Global attractor for the Ginzburg–Landau thermoviscoelastic systems with hinged boundary conditions

This paper is concerned with the following one-dimensional nonlinear system of equations:equation(0.1)utt−P(θ,ε)x−νuxxt+Ruxxxx=f,utt−P(θ,ε)x−νuxxt+Ruxxxx=f,equation(0.2)CVθt−κθxx−θPθεt−νεt2=g, where both ν and R are positive constants. The corresponding free energy density is assumed to be in Ginzburg–Landau form and nonconvex as a function of the order parameter. Results concerning the existence and uniqueness of the global solution, the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit are obtained. Furthermore, we investigate dynamics of the system and prove the existence of global attractor.