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

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^∞×W₀^1,∞×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.