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

Nicholas J. Kass; Mohammad A. Rammaha

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arXiv · 1705.06696

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

Nicholas J. Kass·Mohammad A. Rammaha

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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.

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Nicholas J. Kass, Mohammad A. Rammaha. 2017-05-18. Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type. https://arxiv.org/abs/1705.06696

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