Belkacem Said-Houari

Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneos law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright

A stability result of a Timoshenko system with a delay term in the internal feedback

Abstract In this paper, we consider a Timoshenko system with a delay term in the feedback and prove a stability result. The beam is clamped at the endpoints and has, in addition to an internal damping, a feedback with a delay. Under an appropriate assumption on the weights of the two feedbacks, we prove the well-posedness of the system and establish an exponential decay result for the case of equal-speed wave propagation.

Decay rates and global existence for semilinear dissipative Timoshenko systems

The main goal of this paper is to prove optimal decay estimates for the dissipative Timoshenko system in the one-dimensional whole space, and to prove a global existence theorem for semilinear systems. More precisely, if we restrict the initial data ((φ0,ψ0),(φ1,ψ1)) ∈ ( Hs+1 ( RN ) ∩L1,γ ( RN )) × ( Hs ( RN ) ∩L1,γ ( RN )) with γ ∈ [0,1], then we can derive faster decay estimates than those given in [8]. Then, we use these decay estimates of the linear problem combined with the weighted energy method introduced by Todorova and Yordanov [35] with the special weight given in [11], to tackle a semilinear problem.

Stability result of the Timoshenko system with delay and boundary feedback

Our interest in this paper is to analyse the asymptotic behaviour of a Timoshenko beam system together with two boundary controls, with delay terms in the first and second equation. Assuming the weights of the delay are small enough, we show that the system is well-posed using the semigroup theory. Furthermore, we introduce a Lyapunov functional that gives the exponential decay of the total energy.

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

Abstract In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.

Exponential decay for solutions to semilinear damped wave equation

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].

Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions related to the Kelvin-Voigt damping and a delay term acting on the boundary. If the weight of the delay term in the feedback is less than the weight of the term without delay or if it is greater under an assumption between the damping factor, and the difference of the two weights, we prove the global existence of the solutions. Under the same assumptions, the exponential stability of the system is proved using an appropriate Lyapunov functional. More precisely, we show that even when the weight of the delay is greater than the weight of the damping in the boundary conditions, the strong damping term still provides exponential stability for the system.

Global existence and decay of solutions of a nonlinear system of wave equations

This work is concerned with a system of two 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 show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.

Global existence and asymptotic behavior for a fractional differential equation

This paper is concerned with the global existence and asymptotic behavior of solutions to an initial boundary value problem of hyperbolic type. We investigate the interaction between a polynomial source and a dissipation of fractional order. This fractional dissipation is defined by a temporal nonlocal term.

Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions

Abstract. The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo–Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.