Sharp asymptotic of solutions to some nonlocal parabolic equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910369672331264 |
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| author | Banerjee, Agnid Ghosh, Abhishek |
| author_facet | Banerjee, Agnid Ghosh, Abhishek |
| contents | We show that if $u$ solves the fractional parabolic equation $(\partial_t - Δ)^s u = Vu$ in $B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C\left(1+\|V\|_{C^{1}_{(x,t)}}^{1/2s}\right)$. As $s \to 1$, it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_00341 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp asymptotic of solutions to some nonlocal parabolic equations Banerjee, Agnid Ghosh, Abhishek Analysis of PDEs 35A02, 35B60, 35K05 We show that if $u$ solves the fractional parabolic equation $(\partial_t - Δ)^s u = Vu$ in $B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C\left(1+\|V\|_{C^{1}_{(x,t)}}^{1/2s}\right)$. As $s \to 1$, it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument. |
| title | Sharp asymptotic of solutions to some nonlocal parabolic equations |
| topic | Analysis of PDEs 35A02, 35B60, 35K05 |
| url | https://arxiv.org/abs/2306.00341 |