Large patterns of elliptic systems in infinite cylinders

We consider systems of elliptic equations ∂²_tu + Δ_ru + γ∂_tu + f(u) = 0, u(t,x) ∈ ℝ^N in unbounded cylinders (t,x) ∈ ℝ × Ω with bounded cross-section Ω ⊂ ℝⁿ and Dirichlet boundary conditions. We establish existence of bounded solutions u(t,x) with non-trivial dependence on t ∈ ℝ, ∂_tu(t,x) ≢ 0. Our main assumptions are dissipativity of the nonlinearity f and the existence of at least two t-independent solutions w₁(x), w₂(x) which solve Δ_xw_j + f(w_j) = 0, j = 1,2. The proof exploits the dynamical systems structure of the equations: solutions can be translated along the axis of the cylinder. We first prove existence and compactness of attractors for the dynamical system induced by this translation. We then compute Conley indices for cross-sectional Galerkin approximations to conclude that the attractor does not consist of only the two solutions w_j(x), j = 1,2. We also prove existence of solutions converging for t → +∞ or t → -∞. If the system possesses a gradient-like structure, in addition, solutions will converge on both sides of the cylinder.