Admissibly integral manifolds for semilinear evolution equations

Nguyen Thieu Huy¹ ; Vu Thi Ngoc Ha²

¹ Arbeitsgruppe 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
² 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

Annales Polonici Mathematici, Tome 112 (2014) no. 2, pp. 127-163

Résumé

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.

Détail

DOI : 10.4064/ap112-2-3

Keywords: prove existence integral stable unstable center manifolds admissible classes solutions semilinear integral equation u int evolution family has exponential trichotomy half line whole line nonlinear forcing term satisfies local global varphi lipschitz conditions f varphi x y where varphi belongs classes admissible function spaces these manifolds formed trajectories solutions belonging admissible function spaces which contain wide classes function spaces function spaces type lorentz spaces many other function spaces occurring interpolation theory main methods involve lyapunov perron method rescaling procedures techniques using admissibility function spaces