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In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function.

We consider a Timoshenko system coupled with heat equations modelled by Cattaneo's law. The coupling is through the transverse displacement. Both ends of the beam are dynamic. One end of the beam i...

In this paper, we study the energy decay for two one-dimensional thermoelastic Bresse-type systems in a bounded open interval under mixed homogeneous Dirichlet–Neumann boundary conditions and with ...

The subject of the present paper is to study the stability of a class of laminated Timoshenko-type systems in the whole line $$\mathbb {R}$$ combined with a heat conduction given by Gurtin–Pipkin’s law and acting only on one equation of the laminated Timoshenko-type system. The main result of this paper shows that the thermoelastic dissipation generated by Gurtin–Pipkin’s law is strong enough to stabilize the system at least polynomially, even if only the second or the third equation of the laminated Timoshenko-type system is controlled and the two other ones are free. When only the first equation of the laminated Timoshenko-type system is controlled, we give a necessary and sufficient condition for the polynomial stability. The polynomial decays in the $$L^2$$ -norm of the solution, and its higher-order derivatives with respect to the space variable are specified in terms of the regularity of the initial data and some connections between the coefficients. An application to the particular case of Timoshenko-type systems is also given. The proofs are based on the energy method and Fourier analysis combined with some well-chosen weight functions.

International audience; We consider a quasilinear heat system in the presence of an integral term and establish a general and optimal decay result from which improves and generalizes several stability results in the literature.

We study in this paper the well-posedness and stability of three structures with interfacial slip and two infinite memories effective on the transverse displacement and the rotation angle. We consider a large class of kernels and prove that the system has a unique solution satisfying some regularity properties. Moreover, without restrictions on the values of the parameters, we show that the solution goes to zero at infinity and give an information on its speed of convergence in terms of the growth of kernels at infinity. A numerical analysis of the obtained theoretical results will be also given.

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$ with Dirichlet boundary conditions, where g is a positive nonincreasing function satisfying, for some nonnegative functions $$\xi $$ and G, $$\begin{aligned}g'(t)\le -\xi (t)G(g(t)),\qquad \forall t\ge 0.\end{aligned}$$ Under appropriate conditions on $$\xi $$ and G, we establish some new decay results that generalize and improve many earlier results in the literature such as Mustafa (Math Methods Appl Sci 41(1): 192–204, 2018), Messaoudi et al. (J Integral Equ Appl 30(1): 117–145, 2018) and Guesmia (Math Model Anal 25(3): 351–373, 2020). We consider the equal speeds of propagation case, as well as the nonequal-speed case. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature.

We study in this paper the well-posedness and stability of a linear system of a thermoelastic Cosserat medium with infinite memory, where the Cosserat medium is a continuum in which each point has the degrees of freedom of a rigid body.

The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on one component modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy the same polynomial stability estimates in the L2 -norm of the solution and its higher order derivatives with respect of the space variable. The decay rate depends on the regularity of the initial data. In addition, the presence and absence of the regularity-loss type property are determined by some relations between the parameters of systems. However, when the thermal effect is acting via the first comoponent of the system (transversal displacement), a new stability condition is introduced for both TimoshenkoFourier and Timoshenko-Cattaneo systems. This stability condition is in the form of threshold between polynomial stability and convergence to zero. To prove our results, we use the energy method in Fourier space combined with judicious choices of weight functions to build appropriate Lyapunov functionals., Comment: arXiv admin note: substantial text overlap with arXiv:2102.01735

International audience; In this paper, we consider a linear one-dimensional thermoelastic Bresse system with second sound consisting of three hyperbolic equations and two parabolic equations coupled in a certain manner under mixed homogeneous Dirichlet–Neumann boundary conditions, where the heat conduction is given by Cattaneo’s law. Only the longitudinal displacement is damped via the dissipation from the two parabolic equations, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, non exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. Our method of proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.

International audience; 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.

International audience; In this paper, we consider a viscoelastic plate equation with avelocity-dependent material density and a logarithmic nonlinearity. Using theFaedo-Galaerkin approximations and the multiplier method, we establish theexistence of the solutions of the problem and we prove an explicit and gen-eral decay rate result. These results extend and improve many results in theliterature.

International audience; In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han and Wang [26].

