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We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case