The main goal of this research is the development of a local and global structural stability theory for parabolic partial differential equations and a Floquet theory for time dependent linear parabolic equations. Structural stability was first introduced in the study of ordinary differential equations, in particular for dynamical systems. Many concepts of dynamical systems have been adapted to infinite dimensional systems, especially to partial differential equations which define dynamical systems on non-compact spaces. However, those systems generated by parabolic equations are not reversible. This characteristic creates difficulties not encountered in the study of finite dimensional dynamical systems and new methods must be developed to understand the nature of these systems. A theory for scalar reaction-diffusion equations has recently been obtained. Specific classes of equations such as the Cahn-Hilliard Equation and Phase-Field System will be considered for specific analysis. Other work will study structural stability of flows near periodic solutions for time-periodic parabolic equations. In the same spirit, efforts will be made to carry over the Floquet theory (too technical to describe here) from finite periodic systems to parabolic partial differential equations of similar type.
