Laplacian type:

$u_{tt} - Δ_p u -Δ u_t = 0$

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 $Δ_p$, $2<p<3$, 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 ${W^{1,p}}\left( \Omega \right)$ 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.

Mathematics Subject Classification: Primary: 35L05, 35L20, 35L72; Secondary: 58J45.

 Citation:

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