Том 71
№ 8

All Issues

Hà Phi

Articles: 1
Article (English)

Integral manifolds for semilinear evolution equations and admissibility of function spaces

Hà Phi, Nguyễn Thiếu Huy, Vụ Thì Ngọc Hà

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 772-796

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.