We consider systems of elliptic equations ∂2tu + Δru + γ∂tu + f(u) = 0, u(t,x) ∈ ℝN 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 Δxwj + f(wj) = 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 wj(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.
Bibliographical noteFunding Information:
This work was done during the stay of one of the authors (M. I. Vishik) at the Free University (Berlin) supported by the Alexander von Humboldt Stiftung. The authors would like to thank S. Zelik for useful comments and valuable help in the final editing of this paper.
- Conley index
- Elliptic systems
- Traveling waves