Decay rates for Bresse system with arbitrary nonlinear localized damping

In this paper we consider a vibrating system of Bresse type, in a one-dimensional bounded domain with nonlinear localized damping mechanisms acting in all the three wave equations. We obtain some rates of decay for its solutions with no restrictions around the coefficients as well as the condition of equal-speed wave propagation. A new result concerning an internal observability for the conservative system was also proved in order to reach the asymptotics above.     

Energy decay rate of the thermoelastic Bresse system

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions.      

Decay rates for delayed abstract thermoelastic systems with Cattaneo law

In this paper we consider a thermoelastic type system with Cattaneo’s law and internal time delay. Under a suitable assumption on the weight of delay, we prove that the exponential stability of this system is reduced to an observability estimate for the corresponding uncontrolled system. The proof of the main results uses the methodology introduced by Haraux (Port Math 46:245–258, 1989) and generalized by Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). Illustrating example is given.     

Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials

We prove a theorem about global existence (in time) of the solution to the initial-value problem for a nonliear system of coupled partial differential equations of fourth order describing the thermoelasticity of non-simple materias. We consider such the case of thim system in which some nonlinear coeffcients can depend not only on the temperature and the gradient of displacement and also on the second derivative of displacement. The corresponding global existence theorem has been proved using the L ^p – L ^q time decay estmates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard continuation argument and to continue the local solution to one desired for all t ∈ (0, ∞)     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Cauchy problem for thermoelastic plates

The Cauchy problem for an infinite thermoelastic plate with a non‐homogeneous governing system and homogeneous initial conditions is solved by means of an area potential. This is the first step in the construction of a potential theory for time‐dependent problems for thermoelastic plates, enabling the reduction of various initial‐boundary value problems to their versions for the homogeneous system of equations with homogeneous initial conditions, which, in turn, may then be solved by means of dynamic potentials. Copyright © 2006 John Wiley & Sons, Ltd.     

The Bresse system in thermoelasticity

In this paper, we consider the Bresse system coupled with the Fourier law of heat conduction. We prove that the decay rate of the solution is very slow. In fact, we show that the L²‐norm of the solution decays with the rate of (1 + t)^?1/12 similar to the one obtained for the Timoshenko system. In addition, we found that the wave speed of the first two equations still control the decay rate of the solution with respect to the regularity of the initial data. This seems to be the first result dealing with the behavior of the Cauchy problem in the Bresse–Fourier model. Copyright © 2014 John Wiley & Sons, Ltd.     

Bresse system with infinite memories

In this paper, we consider a one‐dimensional linear Bresse system with infinite memories acting in the three equations of the system. We establish well‐posedness and asymptotic stability results for the system under some conditions imposed into the relaxation functions regardless to the speeds of wave propagations. Copyright © 2014 John Wiley & Sons, Ltd.     

A General Decay Estimate for a Finite Memory Thermoelastic Bresse System

Applications of Mathematics - This work considers a Bresse system with viscoelastic damping on the vertical displacement and heat conduction effect on the shear angle displacement. A general...     

Stability to weak dissipative Bresse system

In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.     

Decay rates for the energy of a singular nonlocal viscoelastic system

This work deals with decay rates for the energy of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. We prove the decay rates for the energy of a singular one‐dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition of relaxation kernels described by the inequality $g_i^{\prime }\left(t\right)\le ?H\left(g_i\left(t\right)\right),\left(i=1,2\right)$ for all t ≥ 0, with H convex.     

Decay rates of the plate equations

We consider the Kirchhoff plate equation and the Bernoulli–Euler plate equation. The energy decay rate in both cases is investigated. Moreover, when we do not have exponential stability in the energy space, we give explicit logarithmic decay estimates valid for regular initial data. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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In this paper, the asymptotic behavior of the weak solution (u_t)_{t\ge0} to the non-local Cauchy problems as stated in (1) is considered. Only using lower bounds of jumping kernel J(x,y) for large |x-y|, it is obtained that \|u_t\|_p\le c(t)\|u_0\|_q with any 1\le q相似文献   

A solution of the steady state thermoelastic equations

Résumé Une solution particulière des equations d'équilibrium thermoélastique est derivée d'une fonction potentielle. Cette solution est combinée avec les solutions connues des equations élastostatiques et appliquée à la solution d'une classe particulière de problèmes aux limites.     

Exact controllability for Bresse system with variable coefficients

This paper is concerned with the internal exact controllability of a generalized Bresse system with variable coefficients, which the controls functions acts in an arbitrarily small subinterval (l₁,l₂) of (0,L). Our computation suggests a minimal time control and a region where the controls are more effective. The variable coefficients can be viewed as a generalization of Laplacian operator. The main result is obtained by applying Hilbert Uniqueness Method proposed by Lions, without using the Holmgren's uniqueness theorem or the hypothesis of equal‐speed waves of propagation. Copyright © 2015 John Wiley & Sons, Ltd.     

On the exponential and polynomial stability for a linear Bresse system

In this paper, we consider a linear one-dimensional Bresse system consisting of three hyperbolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions. Here, we consider that only the longitudinal displacement is damped, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, lack of exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.