Blow-up solutions to the semilinear wave equation with overdamping term

Miaomiao Liu; Bin Guo

doi:10.5802/crmath.432

Partial differential equations

Blow-up solutions to the semilinear wave equation with overdamping term

Miaomiao Liu ¹; Bin Guo ¹

¹ School of Mathematics, Jilin University, Changchun 130012, PR China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 667-672.

Abstract

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

\{\begin{matrix} u_{t t} - Δ u + b (t) u_{t} = M (u), & (t, x) \in R^{+} \times R^{N}, \\ u (0, x) = u_{0} (x), u_{t} (0, x) = u_{1} (x), & x \in R^{N}, \end{matrix} C P

with the focusing nonlinearity $M (u) = {| u |}^{p - 1} u, p > 1 .$ For the focusing nonlinearity $M (u) = \pm {| u |}^{p}, p > 1,$ 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.

Received: 2022-04-14
Revised: 2022-09-22
Accepted: 2022-09-27
Published online: 2023-03-02

DOI: 10.5802/crmath.432

Author's affiliations:

Miaomiao Liu ¹; Bin Guo ¹

¹ School of Mathematics, Jilin University, Changchun 130012, PR China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Miaomiao Liu and Bin Guo},
     title = {Blow-up solutions to the semilinear wave equation with overdamping term},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {667--672},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.432},
     language = {en},
}

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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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