This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t -> +infinity. Our approach is based on integral inequalities and multiplier techniques. These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system. (C) 2007 Elsevier Inc. All rights reserved

The main purpose of this work is to study the damping effect\ud of memory terms associated with singular convolution kernels on\ud the asymptotic behavior of the solutions of second order evolution\ud equations in Hilbert spaces. For kernels that decay exponentially at\ud infinity and possess strongly positive definite primitives, the exponential\ud stability of weak solutions is obtained in the energy norm.\ud It is also shown that this theory applies to several examples of\ud kernels with possibly variable sign, and to a problem in nonlinear\ud viscoelasticity

We study control problems for some integro-differential equations using the Hilbert Uniqueness Method. To do that we follow a harmonic analysis approach. Our results can be applied to concrete examples in viscoelasticity theory. (C) 2009 Elsevier Inc. All rights reserved

International audienceIn this paper we consider an integrodifferential system, which governs the vibration of a viscoelastic one-dimensional object. We assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time T, large enough. We shall prove this result using moment theory and we shall prove that the solution of this problem leads to identification of a Riesz sequence which solves controllability and observability. The results presented here are constructive and can lead to simple numerical algorithms

The main purpose of this paper is to obtain the existence of global solutions to semilinear integro-differential equations in Hilbert spaces for rather general convolution kernels and nonlinear terms with superlinear growth at infinity. The included application to a nonlinear model of heat flow in materials of fading memory type provides motivations for the abstract theory