F-stability, entropy and energy gap for supercritical Fujita equation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911971140435968 |
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| author | Wang, Kelei Wei, Juncheng Wu, Ke |
| author_facet | Wang, Kelei Wei, Juncheng Wu, Ke |
| contents | We study some problems on self similar solutions to the Fujita equation when $p>(n+2)/(n-2)$, especially, the characterization of constant solutions by the energy. Motivated by recent advances in mean curvature flows, we introduce the notion of $F-$functional, $F$-stability and entropy for solutions of supercritical Fujita equation. Using these tools, we prove that among bounded positive self similar solutions, the constant solution has the lowest entropy. Furthermore, there is also a gap between the entropy of constant and non-constant solutions. As an application of these results, we prove that if $p>(n+2)/(n-2)$, then the blow up set of type I blow up solutions is the union of a $(n-1)-$ rectifiable set and a set of Hausdorff dimension at most $n-3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | F-stability, entropy and energy gap for supercritical Fujita equation Wang, Kelei Wei, Juncheng Wu, Ke Analysis of PDEs Differential Geometry We study some problems on self similar solutions to the Fujita equation when $p>(n+2)/(n-2)$, especially, the characterization of constant solutions by the energy. Motivated by recent advances in mean curvature flows, we introduce the notion of $F-$functional, $F$-stability and entropy for solutions of supercritical Fujita equation. Using these tools, we prove that among bounded positive self similar solutions, the constant solution has the lowest entropy. Furthermore, there is also a gap between the entropy of constant and non-constant solutions. As an application of these results, we prove that if $p>(n+2)/(n-2)$, then the blow up set of type I blow up solutions is the union of a $(n-1)-$ rectifiable set and a set of Hausdorff dimension at most $n-3$. |
| title | F-stability, entropy and energy gap for supercritical Fujita equation |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2407.20035 |