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 setvalued 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.09619
 arXiv:
 arXiv:2406.09619
 Bibcode:
 2024arXiv240609619K
 Keywords:

 Mathematics  Analysis of PDEs