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Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

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Abstract

The open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of semilinear damped wave equations or reaction–diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmic type time-weighted Sobolev spaces to obtain lower bound estimates for the lifespan in low spatial dimensions. Then, together with the derived lifespan estimates, new and sharp results on estimates for the lifespan in the critical case are claimed. Finally, we give an application of our results to the semilinear reaction–diffusion system in the critical case.

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Acknowledgements

The first author was supported by the China Postdoctoral Science Foundation (Grant no. 2021T140450 and no. 2021M692084). Tuan Anh Dao was funded by Vingroup JSC and supported by the Postdoctoral Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2021.STS.16. The authors thank Alessandro Palmieri (University of Pisa), Ya-guang Wang (Shanghai Jiao Tong University), Michael Reissig (TU Bergakademie Freiberg) and Yuta Wakasugi (Hiroshima University) for their suggestions in the preparation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Wenhui Chen

  2. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet Road, Hanoi, Vietnam

    Tuan Anh Dao

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, No.18 Hoang Quoc Viet Road, Hanoi, Vietnam

    Tuan Anh Dao

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  1. Wenhui Chen
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Chen, W., Dao, T.A. Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case. Math. Ann. 385, 101–130 (2023). https://doi.org/10.1007/s00208-021-02335-y

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  • DOI: https://doi.org/10.1007/s00208-021-02335-y

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