Regular attractor for a non-linear elliptic system in a cylindrical domain

The system of second-order elliptic equations is considered in the half-cylinder {omega}{sub +}=R{sub +}x{omega}, {omega} subset of R{sup n}. Here u=(u{sup 1},...,u{sup k}) is an unknown vector-valued function, a and {gamma} are fixed positive-definite self-adjoint (kxk)-matrices, f and g(t)=g(t,x) are fixed functions. It is proved under certain natural conditions on the matrices a and {gamma}, the non-linear function f, and the right-hand side g that the boundary-value problem (*) has a unique solution in the space W{sup 2,p}{sub loc}({omega}{sub +},R{sup k}), p>(n+1)/2, that is bounded as t{yields}{infinity}. Moreover, it is established that the problem (*) is equivalent in the class of such solutions to an evolution problem in the space of 'initial data' u{sub 0} element of V{sub 0}{identical_to}Tr{sub t=0}W{sup 2,p}{sub loc}({omega}{sub +},R{sup k}). In the potential case (f={nabla}{sub x} P, g(t,x){identical_to}g(x)) it is shown that the semigroup S{sub t}:V{sub 0}{yields}V{sub 0} generated by (*) possesses an attractor in the space V{sub 0} which is generically the union of finite-dimensional unstable manifolds M{sup +}(z{sub i}) corresponding to the equilibria z{sub i} of S{sub t} (S{sub t}z{sub i}=z{sub i}). In addition, an explicit formula for the dimensions of these manifolds is obtained.