Admissibly integral manifolds for semilinear evolution equations (Q2879367)

The paper is concerned with the semilinear evolution equation of the form NEWLINE\[NEWLINE\displaystyle\frac{dx}{dt}=A(t)x(t)+f(t,x(t)),\,\,\,t\in J,\leqno(E)NEWLINE\]NEWLINE where \(J\) is an interval of \(\mathbb{R}\), \(A(t)\) is an unbounded linear operator on the Banach space \(X\) for every fixed \(t\in J\), and \(f:J\times X\to X\) is a nonlinear operator. By using the Lyapunov-Perron method and some rescaling procedures, the authors establish the existence of stable, unstable and center manifolds of \({\mathcal E}\)-class for equation \((E)\), where the linear part of \((E)\) has an exponential trichotomy on the half-line or on the whole line \(\mathbb{R}\). They assume that the nonlinear term \(f\) is nonuniform Lipschitz continuous, that is, it satisfies the inequality \(\|f(t,x)-f(t,y)\|\leq\varphi(t)\|x-y\|\), where \(\varphi\) is a real positive function which belongs to some admissible function space, such as: functions spaces of \(L_p\) type, the Lorentz spaces \(L_{p,q}\), or other function spaces which occur in the interpolation theory.