Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

Abstract

This paper is concerned with the global existence, exponential stability of solutions and associated nonlinear C0-semigroup as well as the existence of maximal attractors in Hi (i = 1, 2, 4) for a nonlinear one-dimensional thermoviscoelasticity describing a kind of solid-like material. Some new ideas and more delicated estimates are employed to prove the global existence and exponential stability of solutions. The important feature for the existence of maximal attractors in Hi+ (i = 1, 2, 4) is that the metric spaces H1+, H2+ and H4+ we work with are three incomplete metric spaces, as can be seen from the physical constraints, i.e. 𝛉 > 0 and u > 0, with 𝛉 and u being absolute temperature and deformation gradient (strain). For any positive parameters δ1, δ2, …, δ5 verifying some conditions, a sequence of closed subspaces Hiδ ⊂ Hi+ (i = 1, 2, 4) is found, and the existence of maximal attractors in Hiδ (i = 1, 2, 4) is established.