Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929207330734080 |
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| author | Kusaba, Ryunosuke Ozawa, Tohru |
| author_facet | Kusaba, Ryunosuke Ozawa, Tohru |
| contents | We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15234 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation Kusaba, Ryunosuke Ozawa, Tohru Analysis of PDEs 35K91, 35B40, 35C20 We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions. |
| title | Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation |
| topic | Analysis of PDEs 35K91, 35B40, 35C20 |
| url | https://arxiv.org/abs/2306.15234 |