Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

Abstract

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions related to the Kelvin-Voigt damping and a delay term acting on the boundary. If the weight of the delay term in the feedback is less than the weight of the term without delay or if it is greater under an assumption between the damping factor, and the difference of the two weights, we prove the global existence of the solutions. Under the same assumptions, the exponential stability of the system is proved using an appropriate Lyapunov functional. More precisely, we show that even when the weight of the delay is greater than the weight of the damping in the boundary conditions, the strong damping term still provides exponential stability for the system.

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

“…Recently, Gerbi & Said-Houari studied in [21] and [22] problem (1.1) with f 2 = φ = 0, f 1 (u) = |u| p−2 u and a nonlinear boundary damping term of the form ρ(u t ) = |u t | m−2 u t . A local existence result was obtained by combining the Faedo-Galerkin method with the contraction mapping theorem.…”

Exponential Growth of Solution for a Class of Reaction Diffusion Equation with Memory and Multiple Nonlinearities

“…Our technique is different from that in [5,15,[27][28][29] because of the presence of nonlinear term | | −2 and the memory term. Proof.…”

“…We will prove that -norm of the solution grows as an exponential function. An essential tool to the proof is an idea used in [28,29], which based on an auxiliary function (a small perturbation of the total energy), using a differential inequality and obtaining the result. This result extend the early paper.…”

Blow-Up and Asymptotic Behavior of Global Solution of Damped Wave Equation with Dynamic Boundary Conditions

“…最近 Autuori 和 Pucci [25] 还讨论了动力边界的 Kirchhoff 系统。 本文讨论问题(1.1)~(1.4)的整体解的不存在性和衰减性。据作者所知，整体解的不存在性是该类问题的首次 讨论，而引入位势井得到其衰减性与文献 [19]的指数增长性完全不同。本文第二节，我们用经典的凸性引理 [26] 给出了整体解不存在的充分条件。在第三节，我们利用文献 [27]的思路引入位势井 [28] ，利用 Nakao 不等式 …”