Stable and unstable manifolds for bisemigroups

Thursday, June 30, 2011 - 3:00pm - 3:30pm
Mohamed ElBialy (University of Toledo)
We show the existence of local Lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bisemigroups.
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 \tit{Conley-McGehee} 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 illustrate  our theorem by a simple example, namely  the elliptic system  $u\xixid  +  \gD u  = g(u,  u\xid)$
in an infinite cylinder  $\mbbR\times  \gO$.