General decay of energy for a viscoelastic wave equation with a distributed delay term in the nonlinear internal dambing
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General decay of energy for a viscoelastic wave equation with a distributed delay term in the nonlinear internal dambing Mohammed Aili1 · Ammar Khemmoudj1 Received: 17 January 2019 / Accepted: 7 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this article, we consider a viscoelastic wave equation of Kirchhoff type in a bounded domain with a distributed delay term present in the nonlinear internal dambing. By introducing a functional energy and suitable Lyapunov functional, under suitable assumptions, we establish a general decay result from which the exponential and polynomial decay are only special cases. Keywords Viscoelastic Kirchhoff equation · Frictional feedback · Nonlinear distributed delay · Convexity · General decay rate Mathematics Subject Classification 35B40 · 35L20 · 74D99 · 93D15 · 26A51
1 Introduction In this paper, we investigate the following viscoelastic wave equation of Kirchhoff type with a nonlinear internal damping and a nonlinear distributed delay term ⎧ t ⎪ l 2 ⎪ h (t − s) Δu (s) ds ⎪ |u t | u tt − M ∇u2 Δu − Δu tt + ⎪ ⎪ 0 ⎪ τ2 ⎪ ⎪ ⎨ +μ g (u ) + μ (s) g2 (u t (x, t − s)) ds + f (u) = 0 in × R+ , 0 1 t (1.1) τ1 ⎪ ⎪ u = 0 on ∂ × R+ , ⎪ ⎪ ⎪ ⎪ ⎪ u (x, 0) = u 0 (x) , u t (x, 0) = u 1 (x) in , ⎪ ⎩ u t (x, −t) = f 0 (x, t) in × ]0, τ2 [ where is a bounded domain in Rn , n ∈ N∗ , with smooth boundary ∂, l and μ0 are two positive constants, h is a positive non-increasing function defined on R+ and check some
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Ammar Khemmoudj [email protected] Mohammed Aili [email protected]
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Faculty of Mathematics, University of Sciences and Technology Houari Boumedienne, P.O. Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria
123
M. Aili, A. Khemmoudj
assumptions (see (H1)), M (s) = m 0 + m 1 s β for m 0 > 0, m 1 ≥ 0, β ≥ 1 and s ≥ 0. The functions g1 , g2 and f are satisfying some assumptions (see (H2)–(H3)), μ : [τ1 , τ2 ] → R is a bounded function, where τ1 and τ2 are two real numbers satisfying 0 ≤ τ1 < τ2 . Moreover, (u 0 , u 1 , f 0 ) the initial data belong to a suitable function space. Delay effects are very important because most natural phenomena are in many cases very complicated and do not depend only on the current state but also on the past history of the system. The presence of delay can be a source of instability. In recent years, the stabilization of PDEs with delay effects has draw attention for many author and become an active area of research. Nicaise and Pignotti [32] studied the following wave equation with linear frictional damping and internal distributed delay τ2 u tt − Δu + μ0 u t + a (x) μ (s) u t (x, t − s) ds = 0 in × (0, ∞) , τ1
with initial and mixed Dirichlet–Neumann boundary conditions and a is a suitable τ function, they obtained the exponential decay of the solution under the condition a∞ τ12 μ (s) ds < μ0 , where μ : [τ1 , τ2 ] → R is a L ∞ function and μ (x) ≥ 0 almost all x ∈ [τ1 , τ2 ] . In case of τ a delay concentrated at a time τ, the distributed delay term τ12 μ (s) u t (x, t − s) ds becomes simpll
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