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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2412.09876 |
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| _version_ | 1866929628638085120 |
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| author | Barker, Tobias Miura, Hideyuki Takahashi, Jin |
| author_facet | Barker, Tobias Miura, Hideyuki Takahashi, Jin |
| contents | We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded $L^{n(p-1)/2,\infty}(\mathbf{R}^n)$-norm up to the blow-up time. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative $\varepsilon$-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09876 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical norm blow-up rates for the energy supercritical nonlinear heat equation Barker, Tobias Miura, Hideyuki Takahashi, Jin Analysis of PDEs 35K58 We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded $L^{n(p-1)/2,\infty}(\mathbf{R}^n)$-norm up to the blow-up time. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative $\varepsilon$-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations. |
| title | Critical norm blow-up rates for the energy supercritical nonlinear heat equation |
| topic | Analysis of PDEs 35K58 |
| url | https://arxiv.org/abs/2412.09876 |