Admissibly integral manifolds for semilinear evolution equations
Volume 112 / 2014
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)+\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.
