Exponential separation and principal Floquet bundles for linear parabolic equations on RN

We consider linear nonautonomous second order parabolic equations on N. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution - the principal Floquet bundle, the other one consisting of sign-changing solutions. We establish an exponential separation between the two bundles, showing in particular that a class of sign-changing solutions are exponentially dominated by positive solutions.