A Characterization of backward bounded solutions

TL;DR

This paper characterizes backward bounded solutions for semilinear evolution equations, showing their structure as a graph of an upper hemicontinuous set-valued function and establishing the existence of a finite-dimensional Lipschitz manifold limit.

Contribution

It introduces a novel characterization of backward bounded solutions as a graph of an upper hemicontinuous set-valued function without spectral gap assumptions.

Findings

Backward bounded solutions form an invariant graph.

Existence of a finite-dimensional Lipschitz manifold limit.

The manifold limit is contained within the set 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.