Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

Jun Zhou

doi:10.3934/dcdss.2017064

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2017, Volume 10, Issue 5: 1175-1185. Doi: 10.3934/dcdss.2017064

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Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

Jun Zhou

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: August 31, 2016

Revised: January 31, 2017

Published: October 2017

This work is partially supported by the the Basic and Advanced Research Project of CQCSTC grant cstc2016jcyjA0018, NSFC grant 11201380, Fundamental Research Funds for the Central Universities grant XDJK2015A16, XDJK2016E120, Project funded by China Postdoctoral Science Foundation grant 2014M550453,2015T80948

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Abstract

This paper deals with a higher-order wave equation with general nonlinear dissipation and source term

$u''+(-Δ)^mu+g(u')=b|u|^{p-2}u, $

which was studied extensively when $m=1, 2$ and the nonlinear dissipative term $g(u')$ is a polynomial, i.e., $g(u')=a|u'|^{q-2}u'$. We obtain the global existence of solutions and show the energy decay estimate when $m≥1$ is a positive integer and the nonlinear dissipative term $g$ does not necessarily have a polynomial grow near the origin.

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Mathematics Subject Classification: Primary: 35L20, 35L70; Secondary: 58G16.

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