### ON A NONLINEAR PROBLEM MODELLING STATES OF THERMAL EQUILIBRIUM OF SUPERCONDUCTORS

Mohammed Khomssi
2004 Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations   unpublished
Thermal equilibrium states of superconductors are governed by the nonlinear problem i=N i=1 ∂ ∂x i k(u) ∂u ∂x i = λF (u) in Ω , with boundary condition u = 0. Here the domain Ω is an open subset of R N with smooth boundary. The field u represents the thermal state, which we assume is in H 1 0 (Ω). The state u = 0 models the superconductor's state which is the unique physically meaningful solution. In previous works, the superconductor domain is unidirectional while in this paper we consider a
more » ... per we consider a domain with arbitrary geometry. We obtain the following results: A set of criteria that leads to uniqueness of a superconductor state, a study of the existence of normal states and the number of them, and optimal criteria when the geometric dimension is 1. 1. Statement of the problem General model. In the framework of superconductivity, the energy conservation in a physical volume Ω S , having as boundary the closed surface ∂Ω S can be written as ∂ ∂ t Ω S E dv = − Ω S div(− → q) dv + Ω S W dv + Ω S P dv. (1.1) Where the left hand side of the equation is made of the inner quantity of accumulated energy inside Ω S during d t. The first member of the second hand side is the heat flux going by conduction in the closed ∂Ω S , and P is a parasite volume supply of heat of nature, responsible partly, of the thermal perturbation of the environment. The Fourier hypothesis relates the flux density, q at temperature T by q = −K(X, T) · grad T , (1.2) where K is the tensor of thermal conductivity. Note at this stage that, it is well known that the application of first Principle of Thermodynamics Theory to a continuous environment is reduced, without matter transfer to the heat equation. This 2000 Mathematics Subject Classification. 35J60, 34L30, 35Q99.