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Persistence and asymptotic analysis of solutions of nonlinear wave equations

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Abstract

We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai’s, the Camassa–Holm, and the Dullin–Gottwald–Holm equations, as well as some recent shallow water equations with Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentioned in light of our results for the general class.

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Data Availibility Statement

The author declares that data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

Notes

  1. By considerable attention we do not mean just the equation studied in [16, 17], but actually those in [7, 29, 30, 41], which is enclosed in (1.5).

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Acknowledgements

I am grateful to CNPq (grant n\({}^\circ \) 310074/2021-5) and FAPESP (grant n\({}^\circ \) 2020/02055-0) for financial support. I am also thankful to the Institute of Advanced Studies of the Loughborough University for warm hospitality and support for my visit. In addition, it is a pleasure to thank the Mathematical Institute of the Silesian University in Opava for the nice environment, where the manuscript was finalised.

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  1. Institute of Advanced Studies, Loughborough University, Epinal Way, Loughborough, LE11 3TU, UK

    Igor Leite Freire

  2. Departamento de Matemática, Universidade Federal de São Carlos, Rodovia Washington Luís, Km 235, São Carlos, SP, 13565-905, Brasil

    Igor Leite Freire

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  1. Igor Leite Freire
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This paper is dedicated to Professor Antonio Carlos Gilli Martins, who was an example of teacher, inspiration as a professional, and beloved friend. Rest in peace.

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Freire, I.L. Persistence and asymptotic analysis of solutions of nonlinear wave equations. J. Evol. Equ. 24, 6 (2024). https://doi.org/10.1007/s00028-023-00937-4

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