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Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

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Abstract

In this paper, we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. The assumption on the time-dependent coefficients for the damping terms means that these coefficients are summable and nonnegative. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato’s lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case, an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p - q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of semilinear not-damped wave equations with the same kind of nonlinearities.

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Acknowledgements

A. Palmieri is member of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM). This paper was written partially during the stay of A. Palmieri at Tohoku University within the period October to December 2018. He thanks the Mathematical Department of Tohoku University for the hospitality and the great working conditions during his stay. A. Palmieri is supported by the University of Pisa, Project PRA 2018 49. H. Takamura is partially supported by the Grant-in-Aid for Scientific Research (B) (No.18H01132).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Alessandro Palmieri

  2. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Hiroyuki Takamura

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  1. Alessandro Palmieri
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  2. Hiroyuki Takamura
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Correspondence to Alessandro Palmieri.

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Palmieri, A., Takamura, H. Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case. Mediterr. J. Math. 17, 13 (2020). https://doi.org/10.1007/s00009-019-1445-4

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  • DOI: https://doi.org/10.1007/s00009-019-1445-4

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