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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.09410 |
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| _version_ | 1866914249155018752 |
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| author | Hou, Thomas Y. Nguyen, Van Tien Wang, Yixuan |
| author_facet | Hou, Thomas Y. Nguyen, Van Tien Wang, Yixuan |
| contents | We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted $H^k$ stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the $L^2$-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-based stability of blowup with log correction for semilinear heat equation Hou, Thomas Y. Nguyen, Van Tien Wang, Yixuan Analysis of PDEs We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted $H^k$ stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the $L^2$-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting. |
| title | $L^2$-based stability of blowup with log correction for semilinear heat equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.09410 |