Smooth stable invariant manifolds and arbitrary growth rates (Q844982)

The authors continue their work dealing with invariant manifolds of nonautonomous semilinear differential equations \[ \dot x=A(x)+f(t,x) \] with linear parts satisfying a generalized dichotomy condition. In this paper, the generalized exponential dichotomy may exhibit stable and unstable behavior featuring arbitrary growth rates \(e^{c\rho(t)}\) for some nonnegative function \(\rho\). In turn, one has to require that the nonlinearity \(f\), its derivative \(D_2f\) and the Lipschitz constant of \(D_2f\) are decaying in time like \(e^{-a\rho(t)}\) and an appropriate \(a>0\). Using a variant of the Lyapunov-Perron method, the existence and \(C^1\)-smoothness of a stable integral manifold is deduced. Furthermore, for equivariant systems it is shown that the dynamics on the stable manifold in a certain class of graphs is also equivariant. A two-dimensional example illustrates the paper. It would be interesting to relate the results to further generalized dichotomy notions like for instance [\textit{J. S. Muldowney}, Trans. Am. Math. Soc. 283, 465--484 (1984; Zbl 0559.34049)] or [\textit{R. Naulin} and \textit{M. Pinto}, J. Differ. Equations 118, No.~1, 20--35 (1995; Zbl 0836.34047)].