Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

Yanbing Yang; Runzhang Xu

doi:10.3934/cpaa.2019065

Article Contents

2019, Volume 18, Issue 3: 1351-1358. Doi: 10.3934/cpaa.2019065

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Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

Yanbing Yang^1,2,3, and
Runzhang Xu^1, ,

1.
College of Science, Harbin Engineering University, 150001, China
2.
College of Computer Science and Technology, Harbin Engineering University, 150001, China
3.
Department of Mathematics, University of Texas, Arlington, TX 76019, USA

* Corresponding author
* Corresponding author

Received: May 31, 2018

Revised: August 31, 2018

Published: May 2019

The first author was supported by the National Natural Science Foundation of China (11801114), the Heilongjiang Postdoctoral Foundation (LBH-Z15036), the China Scholarship Council (201706685064), the Fundamental Research Funds for the Central Universities. The second author was supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation (2013M540270), the Fundamental Research Funds for the Central Universities

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Abstract

By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level.

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Mathematics Subject Classification: Primary: 35L05, 35L20, 35B44.

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Table 1. Obtained results and open problems for problem (1)-(3)

	Global existence	Asymptotic behavior	Blow up
Subcritical initial energy $E(0)<d$	Reference [4]	Reference [4]	Reference [4]
Critical initial energy $E(0)=d$	Reference [4]	Reference [4]	Reference [4]
Supercritical initial energy $E(0)>d$	Open problem	Open problem	$\omega=0$, $\mu>0$ $\omega=0$, $\mu=0$ Reference [4]	$\omega>0$, $\mu>0$ $\omega>0$, $\mu=0$ $\omega=0$, $\mu>0$ $\omega=0$, $\mu=0$ Present paper

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