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

Nicholas J. Kass; Mohammad A. Rammaha

doi:10.3934/cpaa.2018070

Article Contents

2018, Volume 17, Issue 4: 1449-1478. Doi: 10.3934/cpaa.2018070

This issue Previous Article Asymptotics for the modified witham equation Next Article Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data

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

Nicholas J. Kass^, and
Mohammad A. Rammaha

Department of Mathematics, University of Nebraska-Lincoln, 203 Avery Hall, Lincoln, NE 68588-0130, USA

* Corresponding author: Nicholas J. Kass has been partially supported by NSF grant DMS-1211232
* Corresponding author: Nicholas J. Kass has been partially supported by NSF grant DMS-1211232

Received: April 30, 2017

Revised: November 30, 2017

Published: April 2018

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Abstract

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.

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

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