Long Time Behavior of Solution to Equal Mitosis PDE

In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.