Abstract
In this work, we are interested in a detailed qualitative analysis of the Kawahara equation, which models numerous physical phenomena such as magneto-acoustic waves in a cold plasma and gravity waves on the surface of a heavy liquid. First, we design a feedback control law, which combines a damping component and another one of finite memory type. Then, we are capable of proving that the problem is well-posed under a condition involving the feedback gains of the boundary control and the memory kernel. Afterward, it is shown that the energy associated with this system exponentially decays by employing two different methods: the first one utilises the Lyapunov function and the second one uses a compactness–uniqueness argument which reduces the problem to prove an observability inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availibility
It does not apply to this article as no new data were created or analyzed in this study.
Notes
\(\Vert \omega \Vert ^2\le \dfrac{\ell ^2}{\pi ^2}\Vert \partial _x \omega \Vert ^2\), for \(\omega \in H_0^2(\Omega ),\)
References
Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008)
Amendola, G., Fabrizio, M., Golden, J.M.: Thermodynamics of Materials with Memory: Theory and Applications. Springer, New York (2012)
Araruna, F.D., Capistrano-Filho, R.A., Doronin, G.G.: Energy decay for the modified Kawahara equation posed in a bounded domain. J. Math. Anal. Appl. 385, 743–756 (2012)
Berloff, N., Howard, L.: Solitary and periodic solutions of nonlinear nonintegrable equations. Stud. Appl. Math. 99(1), 1–24 (1997)
Biswas, A.: Solitary wave solution for the generalized Kawahara equation. Appl. Math. Lett. 22, 208–210 (2009)
Bona, J.L., Lannes, D., Saut, J.-C.: Asymptotic models for internal waves. J. Math. Pures Appl. 9(89), 538–566 (2008)
Boyd, J.P.: Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg-de Vries equation. Phys. D 48, 129–146 (1991)
Capistrano-Filho, R.A., Chentouf, B., de Sousa, L.S.: Two stability results for the Kawahara equation with a time-delayed boundary control. Z. Angew. Math. Phys. 74, 16 (2023)
Capistrano-Filho, R.A., de Jesus, I.M.: Massera’s theorems for a higher order dispersive system. Acta Applicandae Mathematicae 185(5), 1–25 (2023)
Capistrano-Filho, R.A., Gonzalez Martinez, V.H.: Stabilization results for delayed fifth-order KdV-type equation in a bounded domain. Math. Control Relat. Fields. https://doi.org/10.3934/mcrf.2023004.
Capistrano-Filho, R.A., de Sousa, L.S.: Control results with overdetermination condition for higher order dispersive system. J. Math. Anal. Appl. 506, 1–22 (2022)
Chentouf, B.: Qualitative analysis of the dynamic for the nonlinear Korteweg-de Vries equation with a boundary memory. Qual. Theory Dyn. Syst. 20(36), 1–29 (2021)
Chentouf, B.: On the exponential stability of a nonlinear Kuramoto-Sivashinsky-Korteweg-de Vries equation with finite memory. Mediterr. J. Math. 19(11), 22 (2022)
Chentouf, B.: Well-posedness and exponential stability of the Kawahara equation with a time-delayed localized damping. Math. Methods Appl. Sci. 45(16), 10312–10330 (2022)
Cui, S.B., Deng, D.G., Tao, S.P.: Global existence of solutions for the Cauchy problem of the Kawahara equation with \(\mathit{L}^{\mathit{2}}\) initial data. Acta Math. Sin. Engl. Ser. 22, 1457–1466 (2006)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Hasimoto, H.: Water waves. Kagaku 40, 401–408 (1970); (Japanese)
Hunter, J.K., Scheurle, J.: Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D 32, 253–268 (1998)
Iguchi, T.: A long wave approximation for capillary-gravity waves and the Kawahara Equations. Acad. Sin. New Ser. 2(2), 179–220 (2007)
Jin, L.: Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation. Math. Comput. Model. 49, 573–578 (2009)
Kaya, D., Al-Khaled, K.: A numerical comparison of a Kawahara equation. Phys. Lett. A 363(5–6), 433–439 (2007)
Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33, 260–264 (1972)
Kakutani, T.: Axially symmetric stagnation-point flow of an electrically conducting fluid under transverse magnetic field. J. Phys. Soc. Jpn. 15, 688–695 (1960)
Lannes, D.: The Water Waves Problem. Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence (2013)
Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 561–1585 (2006)
Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed memory. Differ. Integr. Equ. 21, 935–958 (2008)
Polat, N., Kaya, D., Tutalar, H.I.: An analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method. Appl. Math. Comput. 179, 466–472 (2006)
Pomeau, Y., Ramani, A., Grammaticos, B.: Structural stability of the Korteweg-de Vries solitons under a singular perturbation. Phys. D 31, 127–134 (1988)
Rosier, L.: Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2, 33–55 (1997)
Vasconcellos, C.F., Silva, P.N.: Stabilization of the linear Kawahara equation with localized damping. Asympt. Anal. 58, 229–252 (2008)
Vasconcellos, C.F., Silva, P.N.: Stabilization of the linear Kawahara equation with localized damping. Asympt. Anal. 66, 119–124 (2010)
Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12, 770–785 (2006)
Yusufoglu, E., Bekir, A., Alp, M.: Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine-Cosine method. Chaos Solitons Fract. 37, 1193–1197 (2008)
Acknowledgements
The authors are grateful to the two anonymous reviewers for their constructive comments and valuable remarks. This work is part of the Ph.D. thesis of de Jesus at the Department of Mathematics of the Federal University of Pernambuco. Part of this work was done during the stay of the first author at Virginia Tech. He thanks the departments for their hospitality.
Author information
Authors and Affiliations
Contributions
RdeAC-F, BC and IMdeJ work equality in Conceptualization; formal analysis; investigation; writing—original draft; writing—review and editing.
Corresponding author
Ethics declarations
Conflict of interest
This work does not have any conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Capistrano--Filho was supported by CNPq grants numbers 307808/2021-1, 401003/2022-1, and 200386/2022-0, CAPES grants numbers 88881.311964/2018-01 and 88881.520205/2020-01, and MATHAMSUD 21-MATH-03.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
de A. Capistrano-Filho, R., Chentouf, B. & de Jesus, I.M. Dynamic Stability of the Kawahara Equation Under the Effect of a Boundary Finite Memory. Qual. Theory Dyn. Syst. 23, 28 (2024). https://doi.org/10.1007/s12346-023-00884-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-023-00884-y