Attractors for the one‐dimensional Frémond model of shape memory alloys

Article type: Research Article

Authors: Colli, Pierluigi^; | Shirakawa, Ken

Affiliations: Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Italy E‐mail: pier@dimat.unipv.it | Department of Information Environment Design and Integration, School of Information Environment, Tokyo Denki University, Japan E‐mail: sirakawa@sie.dendai.ac.jp

Note: [] Corresponding author.

Abstract: In this paper, we consider a one‐dimensional Frémond model of shape memory alloys. Let us imagine a wire of a shape memory alloy whose left‐hand side is fixed, and assume that forcing terms, e.g., heat sources and external stress on the right‐hand side, converge to some time‐independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we first show the existence of the global attractor for the limiting autonomous dynamical system, for instance the case of zero external stress, and secondly, characterize the asymptotic stability for nonautonomous case by the limiting global attractor.

Journal: Asymptotic Analysis, vol. 40, no. 2, pp. 109-135, 2004