A semilinear parabolic system arising in the theory of superconductivity

arises in the theory of superconductivity of liquids proposed by Abrahams and Tsuneto in [ 11. System (1) has also been briefly treated as an example by Chueh, Conley and Smaller in (61. Throughout this paper we shall assume that D is a bounded domain in R” (n = 1,2 or 3) with smooth boundary X!. A denotes the Laplacian; all the results of the paper which hold for n > 1 also hold when A is replaced by any second-order symmetric elliptic differential expression with positive spectrum. Because of the very symmetric nature of (l), it is reasonable to expect solutions of (1) to have nice properties. We show in this paper that this is in fact the case. In Section 2 we give a complete description of steady state solutions of (1) satisfying Dirichlet or Neumann boundary conditions; we show that often all steady states are of the form u = u, t‘ = au for some constant a where u is the solution of a single differential equation. It is possible, using standard techniques for scalar nonlinear boundary value problems, to obtain lots of information about the solutions of this single equation and so about solutions of (1). In [6 1 it is shown that all solutions of (1) are attracted towards the set {(u, v): u* + v* < 1). In Section 3 we show that (1) has a Lyapunov function and use this fact and the standard theory of w-limit sets to prove that any solution of (1) tends, as t + 03, towards the family of all steady state solutions of (1). Often, in the case of scalar equations, it can be shown that