Integral manifolds for semilinear evolution equations and admissibility of function spaces (Q1933294)

The authors prove the existence of stable, unstable and center integral manifolds to the integral equation \[ u(t)=U(t,s)u(s)+\int_s ^t U(t,\xi)f(\xi,u(\xi)\,\mathrm{d}\xi\qquad \text{for a.e. }t\geq s,\;t,s\in \mathbb R_+, \] where the evolution familiy \((U(t,s)_{t\geq s\geq 0}\) possesses an exponential trichotomy on \(\mathbb R_+\). Instead of a classical Lipschitz condition of the type \[ \| f(t,x)-f(t,y)\|\leq q\,\| x-y\| \] and sufficiently small \(q\), they assume a \(\phi\)-Lipschitz property \[ \| f(t,x)-f(t,y)\|\leq \phi (t)\,\| x-y\| \] with \(\phi\) belonging to suitable classes of admissible functions, where the smallness of \(q\) is replaced by the smallness of \(\sup_{t\geq 0}\int _t ^{t+1}\phi (\tau)\,\mathrm{d}\tau\).