2018
Том 70
№ 12

# Integral manifolds for semilinear evolution equations and admissibility of function spaces

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.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 6, pp 881-911.

Citation Example: Hà Phi, Nguyễn Thiếu Huy, Vụ Thì Ngọc Hà Integral manifolds for semilinear evolution equations and admissibility of function spaces // Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 772-796.

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