Transversality of Homoclinic Orbits and Exponential Dichotomies for Parabolic Equations*

In this paper we discuss the existence of exponential dichotomies on R of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations, and by making use of the results on exponential dichotomies of this paper, we investigate the transversality of homoclinic orbits for parabolic equations.