The purpose of this paper is to investigate the stabilization of a one-dimensional coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combining with a multiplier method, we prove a polynomial energy decay rate of order 1/t.
The Cauchy problem for a nonlinear elastic wave equations with viscoelastic damping terms is considered on the 3 dimensional whole space. Decay and smoothing properties of the solutions are investigated when the initial data are sufficiently small; and asymptotic profiles as $t \to \infty$ are also derived.
In this paper, we consider a viscoelastic kirchhoff equation with a delay term in the internal feedback. By using the Faedo-Galarkin approximation method we prove the well-posedness of the global solutions. Introducing suitable energy, we prove the general uniform decay results.
The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12] satisfies consistency property corresponding to the additional regularity of the initial data. As a result, sharp estimates in $t$ and approximation formulas by the diffusion waves are established.
In this paper, we consider a transmission problem in a bounded domain with a viscoelastic term and a delay term. Under appropriate hypothesis on the relaxation function and the relationship between the weight of the damping and the weight of the delay, we prove the well-posedness result by using Faedo-Galerkin method. By introducing suitable Lyaponov functionals, we establish a general decay result, from which the exponential and polynomial types of decay are only special cases.
In this paper we study a class of semilinear wave type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis initiated in [16] and then developed in [13, 17].
The goal of the present paper is to study the viscoelastic wave equation with the time delay \[ |u_t|^ρu_{tt}-Δu-Δu_{tt}+\int_0^tg(t-s)Δu(s)ds+μ_1u_t(x,t)+μ_2 u_t(x,t-τ)=b|u|^{p-2}u\] under initial boundary value conditions, where $ρ,~b,~μ_1$ are positive constants, $μ_2$ is a real number, $τ>0$ represents the time delay. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of energy are proved under the general assumption on the relaxation function $g$. This work generalizes and improves earlier results on the stabili…
In this paper, we consider a Timoshenko system with a thermo-viscoelastic damping and a delay term in the internal feedback together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ1, μ2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the the weight of the friction damping term, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Then, by introducing appropriate Lyapunov functionals, under the i…
An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued $C^{0}$- and BV-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using the Lyapunov's indirect method. The singular limit $τ\to…
In this paper we extend the asymptotic analysis in [LLOO], performed on a structure consisting of two linearly elastic bodies connected by a thin soft nonlinear Kelvin-Voigt viscoelastic adhesive layer, to the case in which the total mass of the layer remains strictly positive as its thickness tends to zero. We obtain convergence results by means of a nonlinear version of Trotter's theory of approximation of semigroups acting on variable Hilbert spaces. Differently from the limit models derived in [LLOO], in the present analysis the dynamic effects on the surface to which the layer shrinks do …
We consider an elastic/viscoelastic transmission problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined…
We investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type. The main novelty in this paper is that both the damping and the coupling coefficients are non smooth. First, using a general criteria of Arendt-Batty, combined with an uniqueness result, we prove that our system is strongly stable. Next, using a spectrum approach, we prove the non-exponential (uniform) stability of the system. Finally, using a frequency domain approach, combined with a piecewise multiplier technique and the construction of a new multiplier sati…
We propose a new class of phase field models coupled to viscoelasticity with large deformations, obtained from a diffuse interface mixture model composed by a phase with elastic properties and a liquid phase. The model is formulated in the Eulerian configuration and it is derived by imposing the mass balance for the mixture components and the momentum balance that comes from a generalized form of the principle of virtual powers. The latter considers the presence of a system of microforces and microstresses associated to the microscopic interactions between the mixture's constituents together w…
In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x,t)-Δu(x,t)+\int_0^t g(t-s)Δu(x,s)ds +μ_1 u_t(x,t)+ μ_2 u_t(x,t-τ)=0$$ together with initial-boundary conditions of Dirichlet type in $Ω\times (0,+\infty)$, and prove that for arbitrary real numbers $μ_1$ and $μ_2$, the above mentioned problem has a unique global solution under suitable assumptions on the kernel $g$. This improve the results of the previous literature such as [6] and [13] by removing the restrictio…
In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.
We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible \emph{heat-conducting} viscoelastic rate-type fluids with stress-diffusion, subject to a stick-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the extra stress tensor. In the introductory section we develop the thermodynamic foundations of the proposed model, and we document the role of thermodynamics in obtaining critical structural relations between the quantities of interest. These structural relations are then exploite…
In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto-plasticity, and unidirectional rate-dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis reli…
We investigate the stabilization of a multidimensional system of coupled wave equations with only one Kelvin Voigt damping. Using a unique continuation result based on a Carleman estimate and a general criteria of Arendt Batty, we prove the strong stability of the system in the absence of the compactness of the resolvent without any geometric condition. Then, using a spectral analysis, we prove the non uniform stability of the system. Further, using frequency domain approach combined with a multiplier technique, we establish some polynomial stability results by considering different geometric …
We consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin's-Voigt's rheology derived from a three-dimensional model at a finite-strain setting. As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty and complement the $Γ$-convergence analysis of elastic von Kármán ribbons in [Fr…
The goal of this paper is to study a nonlinear viscoelastic wave equation with strong damping, time-varying delay and dynamical boundary condition. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we then prove a general decay result of the energy, from which the usual exponential and polynomial decay rates are only special cases.
We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. The result of this paper mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preven…
We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin-Voigt and the delay damping are both localized via non smooth coefficients. In the case that the Kelvin-Voigt damping is localized faraway from the tip and the wave is subjected to a locally distributed internal or boundary delay feedback, we prove that the energy of the system decays polynomially of type t^{-4}. However, an exponential decay of the e…
We consider the system of equations describing motion of compressible viscoelastic fluids in a whole space. We investigate the large time behavior of solutions around a motionless state, and obtain the $L^p$ decay estimates of solutions for $1<p\leq\infty$, provided that the initial data is sufficiently close to the motionless state. In addition, we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave.
On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type
Charles University, Faculty of Mathematics and Physics（查理大学数学与物理学院）；University of Warsaw, Faculty of Mathematics, Informatics and Mechanics（华沙大学数学、信息学和机械学院）；Institute of Mathematics of Polish Academy of Sciences（波兰科学院数学研究所）
We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial preturbation belongs to $W^{2,1}$, and is sufficiently small in $H^4\cap L^1$, the solutions grow in time at the same rate as $t^{\frac{1}{2}}$ in $L^1$ due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.
We study the asymptotic behaviour of the wave equation with viscoelastic damping in presence of a time-delayed damping. We prove exponential stability if the amplitude of the time delay term is small enough.
Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases
Laboratory of Pure and Applied Mathematics, University of Laghouat（拉格瓦特大学纯数学与应用数学实验室）；Department of Mathematics and Computer Science, Faculty of Science and Technology, University of Ghardaia（加达亚大学科学与技术学院数学与计算机科学系）
A wide and critical comparison of the capability of Method of Characteristics, Explicit Path-Conservative Finite Volume Method and Semi-Implicit Staggered Finite Volume Method is presented and discussed, in terms of accuracy and efficiency. The viscoelastic behaviour of the pipe wall, the effects of the unsteadiness of the flow on the friction losses, cavitation and cross-sectional changes are considered. The analyses are performed comparing numerical solutions obtained using the three models against experimental data and analytical solutions. Water hammer studies in high density polyethylene …
