Admissibly inertial manifolds for a class of semi-linear evolution equations

Abstract

Consider the semi-linear evolution equation du(t)dt+Au(t)=f(t,u). We prove the existence of a new class of inertial manifolds called admissibly inertial manifolds for this equation. These manifolds are constituted by trajectories of the solutions belonging to rescaledly admissible function spaces which contain wide classes of function spaces like weighted Lp-spaces, the Lorentz spaces Lp,q and many other rescaling function spaces occurring in interpolation theory. The existence of these manifolds is obtained in the case that the partial differential operator A is positive definite and self-adjoint with a discrete spectrum, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions on the domain D(Aθ), 0⩽θ<1, i.e., ‖f(t,x)−f(t,y)‖⩽φ(t)‖Aθ(x−y)‖ and ‖f(t,x)‖⩽φ(t)(1+‖Aθx‖) for φ(⋅) being a real and positive function which belongs to certain classes of admissible function spaces.