Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator i: Large advection

We investigate the effect of large advection on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions. Various asymptotic behaviors of the principal eigenvalues, when the advection coefficient approaches infinity, are established, where spatial or temporal degeneracy could occur in the advection term. Our findings substantially improve the results in [R. Peng and X.-Q. Zhao, Calc. Var. Partial Differential Equations, 54 (2015), pp. 1611-1642] for parabolic operators and also extend the existing results in [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 57 (2008), pp. 627-658; R. Peng and M. Zhou, Indiana Univ. Math J., 67 (2018), pp. 2523-2568] for elliptic operators.