A Characterization of backward bounded solutions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929385739649024 |
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| author | Kwak, Minkyu Lee, Jihoon Lkhagvasuren, Bataa |
| author_facet | Kwak, Minkyu Lee, Jihoon Lkhagvasuren, Bataa |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Characterization of backward bounded solutions Kwak, Minkyu Lee, Jihoon Lkhagvasuren, Bataa Analysis of PDEs 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. |
| title | A Characterization of backward bounded solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.09619 |