Exponential stability of semigroups generated by Volterra integro-differential equations

Abstract

We study abstract Volterra integro-differential equations, which are operator models of problems in the viscoelasticity theory. This class includes Gurtin-Pipkin integro-differential equations describing the heat transfer in medias with memory. In particular, as the kernels of integral operators, the sums of decaying exponentials can serve or the sums of Rabotnov functions with positive coefficients having wide applications in the viscoelasticity theory and the theory of heat transfer. The presented results are based on the approach related with studying one-parametric semi-groups for linear evolution equations. We provide a method for reducing the initial problem for a model integro-differential equation with operator coefficients in the Hilbert space to the Cauchy problem for a first order differential equation. We prove results on existing a strongly continuous contracting semigroup generated by a Volterra integro-differential equation with operator coefficients in a Hilbert space. We establish an exponential decay of the semigroup under known assumptions for the kernels of the integral operators. On the base of the obtained results we establish a well-posedness of initial problem for the Volterra integro-differential equation with appropriate estimates for the solution. The proposed approach can be also employed for studying other integro-differential equations involving integral terms of Volterra convolution type.