Periodic motions of species competition flows and inertial manifolds around them with nonautonomous diffusion

Abstract

Motivated by the competition model of two species with nonautonomous diffusion, we consider fully nonautonomous parabolic evolution equation of the form \(\frac{\textrm{d}u}{\textrm{d}t} + A(t)u(t) = f(t,u)+g(t)\) in which the time-dependent family of linear partial differential operator A(t), the nonlinear term f(t, u), and the external force g is 1-periodic with respect to t. We prove the existence and uniqueness of a periodic solution of the above equation and study the inertial manifold for the solutions nearby that solution. We prove the existence of such an inertial manifold in the cases that the family of linear partial differential operators \((A(t))_{t\in \mathbb {R}}\) generates an evolution family \((U(t,s))_{t\ge s}\) satisfying certain dichotomy estimates, and the nonlinear term f(t, x) satisfies the \(\varphi \)-Lipschitz condition, i.e., \(\left\| f(t,x_1)-f(t,x_2)\right\| \leqslant \varphi (t)\left\| A(t)^{\theta } (x_1-x_2)\right\| \) where \(\varphi (\cdot )\) belongs to some admissible function space on the whole line. Then, we apply our abstract results to the above-mentioned competition model of two species with nonautonomous diffusion.