THE TRAJECTORY ATTRACTOR OF A NONLINEAR ELLIPTIC SYSTEM IN A CYLINDRICAL DOMAIN

In the half-cylinder Omega(+) = R(+) x w, w is an element of R(n), we study a second-order system of elliptic equations containing a non-lin ear function f(u, x(0),x') = (f(l),...,f(k)) and right-hand side g(x(0 ), x') = (g(l),...,g(k)), x(0) is an element of R(+), x' is an element of w. If these functions satisfy certain conditions, then it is prove d that the first boundary-value problem for this system has at least o ne solution belonging to the space [H-2,p(loc)(Omega(+))](k), p > n 1. We study the behaviour of the solutions u(x(0),x') of this system a s x(0) --> +infinity. Along with the original system we study the fami ly of systems obtained from it through shifting with respect to to by all h, h greater than or equal to 0. A semigroup {T(h), h greater than or equal to 0}, T(h) u(x(0) .) = u(x(0) + h .) acts on the set of sol utions K+ of these systems of equations. It is proved that this semigr oup has a trajectory attractor A consisting of the solutions v(x(0), x ') in K+ that admit a bounded extension to the entire cylinder Omega = R x w. Solutions u(x(0),x') is an element of K+ are attracted by the attractor h as x(0) --> +infinity. We give a number of applications an d consider some questions of the theory of perturbations of the origin al system of equations.