Admissible Inertial Manifolds for Delay Equations and Applications to Fisher-Kolmogorov Model

Abstract

For solutions to the delay equations of the form $\dot{u}(t) + Au(t) = f(t,u_{t}), t\in\mathbb{R}$ , we prove the existence of an admissible inertial manifold. Here, the linear differential operator $A$ is positive definite and self-adjoint with a discrete spectrum, and the nonlinear part $f$ is $\varphi$ -Lipschitz for $\varphi$ belonging to an admissible space of functions defined on the whole line. An application to Fisher-Kolmogorov model with time-dependent environmental capacity and finite delay is also given to illustrate our results. Our main method is based on Lyapunov-Perron’s equation in combination with admissibility and duality estimates.