Invariant manifolds of admissible classes for semi-linear evolution equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:246

Article

Nguyen Thieu Huy

关键词: Evolution equations;    Integral equations;    Exponential dichotomy;    Admissibility of function spaces;    Local-stable manifolds;    Invariant manifolds of admissible classes;   

DOI  :  10.1016/j.jde.2008.10.010

来源: Elsevier

【 摘 要 】

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) + integral(t)(s) U(t, xi) f (xi, u(xi))d xi. We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of L-p type, the Lorentz spaces L-p,L-q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U(t, s))(t >= s >= 0) has an exponential dichotomy and the nonlinear forcing term f It, x) satisfies the non-uniform Lipschitz conditions: parallel to f (t, x(1)) - f (t, x(2))parallel to <= phi(t)parallel to x(1) - x(2)parallel to for V being a real and positive function which belongs to certain classes of admissible function spaces. (c) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free