Global existence and energy decay for a transmission problem under a boundary fractional derivative type

Noureddine Bahri; Abderrahmane Beniani; Abdelkader Braik; Svetlin G. Georgiev; Zayd Hajjej; Khaled Zennir; Noureddine Bahri; Abderrahmane Beniani; Abdelkader Braik; Svetlin G. Georgiev; Zayd Hajjej; Khaled Zennir

doi:10.3934/math.20231412

AIMS Mathematics

2023, Volume 8, Issue 11: 27605-27625. doi: 10.3934/math.20231412

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Global existence and energy decay for a transmission problem under a boundary fractional derivative type

1.
Faculty of Science and Technology, Hasiba Benbouali University, Chlef 02000-Algeria
2.
Laboratory of Mathematics and Applications, Hassib Benbouali University, Chlef 02000, Algeria
3.
Department of Mathematics, University Ain Tmouchent, Belhadj Bouchaib, B.P. 284 RP, Ain Temouchent 46000, Algeria
4.
Laboratory of Fundamental and Applicable Mathematics of Oran, University of Oran1 Ahmed Ben Bella, B.P. 1524 El M'naouar, Oran 31000, Algeria
5.
Department of Mathematics, Sorbonne University, 75005, Paris, France
6.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
7.
Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma. B.P. 401 Guelma 24000 Algérie

Received: 10 June 2023 Revised: 16 September 2023 Accepted: 20 September 2023 Published: 28 September 2023
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

The paper considers the effects of fractional derivative with a high degree of accuracy in the boundary conditions for the transmission problem. It is shown that the existence and uniqueness of the solutions for the transmission problem in a bounded domain with a boundary condition given by a fractional term in the second equation are guaranteed by using the semigroup theory. Under an appropriate assumptions on the transmission conditions and boundary conditions, we also discuss the exponential and strong stability of solution by also introducing the theory of semigroups.
- fractional derivatives,
- wave equation,
- transmission problem,
- delay term,
- exponential stability
Citation: Noureddine Bahri, Abderrahmane Beniani, Abdelkader Braik, Svetlin G. Georgiev, Zayd Hajjej, Khaled Zennir. Global existence and energy decay for a transmission problem under a boundary fractional derivative type[J]. AIMS Mathematics, 2023, 8(11): 27605-27625. doi: 10.3934/math.20231412

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Abstract

The paper considers the effects of fractional derivative with a high degree of accuracy in the boundary conditions for the transmission problem. It is shown that the existence and uniqueness of the solutions for the transmission problem in a bounded domain with a boundary condition given by a fractional term in the second equation are guaranteed by using the semigroup theory. Under an appropriate assumptions on the transmission conditions and boundary conditions, we also discuss the exponential and strong stability of solution by also introducing the theory of semigroups.

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