Geodesic
Équations aux dérivées partielles
Blow-up solutions to the semilinear wave equation with overdamping term
Liu, Miaomiao1 ; Guo, Bin1 
1 School of Mathematics, Jilin University, Changchun 130012, PR China
Comptes Rendus. Mathématique, Tome 361 (2023) no. G3, pp. 667-672

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

u tt -Δu+b(t)u t =M(u),(t,x)∈R + ×R N ,u(0,x)=u 0 (x),u t (0,x)=u 1 (x),x∈R N ,CP

with the focusing nonlinearity M(u)=|u| p-1 u,p>1. For the focusing nonlinearity M(u)=±|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.

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DOI : 10.5802/crmath.432

Liu, Miaomiao 1 ; Guo, Bin 1

1 School of Mathematics, Jilin University, Changchun 130012, PR China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     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},
     number = {G3},
     year = {2023},
     doi = {10.5802/crmath.432},
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Liu, Miaomiao; Guo, Bin. Blow-up solutions to the semilinear wave equation with overdamping term. Comptes Rendus. Mathématique, Tome 361 (2023) no. G3, pp. 667-672. doi: 10.5802/crmath.432

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