Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction

2016 ◽  
Vol 23 (3) ◽  
pp. 435-446 ◽  
Author(s):  
Abdelkader Saadallah ◽  
Hamid Benseridi ◽  
Mourad Dilmi ◽  
Salah Drabla
Keyword(s):  
Asymptotic Behavior ◽  
Weak Solution ◽  
Thin Layer ◽  
Existence And Uniqueness ◽  
Limit Problem ◽  
Dynamical Problem ◽  
Asymptotic Convergence ◽  
Nonlinear Friction ◽  
Layer Parameter ◽  
Lamé Coefficients

AbstractIn this paper, we are interested in the study of the asymptotic analysis of a dynamical problem in elasticity with nonlinear friction of Tresca type. The Lamé coefficients of a thin layer are assumed to vary with respect to the thin layer parameter ε and to depend on the temperature. We prove the existence and uniqueness of a weak solution for the limit problem. The proof is carried out by the use of the asymptotic behavior when the dimension of the domain tends to zero.

scholarly journals Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

2020 ◽  
Vol 18 (1) ◽  
pp. 1302-1316
Author(s):  
Guobing Fan ◽  
Zhifeng Yang
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Optimal Solution ◽  
Dispersive Equation ◽  
Weak Dissipation ◽  
Formula Type

Abstract In this paper, we investigate the problem for optimal control of a viscous generalized \theta -type dispersive equation (VG \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

A piezoelectric contact problem with slip dependent friction and damage

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abderrezak Kasri
Keyword(s):  
Weak Solution ◽  
Contact Problem ◽  
Dry Friction ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Normal Compliance ◽  
Electrically Conductive ◽  
Coulomb’S Law ◽  
Electrical Condition ◽  
Coulomb's Law

Abstract The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb’s law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

2021 ◽  
pp. 1-18
Author(s):  
Yuming Qin ◽  
Bin Yang
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Force Term ◽  
Pullback Attractors ◽  
Critical Exponential Growth ◽  
Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

scholarly journals Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Colored Noise ◽  
Unbounded Domains ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Natural Energy ◽  
Random Attractors ◽  
Supercritical Nonlinearity

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

scholarly journals Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Soumia Manaa ◽  
Salah Boulaaras ◽  
Hamid Benseridi ◽  
Mourad Dilmi ◽  
Sultan Alodhaibi
Keyword(s):  
Fluid Flow ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Limit Problem ◽  
Brinkman Equation ◽  
Asymptotic Convergence ◽  
Thin Domain ◽  
Uniqueness Of The Solution

In this paper, we consider the Brinkman equation in the three-dimensional thin domain ℚ ε ⊂ ℝ 3 . The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution u ε , p ε independently of the parameter ε . Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

scholarly journals Weighted fourth order elliptic problems in the unit ball

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Asymptotic Behavior ◽  
Singular Point ◽  
Unit Ball ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Positive Radial Solutions

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">\begin{document}$ B $\end{document}</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">\begin{document}$ x = 0 \in B $\end{document}</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

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