Admissibly integral manifolds for semilinear evolution equations

Nguyen Thieu Huy, and Vu Thi Ngoc Ha. "Admissibly integral manifolds for semilinear evolution equations." Annales Polonici Mathematici 112.2 (2014): 127-163. <http://eudml.org/doc/280415>.

@article{NguyenThieuHuy2014,
abstract = {We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u(t) = U(t,s)u(s) + ∫_s^t U(t,ξ)f(ξ,u(ξ))dξ$ when the evolution family $(U(t,s))_\{t≥s\}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(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.},
author = {Nguyen Thieu Huy, Vu Thi Ngoc Ha},
journal = {Annales Polonici Mathematici},
keywords = {exponential trichotomy; exponential dichotomy; semilinear evolution equations; admissibility of function spaces; admissibly stable and unstable manifolds; center-stable and center-unstable manifolds; admissibly integral manifold},
language = {eng},
number = {2},
pages = {127-163},
title = {Admissibly integral manifolds for semilinear evolution equations},
url = {http://eudml.org/doc/280415},
volume = {112},
year = {2014},
}

TY - JOUR
AU - Nguyen Thieu Huy
AU - Vu Thi Ngoc Ha
TI - Admissibly integral manifolds for semilinear evolution equations
JO - Annales Polonici Mathematici
PY - 2014
VL - 112
IS - 2
SP - 127
EP - 163
AB - We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u(t) = U(t,s)u(s) + ∫_s^t U(t,ξ)f(ξ,u(ξ))dξ$ when the evolution family $(U(t,s))_{t≥s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(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.
LA - eng
KW - exponential trichotomy; exponential dichotomy; semilinear evolution equations; admissibility of function spaces; admissibly stable and unstable manifolds; center-stable and center-unstable manifolds; admissibly integral manifold
UR - http://eudml.org/doc/280415
ER -