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Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations. (English) Zbl 1183.35258
A Ginzburg-Landau-Maxwell model of superconductivity is considered in a bounded, 2-dimensional domain \(\Omega\) with smooth boundary \(\partial\Omega\) and the large-time asymptotic behaviour of solutions is investigated. Under the choice of Coulomb gauge the magnetic potential \(A\) and electric potential \(\Phi\) are used to describe the electromagnetic field. In this model a complex-valued function \(\psi\) describes the state of superconductivity, with \(\left|\psi\right|^{2}\) representing the concentration of superconducting electrons. The dynamics of \(\psi\) is coupled with the magnetic potential \(A\) and the dynamics of the potentials \(\Phi\), \(A\) is coupled with \(\psi\), leading to a hyperbolic-parabolic coupling. The main concern of the paper is to show that solutions tend to the equilibrium state as time goes to infinity. A power law decay estimate is given for this convergence to a unique equilibrium solution.

MSC:
35Q56 Ginzburg-Landau equations
35B40 Asymptotic behavior of solutions to PDEs
82D55 Statistical mechanical studies of superconductors
78A25 Electromagnetic theory, general
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