Stability result for a viscoelastic wave equation in the presence of finite and infinite memories

2021/12/25 by Al-Mahdi, Adel M., Al-Gharabli, Mohammad M., Kafini, Mohammad +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.13256

Abstract

In this paper, we are concerned with the following viscoelastic wave equation u_tt-∇ u +∫₀^t g₁ (t-s)~ div(a₁(x) ∇ u(s))~ ds + ∫₀^{+ ∞} g₂ (s)~ div(a₂(x) ∇ u(t-s)) ~ds = 0, in a bounded domain Ω. Under suitable conditions on a₁ and a₂ and for a wide class of relaxation functions g₁ and g₂. We establish a general decay result. The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data ∇ u₀. This study generalizes and improves previous literature outcomes.

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