Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions

Article type: Research Article

Authors: Shang, Chanyu

Affiliations: Institute of Mathematics, Fudan University, 200433 Shanghai, P. R. China. E-mail: [email protected]

Abstract: This paper deals with the one-dimensional nonlinear thermoviscoelastic system subject to constant temperature boundary conditions, describing phase transitions in shape memory alloys. We shall prove the global existence and uniqueness of the weak solution for initial data (strain, velocity, absolute temperature) (u0, v0, θ0)∈L∞×W01,∞×H1. Furthermore, we investigate the asymptotic behavior as time tends to infinity and establish the following asymptotic properties for the weak solution: As t→∞, v→0 in H1(0, 1), θ→T0 in L∞(0, 1), u(x, t)→u∞(x) a.e., where u∞∈L∞ is a stationary state satisfying f1(u∞)T0+f2(u∞)=0, a.e., in [0, 1]. Here we work in a framework in which u belongs to L∞ to describe phase transitions between different configurations of crystal lattices.

Keywords: shape memory alloys, asymptotic behavior, phase transitions

Journal: Asymptotic Analysis, vol. 55, no. 3-4, pp. 229-251, 2007