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The purpose of this paper is twofold. First we study a class of Banachmanifolds which are not differentiable in traditional sense but they arequasi-differentiable in the sense that a such Banach manifold has an embeddedsubmanifold such that all points in that submanifold are differentiable andtangent spaces at those points can be defined. It follows that differentialcalculus can be performed in that submanifold and, consequently, differentialequations in a such Banach manifold can be considered. Next we study thestructure of phase diagram near center manifold of a parabolic differentialequation in Banach manifold which is invariant or quasi-invariant under afinite number of mutually quasi-commutative Lie group actions. We prove thatunder certain conditions, near the center manifold $\mathcal{M}_c$ theunderline manifold is a homogeneous fibre bundle over $\mathcal{M}_c$, withfibres being stable manifolds of the differential equation. As an application,asymptotic behavior of the solution of a two-free-surface Hele-Shaw problem isalso studied.

Quasi-differentiable Banach manifold and phase-diagram of invariant parabolic differential equation in such manifold