TY - JOUR
T1 - Two stability results for the Kawahara equation with a time-delayed boundary control
AU - Capistrano-Filho, Roberto de A.
AU - Chentouf, Boumediène
AU - de Sousa, Luan S.
AU - Gonzalez Martinez, Victor H.
N1 - Funding Information:
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.
Funding Information:
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.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/1
Y1 - 2023/1
N2 - 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→ ∞. 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.
AB - 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→ ∞. 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.
KW - Boundary time-delay
KW - Critical set
KW - exponential stability
KW - Nonlinear Kawahara equation
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UR - https://www.mendeley.com/catalogue/0827aef9-e7df-3cfc-9d7c-45b6339aa8f6/
U2 - 10.1007/s00033-022-01897-4
DO - 10.1007/s00033-022-01897-4
M3 - Article
AN - SCOPUS:85144118068
SN - 0044-2275
VL - 74
JO - Zeitschrift fur Angewandte Mathematik und Physik
JF - Zeitschrift fur Angewandte Mathematik und Physik
IS - 1
M1 - 16
ER -