Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay

Tae Gab Ha; Seyun Kim; Tae Gab Ha; Seyun Kim

doi:10.3934/math.2023533

AIMS Mathematics

2023, Volume 8, Issue 5: 10513-10528. doi: 10.3934/math.2023533

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Research article

Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay

Tae Gab Ha ^,,
Seyun Kim

Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Republic of Korea

Received: 07 December 2022 Revised: 20 February 2023 Accepted: 22 February 2023 Published: 02 March 2023
MSC : 35A01, 35B35, 35B40, 35L05

This paper is concerned with the wave equation having a nonlinear distributed delay. First, we prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz. Next, we establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data.
- wave equations,
- nonlinear distributed delay,
- existence of solutions,
- energy decay
Citation: Tae Gab Ha, Seyun Kim. Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay[J]. AIMS Mathematics, 2023, 8(5): 10513-10528. doi: 10.3934/math.2023533

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Abstract

This paper is concerned with the wave equation having a nonlinear distributed delay. First, we prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz. Next, we establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data.

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