Blow-up solutions to the semilinear wave equation with overdamping term
This article deals with the Cauchy problem to the following damped wave equation \begin{equation*} {\left\lbrace \begin{array}{ll} u_{tt}-\Delta 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{array}\right.}CP \end{equation*} with the focus...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-03-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.432/ |
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| author | Liu, Miaomiao Guo, Bin |
| author_facet | Liu, Miaomiao Guo, Bin |
| author_sort | Liu, Miaomiao |
| collection | DOAJ |
| description | This article deals with the Cauchy problem to the following damped wave equation
\begin{equation*}
{\left\lbrace \begin{array}{ll} u_{tt}-\Delta 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{array}\right.}CP
\end{equation*}
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. |
| format | Article |
| id | doaj-art-e66fbca70cdd4ff1bbef41ed6649ca05 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-e66fbca70cdd4ff1bbef41ed6649ca052025-02-07T11:07:00ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-03-01361G366767210.5802/crmath.43210.5802/crmath.432Blow-up solutions to the semilinear wave equation with overdamping termLiu, Miaomiao0Guo, Bin1School of Mathematics, Jilin University, Changchun 130012, PR ChinaSchool of Mathematics, Jilin University, Changchun 130012, PR ChinaThis article deals with the Cauchy problem to the following damped wave equation \begin{equation*} {\left\lbrace \begin{array}{ll} u_{tt}-\Delta 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{array}\right.}CP \end{equation*} 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.432/ |
| spellingShingle | Liu, Miaomiao Guo, Bin Blow-up solutions to the semilinear wave equation with overdamping term Comptes Rendus. Mathématique |
| title | Blow-up solutions to the semilinear wave equation with overdamping term |
| title_full | Blow-up solutions to the semilinear wave equation with overdamping term |
| title_fullStr | Blow-up solutions to the semilinear wave equation with overdamping term |
| title_full_unstemmed | Blow-up solutions to the semilinear wave equation with overdamping term |
| title_short | Blow-up solutions to the semilinear wave equation with overdamping term |
| title_sort | blow up solutions to the semilinear wave equation with overdamping term |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.432/ |
| work_keys_str_mv | AT liumiaomiao blowupsolutionstothesemilinearwaveequationwithoverdampingterm AT guobin blowupsolutionstothesemilinearwaveequationwithoverdampingterm |