Convergence to Equilibrium for Parabolic-Hyperbolic Time-Dependent Ginzburg–Landau–Maxwell Equations

Abstract

We consider a Ginzburg–Landau–Maxwell model which describes the behavior of a two-dimensional superconducting material. The state variables are the complex-valued order parameter $\psi$, the magnetic potential $\tilde{\mathbf{A}}$, and the electric potential $\Phi$. Under the choice of Coulomb (i.e., London) gauge, the resulting system is a parabolic-hyperbolic coupled system of nonlinear partial differential equations subject to suitable boundary and initial conditions. Global well-posedness results were proved in [M. Tsutsumi and H. Kasai, Nonlinear Anal., 37 (1999), pp. 187–216], while the existence of global attractor and exponential attractors was proved in [V. Berti and S. Gatti, Quart. Appl. Math., 64 (2006), pp. 617–639]. In this paper we use an extended Łojasiewicz–Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect...