Boundary value problems for partial differential equations with exponential dichotomies (Q1201108)

Evolution problems of the type \[ {\partial \over \partial t} w = A(t)w + H(t,x,w) \] are considered, where \(w\) is a \(\mathbb{C}^ n\) valued function of \(x\) and \(t\), \(x \in T^ m\), \(t \in [t_ 0,t_ 1]\), \(H\) is some sufficiently smooth function of its arguments, \(A\) is a linear differential operator in \(x\) with time dependent coefficients. The notion of exponential dichotomies is extended to partial differential evolution equations on the \(n\)-torus. There are given some geometric criteria for the existence of solutions to certain nonlinear Dirichlet boundary value problems.