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Two stability results for the Kawahara equation with a time-delayed boundary control

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Abstract

In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy functional and using Lyapunov’s approach, to ensure that the energy of the Kawahara system goes to 0 exponentially as \(t \rightarrow \infty \). The second result is achieved by employing a compactness–uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the novelty of this work is to characterize the critical lengths phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the Möbius transformations.

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Notes

  1. See for instance [1, 6, 32] and references therein, for a rigorous justification of various asymptotic models for surface and internal waves.

  2. \(\Vert u\Vert ^{2}_{L^{2}(0,L)} \le \frac{L^{2}}{\pi ^{2}}\Vert \partial _xu\Vert _{L^{2}(0,L)}\) for \(u \in H_{0}^{2}(0,L)\).

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Acknowledgements

The authors are grateful to the editor and the two anonymous reviewers for their constructive comments and valuable remarks. Capistrano-Filho was supported by CNPq grant 307808/2021-1, CAPES grants 88881.311964/2018-01 and 88881.520205/2020-01, MATHAMSUD grant 21-MATH-03 and Propesqi (UFPE). De Sousa acknowledges support from CAPES-Brazil and CNPq-Brazil. Gonzalez Martinez was supported by FACEPE grants BFP-0065-1.01/21 and BFP-0099-1.01/22. This work is part of the PhD thesis of de Sousa at the Department of Mathematics of the Universidade Federal de Pernambuco.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), Recife, PE, 50740-545, Brazil

    Roberto de A. Capistrano-Filho, Luan S. de Sousa & Victor H. Gonzalez Martinez

  2. Department of Mathematics, Faculty of Science, Kuwait University, Safat, 13060, Kuwait City, Kuwait

    Boumediène Chentouf

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  1. Roberto de A. Capistrano-Filho
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  3. Luan S. de Sousa
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Correspondence to Boumediène Chentouf.

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Capistrano-Filho, R.d.A., Chentouf, B., de Sousa, L.S. et al. Two stability results for the Kawahara equation with a time-delayed boundary control. Z. Angew. Math. Phys. 74, 16 (2023). https://doi.org/10.1007/s00033-022-01897-4

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