Admissibly integral manifolds for semilinear evolution equations

Tom 112 / 2014

Nguyen Thieu Huy, Vu Thi Ngoc Ha Annales Polonici Mathematici 112 (2014), 127-163 MSC: Primary 34C45, 34D09; Secondary 34G20, 35B40. DOI: 10.4064/ap112-2-3

Streszczenie

We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u(t)=U(t,s)u(s)+\int _s^tU(t,\xi )f(\xi ,u(\xi ))\,d\xi $ when the evolution family $(U(t,s))_{t\ge s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term $f$ satisfies the (local or global) $\varphi $-Lipschitz conditions, {\it i.e.,} $\| f(t,x)-f(t,y)\| \le \varphi (t)\| x-y\| $ where $\varphi (t)$ belongs to some classes of admissible function spaces. These manifolds are formed 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 $L_{p,q}$ and many other function spaces occurring in interpolation theory. Our main methods involve the Lyapunov–Perron method, rescaling procedures, and techniques using the admissibility of function spaces.

Autorzy

  • Nguyen Thieu HuyArbeitsgruppe Angewandte Analysis
    Fachbereich Mathematik
    Technische Universität Darmstadt
    Schlossgartenstr. 7
    64289 Darmstadt, Germany
    and
    School of Applied Mathematics and Informatics
    Hanoi University of Science and Technology
    Vien Toan ung dung va Tin hoc
    Dai hoc Bach khoa Hanoi
    1 Dai Co Viet, Hanoi, Vietnam
    e-mail
    e-mail
  • Vu Thi Ngoc HaSchool of Applied Mathematics and Informatics
    Hanoi University of Science and Technology
    Vien Toan ung dung va Tin hoc
    Dai hoc Bach khoa Hanoi
    1 Dai Co Viet, Hanoi, Vietnam
    e-mail
    e-mail

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