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Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms

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Abstract

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type. The proof of the blow-up results is based on an iteration argument. We find as critical curve for the pair of exponents (p, q) in the nonlinear terms the same one found for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

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Acknowledgements

The first author 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 the first author at Tohoku University within the period October to December 2018. He thanks the Mathematical Institute of Tohoku University for the worm hospitality and the excellent working conditions during this period. The first author is supported by the University of Pisa, Project PRA 2018 49. The second author is partially supported by the Grant-in-Aid for Scientific Research (B)(No.18H01132).

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  1. Department of Mathematics, University of 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 in the scattering case with mixed nonlinear terms. Nonlinear Differ. Equ. Appl. 27, 58 (2020). https://doi.org/10.1007/s00030-020-00662-8

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