Global integral manifolds with exponential tracking for nonautonomous equations

Nonautonomous evolution equations of the form du/dt + Au = F(u, t) in a Hilbert space are studied. Here A is a linear operator semibounded below and F(u, t) satisfies the global Lipschitz condition with respect to u. The existence of a finite-dimensional integral manifold with exponential tracking is proved under the assumption that the spectral gap is sufficiently wide. The results are applied to a reaction-diffusion system with time-dependent nonlinear interaction function and an external force.