Advanced Mathematical Methods for Scientists and Engineers

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Numerical Approximation Of Partial Differential Equations

General decay of solutions of a viscoelastic equation

We consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While exponential stability under the Fourier law of heat conduction holds, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier’s law by changing to the Cattaneo law causes a loss of the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property.

/pdf/on-the-stability-of-damped-timoshenko-systems-cattaneo-4dd3c3jltp.pdf

On the stability of damped Timoshenko systems - Cattaneo versus Fourier law

In a bounded domain, we consider u tt - Δ u + ∫ 0 t g ( t - τ ) Δ u d τ = | u | γ u , where γ > 0 , and g is a nonnegative and decaying function. We prove a local existence theorem and show, for certain initial data and suitable conditions on g and γ , that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g .

Existence and decay of solutions of a viscoelastic equation with a nonlinear source

In this paper the nonlinear viscoelastic wave equation



associated with initial and Dirichlet boundary conditions is considered. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p > m. Also the case of a stronger damping is considered and it is showed that solutions exist globally for any initial data, in the appropriate space, provided that m ≥ p.

/pdf/blow-up-and-global-existence-in-a-nonlinear-viscoelastic-1pzlbbuaeb.pdf

Blow up and global existence in a nonlinear viscoelastic wave equation

Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation

In a bounded domain, we consider u t t − Δ u + ∫ 0 t g ( t − τ ) Δ u d τ = | u | γ u , where γ > 0 and g is a nonnegative and decaying function We prove that, for certain class of relaxation functions and certain initial data, the rate of decay of energy is similar to that of g  This result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered

Salim A. Messaoudi

Papers

General decay of solutions of a viscoelastic equation

Existence and decay of solutions of a viscoelastic equation with a nonlinear source

Blow up and global existence in a nonlinear viscoelastic wave equation

Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation

General decay of the solution energy in a viscoelastic equation with a nonlinear source