Partial differential equations
Blow-up solutions to the semilinear wave equation with overdamping term
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 667-672.

This article deals with the Cauchy problem to the following damped wave equation

 $\left\{\begin{array}{cc}{u}_{tt}-\Delta u+b\left(t\right){u}_{t}=M\left(u\right),\phantom{\rule{3.33333pt}{0ex}}\hfill & \left(t,x\right)\in {R}^{+}×{R}^{N},\hfill \\ u\left(0,x\right)={u}_{0}\left(x\right),\phantom{\rule{3.33333pt}{0ex}}{u}_{t}\left(0,x\right)={u}_{1}\left(x\right),\phantom{\rule{3.33333pt}{0ex}}\hfill & x\in {R}^{N},\hfill \end{array}\right\CP$

with the focusing nonlinearity $M\left(u\right)={|u|}^{p-1}u,\phantom{\rule{3.33333pt}{0ex}}p>1.$ For the focusing nonlinearity $M\left(u\right)=±{|u|}^{p},\phantom{\rule{3.33333pt}{0ex}}p>1,$ Ikeda and Wakasugi in [8] have showed that the solution to Problem (CP) exists globally for small data and fails to exist globally for large data. Meanwhile, they also proposed an open problem [8, Remark 1.3]. In this note, we give a positive answer to this open problem by using a method different from the test-function method. In addition, an inverse Hölder inequality associated with the solution and a differential inequality argument are used to establish a lower bound for the blow-up time.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.432

Miaomiao Liu 1; Bin Guo 1

1 School of Mathematics, Jilin University, Changchun 130012, PR China
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Miaomiao Liu; Bin Guo. Blow-up solutions to the semilinear wave equation with overdamping term. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 667-672. doi : 10.5802/crmath.432. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.432/

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