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On the Cauchy Problem for the Nonlinear Wave Equation with Damping and Potential

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Harmonic Analysis and Partial Differential Equations

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Abstract

In this note, we study the Cauchy problem for the nonlinear wave equation with damping and potential terms. The aim of this study is to generalize the result in Georgiev et al. (J. Differ. Equ. 267(5):3271–3288, 2019) into two directions. One is to relax the condition which characterizes the behavior of the coefficient of the damping term at spatial infinity as in (6). The other is to treat the slowly decreasing initial data. The decaying rate of the data affects the global behavior of the solutions even if the nonlinear exponent lies in the super-critical regime (see Theorem 5 below).

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Acknowledgements

The authors are grateful to the referee for useful comments which make the original manuscript be improved. The first author was partially supported by Grant-in-Aid for Science Research (No.19H01795), JSPS. The second author was partially supported by Grant-in-Aid for Science Research (No.16H06339 and No.19H01795), JSPS.

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Authors and Affiliations

  1. Faculty of Science and Engineering, Muroran Institute of Technology, Muroran, Japan

    Masakazu Kato

  2. Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan

    Hideo Kubo

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  1. Masakazu Kato
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  2. Hideo Kubo
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Correspondence to Masakazu Kato .

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Editors and Affiliations

  1. Department of Mathematics, Ghent University, Ghent, Belgium

    Michael Ruzhansky

  2. Department of Mathematics, University of Stuttgart, Stuttgart, Germany

    Jens Wirth

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Kato, M., Kubo, H. (2022). On the Cauchy Problem for the Nonlinear Wave Equation with Damping and Potential. In: Ruzhansky, M., Wirth, J. (eds) Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-24311-0_3

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