Lack of exponential stability to Timoshenko system with viscoelastic Kelvin–Voigt type

We study the Timoshenko systems with a viscoelastic dissipative mechanism of Kelvin–Voigt type. We prove that the model is analytical if and only if the viscoelastic damping is present in both the shear stress and the bending moment. Otherwise, the corresponding semigroup is not exponentially stable no matter the choice of the coefficients. This result is different to all others related to Timoshenko model with partial dissipation, which establish that the system is exponentially stable if and only if the wave speeds are equal. Finally, we show that the solution decays polynomially to zero as \({t^{-1/2}}\) , no matter where the viscoelastic mechanism is effective and that the rate is optimal whenever the initial data are taken on the domain of the infinitesimal operator.     

Existence and optimal decay rates of the compressible non-isentropic Navier–Stokes–Poisson models with external forces

In this paper, we consider the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations with the potential external force. Under the smallness assumption of the external force in some Sobolev space, the existence of the stationary solution is established by solving a nonlinear elliptic system. Next, we show global well-posedness of the initial value problem for the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations, provided the prescribed initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L²$L^2$-decay estimates for the semigroup generated by the linearized equation, we give the optimal L²$L^2$-convergence rates of the solutions toward the stationary solution.     

Global well-posedness and stability of semilinear Mindlin–Timoshenko system

We study long-term behavior of Reissner–Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. This work complements [28] which established local well-posedness of this problem, global well-posedness when damping dominates the sources (in an appropriate sense) and a blow-up in the complementary scenario assuming negative “total” initial energy. The current paper develops the potential well theory for the RMT system: it proves global existence for potential well solutions without restricting the source exponents, derives explicit energy decay rates dependent on the order of the damping exponents, and verifies a blow-up result for positive total initial energy.     

Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

Trajectory and Global Attractors for a Modified Kelvin–Voigt Model

Journal of Applied and Industrial Mathematics - We study the qualitative behavior of weak solutions to an autonomous modified Kelvin–Voigt model on the base of the theory of attractors for...     

Exponential stability for the wave equations with local Kelvin–Voigt damping

We consider the wave equations with local viscoelastic damping distributed around the boundary of a bounded open set We show that the energy of the wave equations goes uniformly and exponentially to zero for all initial data of finite energy. This author supported partially the National Natural Sciences Foundation grant 10271111. Received: February 8, 2005; revised: July 3, 2005     

Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

Weak Solvability of Kelvin–Voigt Model of Thermoviscoelasticity

In this paper we study the weak solvability of an initial-boundary value thermoviscoelasticity problem for one Kelvin–Voigt mathematical model which describes the flow of weakly concentrated aqueous polymer solutions.     

Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping

We consider the KdV–Burgers equation _ut+_uxxx−_uxx+λu+u_ux=0$u_t+u_{xxx}-u_{xx}+\lambda u+u{u}_x=0$ and its linearized version _ut+_uxxx−_uxx+λu=0$u_t+u_{xxx}-u_{xx}+\lambda u=0$ on the whole real line. We investigate their well-posedness their exponential stability when λ is an indefinite damping.     

On permanence of Lotka–Volterra systems with delays and variable intrinsic growth rates

For competitive Lotka–Volterra systems, Ahmad and Lazer’s work [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Annali di Matematica 185 (2006) S47–S67] on total permanence of systems without delays has been extended to delayed systems [Z. Hou, On permanence of all subsystems of competitive Lotka–Volterra systems with delays, Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301]. In this paper, existence and boundedness of nonnegative solutions and permanence are considered for general Lotka–Volterra systems with delays including competitive, cooperative, predator–prey and mixed type systems. First, a condition is established for the existence and boundedness of solutions on a half line. Second, a necessary condition on the limits of the average growth rates is provided for permanence of all subsystems. Then the result for competitive systems is also proved for the general systems by using the same techniques. Just as for competitive systems, the eminent finding is that permanence of the system and all of its subsystems is completely irrelevant to the size and distribution of the delays.     

Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited

This paper is a continuation of our work in Grobbelaar-Van Dalsen (Appl Anal 90:1419–1449, 2011) where we showed the strong stability of models involving the thermoelastic Mindlin–Timoshenko plate equations with second sound. For the case of a plate configuration consisting of a single plate, this was accomplished in radially symmetric domains without applying any mechanical damping mechanism. Further to this result, we establish in this paper the non-exponential stability of the model for a particular configuration under mixed boundary conditions on the shear angle variables and Dirichlet boundary conditions on the displacement and thermal variables when the heat flux is described by Fourier’s law of heat conduction. We also determine the rate of polynomial decay of weak solutions of the model in a radially symmetric region under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables.     

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method. The material behaviour is modelled by using a general Kelvin?CVoigt constitutive law.