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Volume 36, Issue 3
Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping

Fei Guo, Jinling Liang & Changwang Xiao

DOI: 10.4208/jpde.v36.n3.1

J. Part. Diff. Eq., 36 (2023), pp. 235-261.

Published online: 2023-08

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  • Abstract

This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension $n=1$ and the nonlinear power is bigger than 2, the life-span $\widetilde T(\varepsilon)$ and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index $K$, which depends on the time-dependent damping and the nonlinear term, the life-span $\widetilde T(\varepsilon)$  can be estimated below by $\varepsilon^{-\frac{p}{1-K}}$, $e^{\varepsilon^{-p}}$ or $+\infty$, where $\varepsilon$ is the scale of the compact support of the initial data.

  • Keywords

Semilinear wave equation, time-dependent damping, life-span, global iteration method.

  • AMS Subject Headings

35A09, 35B44, 35L05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-36-235, author = {Guo , FeiLiang , Jinling and Xiao , Changwang}, title = {Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping}, journal = {Journal of Partial Differential Equations}, year = {2023}, volume = {36}, number = {3}, pages = {235--261}, abstract = {

This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension $n=1$ and the nonlinear power is bigger than 2, the life-span $\widetilde T(\varepsilon)$ and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index $K$, which depends on the time-dependent damping and the nonlinear term, the life-span $\widetilde T(\varepsilon)$  can be estimated below by $\varepsilon^{-\frac{p}{1-K}}$, $e^{\varepsilon^{-p}}$ or $+\infty$, where $\varepsilon$ is the scale of the compact support of the initial data.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n3.1}, url = {http://global-sci.org/intro/article_detail/jpde/21887.html} }
TY - JOUR T1 - Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping AU - Guo , Fei AU - Liang , Jinling AU - Xiao , Changwang JO - Journal of Partial Differential Equations VL - 3 SP - 235 EP - 261 PY - 2023 DA - 2023/08 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n3.1 UR - https://global-sci.org/intro/article_detail/jpde/21887.html KW - Semilinear wave equation, time-dependent damping, life-span, global iteration method. AB -

This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension $n=1$ and the nonlinear power is bigger than 2, the life-span $\widetilde T(\varepsilon)$ and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index $K$, which depends on the time-dependent damping and the nonlinear term, the life-span $\widetilde T(\varepsilon)$  can be estimated below by $\varepsilon^{-\frac{p}{1-K}}$, $e^{\varepsilon^{-p}}$ or $+\infty$, where $\varepsilon$ is the scale of the compact support of the initial data.

Guo , FeiLiang , Jinling and Xiao , Changwang. (2023). Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping. Journal of Partial Differential Equations. 36 (3). 235-261. doi:10.4208/jpde.v36.n3.1
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