Attraction Property of Admissible Integral Manifolds and Applications to Fisher-Kolmogorov Model

Abstract

In this paper we investigate the attraction property of an unstable manifold of admissible classes for solutions to the semi-linear evolution equation of the form $u(t) = U(t,s) u(s) + \int_s^t U(t, \xi) f(\xi, u(\xi)) \, d\xi$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces, the Lorentz spaces $L_{p, q}$ and many other function spaces occurring in interpolation theory. We then apply our abstract results to study Fisher-Kolmogorov model with time-dependent environmental capacity.