Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

Yue Pang; Xingchang Wang; Furong Wu

doi:10.3934/dcdss.2021115

Article Contents

2021, Volume 14, Issue 12: 4439-4463. Doi: 10.3934/dcdss.2021115 open access

This issue Previous Article Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation Next Article Anisotropic singular double phase Dirichlet problems

Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

College of Mathematical Sciences, Harbin Engineering University, Heilongjiang, Harbin 150001, China

^* Corresponding author: Xingchang Wang
^* Corresponding author: Xingchang Wang

Received: July 31, 2021

Revised: August 31, 2021

Early access: October 2021

Published: December 2021

The first author is supported by the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072021CF2404), the second author is supported by NSFC grant 11871017

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Abstract

We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.

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Mathematics Subject Classification: Primary: 35L05, 35A01; Secondary: 35B45.

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Table 1. The values for $ m $ and $ m' $

The value					Results
$ n $	$ \frac{2n}{n-2} $	$ m\in\left(n,\frac{2n}{n-2}\right) $	$ m'\in\left(n,\frac{2n}{n-2}\right) $	$ \frac{1}{m}+\frac{1}{m'}=\frac{1}{2} $	Results
$ \le 2 $	$ \infty $	$ (n, \infty) $	$ (n,\infty) $	$ (m,m')=(3,6) $	valid
$ 3 $	$ 6 $	$ (3,6) $	$ (3,6) $	$ (m,m')=\left(5,\frac{10}{3}\right) $	valid
$ 4 $	$ 4 $	$ - $	$ - $	$ - $	invalid
$ 5 $	$ \frac{10}{3} $	$ - $	$ - $	$ - $	invalid
$ \vdots $	$ \vdots $	$ \vdots $	$ \vdots $	$ \vdots $	invalid

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