Electron. J. Differential Equations, Vol. 2022 (2022), No. 07, pp. 1-18.

Asymptotic behavior of solutions to 3D Kelvin-Voigt-Brinkman-Forchheimer equations with unbounded delays

Le Thi Thuy

In this article we consider a 3D Kelvin Voigt Brinkman Forchheimer equations involving unbounded delays in a bounded domain $\Omega \subset \mathbb{R}^3$. First, we show the existence and uniqueness of weak solutions by using the Galerkin approximations method and the energy method. Second, we prove the existence and uniqueness of stationary solutions by employing the Brouwer fixed point theorem. Finally, we study the stability of stationary solutions via the direct classical approach and the construction of a Lyapunov function. We also give a sufficient condition for the polynomial stability of the stationary solution for a special case of unbounded variable delay.

Submitted November 3, 2020. Published January 17, 2022.
Math Subject Classifications: 35B41, 35K65, 35D05
Key Words: Kelvin-Voigt-Brinkman-Forchheimer equation; delay equation; stationary solution; local stability; asymptotically stable; polynomial stable.

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Le Thi Thuy
Department of Mathematics
Electric Power University
235, Hoang Quoc Viet
Bac Tu Liem, Hanoi, Vietnam
email: thuylt@epu.edu.vn, thuylephuong@gmail.com

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