A Characterization of backward bounded solutions

Abstract

We prove that the collection ${\mathcal{M}}_{-\infty }$ of backward bounded solutions for a semilinear evolution equation is the graph of an upper hemicontinuous set-valued function from the low Fourier modes to the higher Fourier modes, which is invariant and contains the global attractor. We also show that there exists a limit ${\mathcal{M}}_{\infty }$ of finite dimensional Lipschitz manifolds ${\mathcal{M}}_t$ generated by the time $t$-maps ($t>0$) from the flat manifold ${\mathcal{M}}_0$ with the Hausdorff distance and we find ${\mathcal{M}}_{\infty }\subset {\mathcal{M}}_{-\infty }$. No spectral gap conditions are assumed.