International audience; The model under consideration in this paper describes a vibrating structure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval under mixed homogeneous Dirichlet-Neumann boundary conditions. The first two equations are related to Timoshenko-type systems and the third one is subject to the dynamics of the slip. The main problem we discuss here is stabilizing the system by a viscoelastic damping generated by an infinite memory and acting only on one equation. First, we prove the existence, uniqueness and regularity of solutions using the semigroup theory. After that, we combine the energy method and the frequency domain approach to show that the infinite memory is capable alone to guarantee the strong and polynomial stability of the model, that is bringing it back to its equilibrium state with a decay rate of type t −d , where d is a positive constant depending on the regularity of initial data. Moreover, we prove that, when the infinite memory is effective on the first equation, the model is not exponentially stable independently of the values of the parameters. However, when the infinite memory is effective on the second or the third equation, we prove that the exponential stability is equivalent to the equality between the three speeds of wave propagations. An extension of our results to the frictional damping case is also given. Our results improve and extend some existing results in the literature subject to other types of controls.

International audience; In this paper, we consider a vibrating system of Timoshenko-type in a one-dimensional bounded domain with complementary frictional damping and infinite memory acting on the transversal displacement. We show that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case. We establish in each case a general decay estimate of the solutions. In the particular case when the wave propagation speeds are different and the frictional damping is linear, we give a relationship between the smoothness of the initial data and the decay rate of the solutions. By the end of the paper, we discuss some applications to other Timoshenko-type systems.

International audience; In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.

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The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds ofwave propagations are the same, and to the polynomial one with decay rate otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.

International audience; In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain with discrete time delay and complementary frictional damping and infinite memory controls all acting on the transversal displacement. We show that the system is well-posed in the sens of semigroup and that, under appropriate assumptions on the weights of the delay and the history data, the stability of the system holds in case of the equal-speed propagation as well as in the opposite case in spite of the presence of a discrete time delay, where the decay rate of solutions is given in terms of the smoothness of the initial data and the growth of the relaxation kernel at infinity. The results of this paper extend the ones obtained by the present author and Messaoudi in (Acta Math Sci 36:1–33, 2016) to the case of presence of discrete delay.

International audience; The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was proved by Guesmia and Kafini (Math Methods Appl Sci 38:2389-2402, 2015) under three infinite memories, Guesmia and Kirane (Z Angew Math Phys 67:1-39, 2016) under two infinite memories, and De Lima Santos et al. (Q Appl Math 73:23-54, 2015) under one infinite memory acting on the shear angle displacements. The subject of this paper is to complete these results by proving that the asymptotic stability of Bresse systems holds also under one infinite memory acting on the longitudinal displacements.

International audience; In this work, we consider the Lamé system in 3-dimension bounded domain with infinite memories. We prove, under some appropriate assumptions , that this system is well-posed and stable, and we get a general and precise estimate on the convergence of solutions to zero at infinity in terms of the growth of the infinite memories.

In this paper, we consider a Timoshenko system in the presence of an infinite memory, where the relaxation function satisfies a relation of the form Under the same hypothesis on g and ξ imposed for the finite memory case, we establish some general decay results for the equal and nonequal speed propagation cases. Our results improve in some situations some known decay rates. Also, some applications to other problems are discussed. Copyright © 2013 John Wiley & Sons, Ltd.

International audience; In this paper, we consider one-dimensional linear Bresse systems in a bounded open domain under Dirichlet-Neumann-Neumann boundary conditions with two infinite memories acting only on two equations. First, we establish the well-posedness in the sense of semigroup theory. Then, we prove two (uniform and weak) decay estimates depending on the speeds of wave propagations, the smoothness of initial data and the arbitrarily growth at infinity of the two relaxation functions.

International audience; In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain under Dirichlet-Dirichlet or Dirichlet-Neumann boundary conditions with one or two discrete time delays and one or two internal frictional dampings. First, we show that the system is well posed in the sens of semigroup theory. Second, we prove the exponential stability regardless to the speeds of wave propagation of the system if the weights of the time delays are smaller than the ones of the corresponding dampings, respectively. However, when the weight of one time delay is not smaller than the one of the corresponding damping, we prove the exponential stability in case of equal-speed wave propagation, and the polynomial stability in the opposite case.

This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247–263, 2008) and Liu (Nonlinear Anal. 71:2257–2267, 2009) in which only the exponential and polynomial decay rates are considered.

International audience; In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.

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.

International audience; In this paper, we consider coupled wave-wave, Petrovsky-Petrovsky and wave-Petrovsky systems in N-dimensional open bounded domain with complementary frictional damping and infinite memory acting on the first equation. We prove that these systems are well- posed in the sense of semigroups theory and provide a weak stability estimate of solutions, where the decay rate is given in terms of the general growth of the convolution kernel at infinity and the arbitrary regularity of the initial data. We finish our paper by considering the uncoupled wave and Petrovsky equations with complementary frictional damping and infinite memory, and showing a strong stability estimate depending only on the general growth of the convolution kernel at infinity.