On the Porous-Elastic System with Thermoelasticity of Type III and Distributed Delay: Well-Posedness and Stability

Djamel Ouchenane,Salah Mahmoud Boulaaras,Mohamed Abdalla,Abdelbaki Choucha,Bahri Belkacem Cherif,Ioan Rasa

doi:10.1155/2021/9948143

Abstract

The paper deals with a one-dimensional porous-elastic system with thermoelasticity of type III and distributed delay term. This model is dealing with dynamics of engineering structures and nonclassical problems of mathematical physics. We establish the well posedness of the system, and by the energy method combined with Lyapunov functions, we discuss the stability of system for both cases of equal and nonequal speeds of wave propagation.

Highlights

Let H = ð0, 1Þ × ðτ1, τ2Þ × ð0,∞Þ, τ1, τ2 > 0
This paper studied the asymptotic behavior of a onedimensional thermoelastic system with distributed time delay; namely, an integral damping term on a time interval 1⁄2t − τ2, t − τ1 is taken into account
Beside the distributed delay term, a standard undelayed damping is included in the model ð−μ1φtÞ

Summary

Introduction

= δψxx − bφx − ξψ − = lθxx − γψtx + kθtxx, jμ2ðsÞjψtðx, t − sÞds − βθtx, ð14Þ with the initial data φðx, 0Þ = φ0ðxÞ, φtðx, 0Þ = φ1ðxÞ, ψðx, 0Þ = ψ0ðxÞ, ψtðx, 0Þ = ψ1ðxÞ, θðx, 0Þ = θ0ðxÞ, θtðx, 0Þ = θ1ðxÞ, ð15Þ ψtðx,−tÞ = −f0ðx, tÞ, x ∈ ð0, 1Þ and boundary conditions φxð0, tÞ = φxð, tÞ = ψð0, tÞ = ψð, tÞ = θxð0, tÞ = θxð, tÞ = 0, t ≥ 0: ð16Þ. >>>>>>>: ρ3θtt = lθxx − γψtx + kθtxx, sztðx, ρ, s, tÞ + zρðx, ρ, s, tÞ ð19Þ where ðx, ρ, s, tÞ ∈ ð0, 1Þ × H , ð20Þ with the boundary and the initial conditions ð21Þ φðx, 0Þ = φ0ðxÞ, φtðx, 0Þ = φ1ðxÞ, ψðx, 0Þ = ψ0ðxÞ, ð22Þ ψtðx, 0Þ = ψ1ðxÞ, θðx, 0Þ = θ0ðxÞ, θtðx, 0Þ = θ1ðxÞ, x ∈ ð0, 1Þ, ð23Þ zðx, ρ, s, 0Þ = −f0ðx, ρsÞ = h0ðx, ρsÞ, x ∈ ð0:1Þ, ρ ∈ ð0:1Þ, s ∈ ð0, τ2Þ: ð24Þ.

Well-Posedness

Stability Results

Conclusion