Inertial manifolds for the nonlocal parabolic problem

Abstract

This paper provides an abstract framework for studying inertial manifolds associated with a class of nonlocal parabolic problems. In particular, by suitably modifying the nonlocal term outside the absorbing ball and changing the scale of time, we derive a corresponding spectral gap condition. As applications, we establish the existence of inertial manifolds for two classes of two-dimensional modified nonlocal parabolic equations on a square domain, whose diffusion coefficients depend on the $L^2$-norm of the solution and of its gradient, respectively.