Stable and unstable manifolds for hyperbolic bi-semigroups

We show the existence of local Lipschitzian stable and unstable manifolds for the ill-posed problem of perturbations of hyperbolic bi-semigroups. We do not assume backward nor forward uniqueness of solutions. We do not use cut-off functions because we do not assume global smallness conditions on the nonlinearities. We introduce what we call dichotomous flows which recovers the symmetry between the past and the future. Thus, we need to prove only a stable manifold theorem. We modify the Conley–McGehee–Moeckel approach to avoid appealing to Wazewski principle and Brouwer degree theory. Hence we allow both the stable and unstable directions to be infinite dimensional. We apply our theorem to the elliptic system uξξ+Δu=g(u,uξ) in an infinite cylinder R×Ω.