Salim A. Messaoudi

Blow Up in a Nonlinearly Damped Wave Equation

In this paper we consider the nonlinearly damped semilinear wave equation utt – Δu + aut |ut|m – 2 = bu|u|p – 2 associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2].

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

Note on intrinsic decay rates for abstract wave equations with memory

In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g′ + H(g) ⩽ 0, s ⩾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [“Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,” Differential and Integral Equations 6, 507–533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) ∼ sp) and for the full range of admissible parameters p ∈ [1, 2). While such result has been known for certain restrictive ranges of the parameters p ∈ [1, 3/2...

General stability result for viscoelastic wave equations

In this paper, we consider a viscoelastic equation and establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature.

A blow-up result in a Cauchy viscoelastic problem

In this work we consider a Cauchy problem for a nonlinear viscoelastic equation. Under suitable conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.

On the control of solutions of a viscoelastic equation

Abstract In this paper we consider the semilinear viscoelastic equation u tt - Δ u + ∫ 0 t g ( t - τ ) Δ u ( τ ) d τ + | u | γ u = 0 , in a bounded domain, and establish a uniform decay result under weaker conditions on the relaxation function g.

On formation of singularities in one-dimensional nonlinear thermoelasticity

It is well known that smooth motions of nonlinear elastic bodies generally will break down in finite time due to the formation of shock waves. On the other hand, for thermoelastic materials, the conduction of heat provides dissipation that competes with the destabilizing effects of nonlinearity in the elastic response. The work of Coleman & Gurtin [2] on the growth and decay of acceleration waves provides a great deal of insight concerning the interplay between dissipation and nonlinearity in one-dimensional nonlinear thermoelastic bodies. Assuming that the elastic modulus, specific heat, and thermal conductivity are strictly positive, the stress-temperature modulus is nonzero, and that the elastic response is genuinely nonlinear they show that acceleration waves of small initial amplitude decay but waves of large initial amplitude can explode in finite time. In other words, thermal diffusion manages to restrain waves of small amplitudes but nonlinearity in the elastic response is dominant for waves of large amplitudes.

A note on blow up of solutions of a quasilinear heat equation with vanishing initial energy

Abstract In this work we consider an initial boundary value problem related to the equation u t − div |∇u| m−2 ∇u =f(u) and prove, under suitable conditions on f, a blow up result for solutions with vanishing or negative initial energy.

Uniform stabilization of solutions of a nonlinear system of viscoelastic equations

In this article, we are concerned with a system of two coupled viscoelastic equations which describes the interaction between two different fields arising in viscoelasticity. Under weaker conditions on the relaxation functions and for more general forms of nonlinearities, we extend and generalize some existing results, concerning the uniform decay for a single equation, to the case of a system.

Initial inverse problem in heat equation with Bessel operator

Abstract We investigate the inverse problem involving recovery of initial temperature from the information of final temperature profile in a disc. This inverse problem arises when experimental measurements are taken at any given time, and it is desired to calculate the initial profile. We consider the usual heat equation and the hyperbolic heat equation with Bessel operator. An integral representation for the problem is found, from which a formula for initial temperature is derived using Picards criterion and the singular system of the associated operators.