The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

Takiko Sasaki; Shu Takamatsu; Hiroyuki Takamura; Takiko Sasaki; Shu Takamatsu; Hiroyuki Takamura

doi:10.3934/math.20231300

AIMS Mathematics

2023, Volume 8, Issue 11: 25477-25486. doi: 10.3934/math.20231300

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Research article

The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

1.
Department of Mathematical Engineering, Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo 135-8181, Japan
2.
Master course, Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
3.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Received: 23 June 2023 Revised: 17 August 2023 Accepted: 23 August 2023 Published: 31 August 2023
MSC : primary 35L71, secondary 35B44

This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.
- semilinear wave equation,
- one dimension,
- classical solution,
- lifespan
Citation: Takiko Sasaki, Shu Takamatsu, Hiroyuki Takamura. The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative[J]. AIMS Mathematics, 2023, 8(11): 25477-25486. doi: 10.3934/math.20231300

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Abstract

This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.

References

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