LARGE PATTERNS OF ELLIPTIC-SYSTEMS IN INFINITE CYLINDERS

We consider systems of elliptic equations partial derivative(t)(2)u Delta(x)u + gamma partial derivative(t)u + f(u) = 0, u(t,x) is an elem ent of R-N in unbounded cylinders (t,x) is an element of R x Omega wit h bounded cross-section Omega subset of R-n and Dirichlet boundary con ditions. We establish existence of bounded solutions u(t, x) with non- trivial dependence on t is an element of R, partial derivative(t)u(t,x ) not equivalent to 0. Our main assumptions are dissipativity of the n onlinearity f and the existence of at least two t-independent solution s w(1)(x), w(2)(x) which solve Delta(x)w(j) + f(w(j)) = 0, j = 1,2. Th e proof exploits the dynamical systems structure of the equations: sol utions can be translated along the axis of the cylinder. We first prov e existence and compactness of attractors for the dynamical system ind uced by this translation. We then compute Conley indices for cross-sec tional Galerkin approximations to conclude that the attractor does not consist of only the two solutions w(j)(x): j = 1,2. We also prove exi stence of solutions converging for t --> +infinity or t --> -infinity. If the system possesses a gradient-like structure, in addition, solut ions will converge on both sides of the cylinder. (C) Elsevier, Paris.