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

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

with the focusing nonlinearity $M\left(u\right)={\left|u\right|}^{p-1}u,\phantom{\rule{3.33333pt}{0ex}}p>1.$ For the focusing nonlinearity $M\left(u\right)=\pm {\left|u\right|}^{p},\phantom{\rule{3.33333pt}{0ex}}p>1,$ 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:

Miaomiao Liu ^{1};
Bin Guo ^{1}

@article{CRMATH_2023__361_G3_667_0, 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}, }

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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