Asymptotic and global analysis of principal eigenvalues for linear time-periodic parabolic systems

The paper is concerned with the effects of the spatio-temporal heterogeneity on the principal eigenvalues of some linear time-periodic parabolic systems. Various asymptotic behaviors of the principal eigenvalue and its monotonicity, as a function of the diffusion rate and frequency, are derived. In particular, some singular behaviors of the principal eigenvalues are characterized when both the diffusion rate and frequency approach zero, with some scalar time-periodic Hamilton-Jacobi equation as the limiting equation. Furthermore, we completely classify the topological structures of the level sets for the principal eigenvalues in the plane of the diffusion rate and frequency. Our results not only generalize the findings in [28] for scalar periodic-parabolic operators, but also reveal more rich global information, for time-periodic parabolic systems, on the dependence of the principal eigenvalues upon the spatio-temporal heterogeneity.