Well-posedness is studied for a special system of two-point boundary value
problem for evolution equations which is called a forward-backward evolution
equation (FBEE, for short). Two approaches are introduced: A decoupling method
with some brief discussions, and a method of continuation with some substantial
discussions. For the latter, we have introduced Lyapunov operators for FBEEs,
whose existence leads to some uniform a priori estimates for the mild solutions
of FBEEs, which will be sufficient for the well-posedness. For some special
cases, Lyapunov operators are constructed. Also, from some given Lyapunov
operators, the corresponding solvable FBEEs are identified.Comment: 52 page
