The sharp decay rate of thermoelastic transmission system with infinite memories
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The sharp decay rate of thermoelastic transmission system with infinite memories Lakhdar Kassah Laouar1 · Khaled Zennir2,3
· Salah Boulaaras2,4
Received: 20 September 2018 / Accepted: 11 March 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract The main contributions here are to show the lack of exponential stability and to prove that the t −1 is the sharp decay rate of problem (1.1). That is to show that for this types of materials, the dissipation produced by the weak-infinite memories are not strong enough to produce an exponential decay of the solution under usual/non usual conditions on the the relaxation functions’s growth. This work extends the previous results by Bayoud et al. (Appl Sci 20:18– 35, 2018), Zennir and Feng (Math Methods Appl Sci 112:6895–6906, 2018. https://doi.org/ 10.1002/mma.5201) to the weak-viscoelasticities in two parts. In order to fill this gaps, we use an appropriate estimates. Keywords Infinite memory · Thermoelastic transmission problem · Polynomial decay · Exponential stability · Semigroup · Sharpness Mathematics Subject Classification 35L05 · 35B40 · 93D20 · 74F05
1 Introduction The extent to which different types of decay rate of transmission system get our attention to study the role of the damped case, it is largely requested, especially with the rapid adoption of
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Khaled Zennir [email protected] Lakhdar Kassah Laouar [email protected] Salah Boulaaras [email protected]
1
Department of Mathematics, University of Constantine 1, Constantine, Algeria
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Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Buraydah, Kingdom of Saudi Arabia
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Laboratory LAMAHIS, Department of Mathematics, University 20 Août 1955- Skikda, 21000 Skikda, Algeria
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Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1 Ahmed Benbella, Oran, Algeria
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weak-viscoelasatic terms. While there is no novelty in the idea of inter-relation between the problems of thermoelastic and of viscoelastic in the transmission system with 1 − d mixed type I and type I I , nevertheless recent few works in only one part. This research addressed the needs of mathematical physics interests for the thermoelastic transmission problem with weak- infinite memories, those which are acting in the first and second parts (see [2,24,25]). The problem we will consider here, for t > 0, is ⎧ t ⎪ − a u ⎪ ρ u − α (t) μ1 (t − s)u x x (s)ds + β1 θx = 0, x ∈ (−L, 0), ⎪ 1 1 xx 1 ⎪ ⎪ −∞ ⎪ ⎪ − lθ + β u = 0, ⎪ c w x ∈ (−L, 0), ⎪ 1 x x 1 x 1 ⎪ t ⎪ ⎪ ⎪ ⎪ μ2 (t − s)vx x (s)ds + β2 qx = 0, x ∈ (0, L), ρ2 v − a2 vx x − α2 (t) ⎪ ⎪ ⎪ −∞ ⎪ ⎨ x ∈ (0, L), c2 w2 − kw2,x x + β2 vx = 0, (1.1) ⎪ ⎪ ⎪ ⎪ u(0, t) = v(0, t), ⎪ ⎪ ⎪ ⎪ θ (0, t) = q(0, t), ⎪ ⎪ ⎪ ⎪ ⎪ w1 (0, t) = w2 (0, t), ⎪ ⎪ ⎪ ⎪ lθx (0, t) = kw2,x (0, t), ⎪ ⎩ a1 u x (0, t) − a2 vx (0, t) = β1 θ (0, t) + β2 q(0, t), where u, v are the displacement of the system at time t in (−L, 0) and (0, L) and θ, q are respectively the temperature difference
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