Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Abstract

Consider an evolution family U = (U (t, s))t ≥ s ≥ 0 on a half-line R+ and a semi-linear integral equation u (t) = U (t, s) u (s) + ∫st U (t, ξ) f (ξ, u (ξ)) d ξ. We prove the existence of stable manifolds of solutions to this equation in the case that (U (t, s))t ≥ s ≥ 0 has an exponential dichotomy and the nonlinear forcing term f (t, x) satisfies the non-uniform Lipschitz conditions: {norm of matrix} f (t, x1) - f (t, x2) {norm of matrix} ≤ φ (t) {norm of matrix} x1 - x2 {norm of matrix} for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp, q and many other function spaces occurring in interpolation theory. © 2009 Elsevier Inc. All rights reserved.