Decay estimate for a viscoelastic plate equation with strong time-varying delay
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cay estimate for a viscoelastic plate equation with strong time-varying delay · Soh Edwin Mukiawa1
Cyril Dennis Enyi1
Received: 18 November 2019 / Accepted: 25 August 2020 © Università degli Studi di Ferrara 2020
Abstract In this paper, we consider a viscoelastic plate equation in the presence of a strong time-varying delay, namely; u tt + 2 u −
t
g(t − s)2 u(s)ds − μ1 u t − μ2 u t (·, t − τ (t)) = 0.
0
We proved a general and optimal decay result for the associated energy functional of the system. The following are novel achievements of the present work: A more general relaxation function g with minimal conditions is used, this allowed us to obtain exponential and polynomial decay rates as special cases. Furthermore, our results hold with lesser restriction √ on the values of μ1 and μ2 as against previous results which assume that |μ2 | < 1 − dμ1 . This work improves and generalizes previous results in the literature. Keywords General decay · Optimal decay · Strong damping · Viscoelastic plate equation Mathematics Subject Classification 35B35 · 35B40 · 93D15
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Cyril Dennis Enyi [email protected] Soh Edwin Mukiawa [email protected]
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College of Sciences, Department of Mathematics, University of Hafr Al Batin, P.O. Box 1803, Hafr Al Batin 31991, Saudi Arabia
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ANNALI DELL’UNIVERSITA’ DI FERRARA
1 Introduction In the present work, we study the following problem; t ⎧ 2 ⎪ ⎪ u + u − g(t − s)2 u(s)ds − μ1 u t − μ2 u t (·, t − τ (t)) = 0, ⎪ tt ⎪ ⎪ 0 ⎨ in × (0, +∞), ⎪ ⎪ ⎪ u| = u| = 0, u| = u , u | = u , 0 t t=0 1 ⎪ t=0 ∂ ⎪ ⎩ ∂ u t (x, t) = f 0 (x, t), t ∈ [−τ (0), 0), x ∈ , where is a bounded domain of Rn , n ≥ 1, with smooth boundary ∂, τ (t) > 0 is the time-varying delay, μ1 , μ2 are constants and g is a given relaxation function. Time delays occur in systems modelling many phenomena in areas not limited to, biosciences, medicine, physics, robotics, economics, chemical, thermal,and structural engineering, these phenomena naturally not only depend on the present state but also on some past history of occurrences. The present problem models the displacement of viscoelastic plates with fixed boundary (that is u|∂ = u|∂ = 0) in the presence of strong damping and strong time-varying delay damping, see also the works in [2,8,25] for the description of plates/beams models related to the present problem. Viscoelastic materials are materials that deform gradually upon application of an external force, but return to their original state once the deforming force is removed. Mechanical properties of materials depend on their rate of deformation, and are in the general sense examined in the light of the relationship that exists between stress and strain (or load-deformation). For example, material stiffness increases according to loading rates. For a deformable elastic structure, if a viscoelastic material is introduced, it changes the Young modulus, the mass density and the damping coefficients. New mathematical challenges are inevitable by this process, thus it attracts the attention of rese
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