scholarly journals General and optimal decay for a quasilinear parabolic viscoelastic system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Abderrahmane Youkana ◽  
Salim A. Messaoudi
Keyword(s):  
Convex Function ◽  
Decay Rate ◽  
General Assumption ◽  
Nonincreasing Function ◽  
Viscoelastic System ◽  
General Decay Rate ◽  
Relaxation Functions ◽  
Optimal Decay ◽  
Stability Results ◽  
Quasilinear Parabolic

<p style='text-indent:20px;'>In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying <inline-formula><tex-math id="M1">\begin{document}$ g'(t) \leq - \xi(t) H(g(t)) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ H $\end{document}</tex-math></inline-formula> is an increasing, convex function and <inline-formula><tex-math id="M3">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when <inline-formula><tex-math id="M4">\begin{document}$ H(t) = t^p, \ t\geq 0, \forall p&gt;1 $\end{document}</tex-math></inline-formula>, instead the parameter <inline-formula><tex-math id="M5">\begin{document}$ p \in [1, \frac{3}{2}[ $\end{document}</tex-math></inline-formula>.</p>

scholarly journals General decay rate of a weakly dissipative viscoelastic equation with a general damping

2020 ◽  
Vol 40 (6) ◽  
pp. 647-666
Author(s):  
Khaleel Anaya ◽  
Salim A. Messaoudi
Keyword(s):  
Decay Rate ◽  
Wide Class ◽  
Nonlinear Damping ◽  
Numerical Results ◽  
General Decay ◽  
General Decay Rate ◽  
Relaxation Functions ◽  
Viscoelastic Equation

In this paper, we consider a weakly dissipative viscoelastic equation with a nonlinear damping. A general decay rate is proved for a wide class of relaxation functions. To support our theoretical findings, some numerical results are provided.

scholarly journals Stability results of some first order viscous hyperbolic systems

2019 ◽  
Vol 25 ◽  
pp. 33
Author(s):  
Serge Nicaise
Keyword(s):  
Spectral Analysis ◽  
Decay Rate ◽  
Exponential Decay ◽  
Hyperbolic Systems ◽  
Natural Assumption ◽  
Abstract Problem ◽  
First Order ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Stability Results

In this paper, we first introduce an abstract viscous hyperbolic problem for which we prove exponential decay under appropriated assumptions. We then give some illustrative examples, like the linearized viscous Saint-Venant system. In order to achieve the optimal decay rate, we also perform a detailed spectral analysis of our abstract problem under a natural assumption satisfied by various examples. We finally consider the boundary stabilizability of the linearized viscous Saint-Venant system on trees.

scholarly journals GENERAL DECAY OF SOLUTION FOR COUPLED SYSTEM OF VISCOELASTIC WAVE EQUATIONS OF KIRCHHOFF TYPE WITH DENSITY IN Rn

2017 ◽  
pp. 1073
Author(s):  
Abbes Benaissa ◽  
Abderrahmane Beniani ◽  
Khaled Zennir
Keyword(s):  
Decay Rate ◽  
Density Function ◽  
Wave Equations ◽  
Coupled System ◽  
General Decay ◽  
General Decay Rate ◽  
Kirchhoff Type ◽  
Relaxation Functions ◽  
Viscoelastic Wave

A system of viscoelastic wave equations of Kirchhoff type is considered. For a wider class of relaxation functions, we use spaces weighted by the density function to establish a very general decay rate of the solution.

scholarly journals Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Baowei Feng ◽  
Mostafa Zahri
Keyword(s):  
Convex Function ◽  
Decay Rate ◽  
Relaxation Function ◽  
Energy Decay ◽  
Kirchhoff Plate ◽  
Nonlinear Viscoelastic ◽  
Optimal Decay ◽  
General Energy ◽  
Optimal Decay Rate

This paper is concerned with a nonlinear viscoelastic Kirchhoff plate uttt−σΔuttt+Δ2ut−∫0tgt−sΔ2usds=divF∇ut. By assuming the minimal conditions on the relaxation function g: g′t≤ξtGgt, where G is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for Gt=tp with the range p∈1, 2, which improves earlier decay results with the range p∈1,3/2. At last, we give some numerical illustrations and related comparisons.

scholarly journals New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammad M. Al-Gharabli ◽  
Adel M. Al-Mahdi ◽  
Salim A. Messaoudi
Keyword(s):  
Wide Class ◽  
Relaxation Function ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
General Assumption ◽  
General Decay ◽  
Source Terms ◽  
Relaxation Functions ◽  
Viscoelastic Equations ◽  
General Source

Abstract This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function $k_{i}$ k i , namely, $$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$ k i ′ ( t ) ≤ − ξ i ( t ) Ψ i ( k i ( t ) ) , i = 1 , 2 . We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when $k_{i}(s) = s^{p}$ k i ( s ) = s p and p covers the full admissible range $[1, 2)$ [ 1 , 2 ) .

On the global polynomial stabilization and observation with optimal decay rate

2021 ◽  
Vol 153 ◽  
pp. 111447
Author(s):  
Chaker Jammazi ◽  
Mohamed Boutayeb ◽  
Ghada Bouamaied
Keyword(s):  
Decay Rate ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Polynomial Stabilization

scholarly journals Asymptotic behavior for a viscoelastic Kirchhoff equation with distributed delay and Balakrishnan–Taylor damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdelbaki Choucha ◽  
Salah Boulaaras
Keyword(s):  
Asymptotic Behavior ◽  
Decay Rate ◽  
Energy Method ◽  
Distributed Delay ◽  
Type Equation ◽  
General Decay ◽  
Kirchhoff Equation ◽  
General Decay Rate ◽  
Kirchhoff Type ◽  
Nonlinear Viscoelastic

AbstractA nonlinear viscoelastic Kirchhoff-type equation with Balakrishnan–Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

scholarly journals General Decay Rate of Solution for Love-Equation with Past History and Absorption

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1632
Author(s):  
Khaled Zennir ◽  
Mohamad Biomy
Keyword(s):  
Decay Rate ◽  
Source Term ◽  
Blow Up ◽  
Past History ◽  
Point Of View ◽  
General Decay ◽  
General Decay Rate ◽  
Infinite Memory ◽  
New Class ◽  
The Relationship

In the present paper, we consider an important problem from the point of view of application in sciences and engineering, namely, a new class of nonlinear Love-equation with infinite memory in the presence of source term that takes general nonlinearity form. New minimal conditions on the relaxation function and the relationship between the weights of source term are used to show a very general decay rate for solution by certain properties of convex functions combined with some estimates. Investigations on the propagation of surface waves of Love-type have been made by many authors in different models and many attempts to solve Love’s equation have been performed, in view of its wide applicability. To our knowledge, there are no decay results for damped equations of Love waves or Love type waves. However, the existence of solution or blow up results, with different boundary conditions, have been extensively studied by many authors. Our interest in this paper arose in the first place in consequence of a query for a new decay rate, which is related to those for infinite memory ϖ in the third section. We found that the system energy decreased according to a very general rate that includes all previous results.

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