On the existence and regularity of admissibly inertial manifolds with sectorial operators

Motivated by a predator–prey model with cross-diffusion, we consider the evolution equation of the form where the linear operator is a sectorial operator having a gap in its spectrum. We prove the existence of an admissibly inertial manifold for such an evolution equation in the case of the spectrum of contains an isolated subset which is sufficiently far from the rest, and the nonlinear term f satisfies φ-Lipschitz condition for φ belonging to some admissible space. Next, we will study the regularity of such admissibly inertial manifolds. We then apply the obtained result to the above-mentioned predator–prey model.