A Mixture With Local Kelvin-Voigt Damping

Abstract

In this paper, we will further explore a theory of viscoelastic mixtures proposed by [Iesan, 2004] and analyzed by [Quintanilla, 2005]. In the first of these papers, the author Iesan introduced a theory for a mixture between a viscoelastic material and an elastic material. In this theory, the dissipation effects are determined by the viscosity of rate type of a constituent and the relative velocity. [Quintanilla, 2005] states the linear equations of the thermomechanical deformations. Then, he studies several qualitative properties such as well-posedness and exponential stability. An important result of the paper of [Quintanilla, 2005] is a proof of exponential decay of solution when only one of the dissipative mechanisms is present in the case of anti-shear deformations. We concentrate our attention on a similar case where only one of the dissipative mechanisms is present and is localized. This means that the dissipation can be zero in a part of the domain, but have positive measure support.