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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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