TY - JOUR
T1 - A remark on the blowing up of solutions to Nakao's problem
AU - Kita, Kosuke
AU - Kusaba, Ryunosuke
N1 - Funding Information:
The authors wish to express their thanks to Prof. Masahiro Ikeda for his valuable comments. The first author was partially supported by Grant-in-Aid for JSPS Fellows # 20J11425 . We would like to thank the anonymous referees for their very helpful suggestions and comments which lead to the improvement of this paper.
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/9/1
Y1 - 2022/9/1
N2 - This paper is devoted to studying blow-up of solutions to some system of a semilinear damped wave equation and a semilinear wave equation for small data. In particular, we prove that if the exponents of the nonlinear terms satisfy suitable conditions, all solutions blow up in finite time even for small initial data, and we are also concerned with the upper estimate of lifespan of blowing-up solutions. A study on blowing up of solutions for this problem was originated by Wakasugi (2017), which used the test function method, and was subsequently partially improved by Chen-Reissig (2021) with an iteration argument. However, there is a gap between these two results. Our main aim in this paper is to bridge this gap while also providing a more concise proof.
AB - This paper is devoted to studying blow-up of solutions to some system of a semilinear damped wave equation and a semilinear wave equation for small data. In particular, we prove that if the exponents of the nonlinear terms satisfy suitable conditions, all solutions blow up in finite time even for small initial data, and we are also concerned with the upper estimate of lifespan of blowing-up solutions. A study on blowing up of solutions for this problem was originated by Wakasugi (2017), which used the test function method, and was subsequently partially improved by Chen-Reissig (2021) with an iteration argument. However, there is a gap between these two results. Our main aim in this paper is to bridge this gap while also providing a more concise proof.
KW - Blow-up
KW - Damped wave equation
KW - Semilinear hyperbolic system
KW - Wave equation
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U2 - 10.1016/j.jmaa.2022.126199
DO - 10.1016/j.jmaa.2022.126199
M3 - Article
AN - SCOPUS:85127214707
SN - 0022-247X
VL - 513
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
M1 - 126199
ER -