Integral manifolds for semilinear evolution equations and admissibility of function spaces

  • Phi Hà Hanoi Univ. Education, Vietnam
  • Thiếu Huy Nguyễn Hanoi Univ. Sci. and Technology, Vietnam; Techn. Univ. Darmstadt, Germany
  • Thì Ngọc Hà Vụ Hanoi Univ. Sci. and Technology, Vietnam

Abstract

We prove the existence of integral (stable, unstable, center) manifolds for the solutions to the semilinear integral equation $u(t) = U(t,s)u(s) + \int^t_s U(t,\xi)f (\xi,u(\xi))d\xi$ in the case where the evolution family $(U(t, s))_{t leq s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz conditions, i.e., $||f (t, x) — f (t, y) \leq \varphi p(t)||x — y||$, where $\varphi (t)$ belongs to some classes of admissible function spaces. Our main method invokes the Lyapunov-Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.
Published
25.06.2012
How to Cite
HàP., NguyễnT. H., and VụT. N. H. “Integral Manifolds for Semilinear Evolution Equations and Admissibility of Function Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 6, June 2012, pp. 772-96, http://umj.imath.kiev.ua/index.php/umj/article/view/2616.
Section
Research articles