Decay rate of a weakly dissipative viscoelastic plate equation with infinite memory
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Arabian Journal of Mathematics
Khaleel Anaya · Salim A. Messaoudi · Kassem Mustapha
Decay rate of a weakly dissipative viscoelastic plate equation with infinite memory
Received: 12 June 2020 / Accepted: 2 September 2020 © The Author(s) 2020
Abstract In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in space, combined with time-stepping finite differences. Mathematics Subject Classfication
35B40 · 35L90 · 45K05 · 65M06 · 65M60
1 Introduction The modeling of generalized Kirchhoff viscoelastic plate, where a bending moment relation with infinite memory is considered, can be described by the following weakly dissipative viscoelastic equation: ⎧ ∞ ⎨u + Δ2 u + g(s)Δu(t − s)ds = 0 in Ω × (0, ∞), (1.1) 0 ⎩ u(x, −t) = u 0 (x, t), u (x, 0) = u 1 (x), in Ω, subject to the homogeneous conditions u = Δu = 0 on ∂Ω, where the physical domain Ω ⊂ Rn is a bounded domain with a smooth (or piecewise smooth) boundary ∂Ω. The history condition u(x, −t) = u 0 (x, t) means that we are taking into account all the deformation the material has undergone before the instant t = 0. The initial velocity u 1 and the non-negative relaxation function g are given. The asymptotic or decaying behaviour, of different types of viscoelastic equations, including finite and infinite memories also linear and nonlinear dampings, were the subject of study of many researchers since the pioneer work of Dafermos [4,5]. The achieved energy decaying in the literature varies between (fractional) polynomial and exponential rates. This depends on the damping term, the memory term, the relaxation function, and the differential operator. In the presence of memory, the main common challenge was showing the energy decaying for most general relaxation function g, we refer the readers to [1–3,6,7,9–11,13]. K. Anaya (B) · K. Mustapha Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of Saudi Arabia E-mail: [email protected] K. Mustapha E-mail: [email protected] S. A. Messaoudi Department of Mathematics, University of Sharjah, 27272 Sharjah, UAE E-mail: [email protected]
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Assuming that the relaxation function g in (1.1) satisfies g (t) ≤ −δg(t), for t ≥ 0 and for some positive constant δ, Revira et al. [15] showed an exponential convergence of the energy decay. The main focus of this work was on investigating the energy decay of problem (1.1) but for a wider class of g, see (2.2) below. Owing to the presence of the weakly dissipative term, a second energy functional is introduced to achieve our goal. To make use of the convolution properties and the history condition, the viscoelastic (infinite memory) term is decomposed accordingly. For numerical illustrations of our
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