Stationary solutions to a Strain-gradient type thermoviscoelastic system

Abstract

In this paper we study a strain-gradient type thermoviscoelastic system. We focus on the stationary states and their dynamical stability. The adiabatic stationary state is formulated as a nonlinear eigenvalue problem with non-local terms associated with the total energy conservation. One of the purposes of this paper is to extend the results obtained in Suzuki-Tasaki [34]. We reveal a unified structure, called semi-dualities, of the thermoviscoelastic system of viscosity-capillarity type with temperature-dependent viscous and elastic moduli. We describe a physical background and outline the thermodynamic derivation of the system. Based on the semi-dual structure we construct a series of general results concerning the stationary states and their stability. The application of these results together with the bifurcation theory allows us to analyze the total set of the stationary solutions in more detail.