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Wave equations with a damping term degenerating near low and high frequency regions

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Abstract

We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter \(\theta \in (0,1)\). This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in (Charão et al. in Math Methods Appl Sci 44:14003-14024, 2021; Charão and Ikehata in Angew Math Phys 71:26, 2020; Piske et al. in J Diff Eqns 311:188-228, 2022). The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense, particularly, to the case \(0<\theta <1/ 2\).

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Acknowledgements

The authors thank the reviewers for their careful reading and valuable advice. The work of the first author (R. C. CHARÃO) was partially supported by PRINT/CAPES- Process 88881.310536/2018-00. The work of the second author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C) 20K03682 of JSPS.

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Authors and Affiliations

  1. Department of Mathematics, Federal University of Santa Catarina, Florianopolis, SC, 88040-270, Brazil

    Ruy Coimbra Charão

  2. Division of Educational Sciences, Department of Mathematics, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima, 739-8524, Japan

    Ryo Ikehata

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  1. Ruy Coimbra Charão
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Correspondence to Ruy Coimbra Charão.

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Charão, R.C., Ikehata, R. Wave equations with a damping term degenerating near low and high frequency regions. J. Pseudo-Differ. Oper. Appl. 15, 19 (2024). https://doi.org/10.1007/s11868-024-00589-z

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  • DOI: https://doi.org/10.1007/s11868-024-00589-z

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