Abstract
We consider positive solutions for the fractional heat equation with critical exponent
where \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^n\), \(n > 4s\), \(s\in (0, 1)\), \(u:{\mathbb {R}}^n\times [0, \infty )\rightarrow {\mathbb {R}}\) and \(u_0\) is a positive smooth initial datum with \(u_0|_{{\mathbb {R}}^n{\setminus } \Omega } = 0\). We prove the existence of \(u_0\) such that the solution blows up precisely at prescribed distinct points \(q_1,\ldots , q_k\) in \(\Omega \) as \(t\rightarrow +\infty \). The main ingredient of the proofs is a new inner–outer gluing scheme for the fractional parabolic problems.
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Acknowledgements
J. Wei is partially supported by NSERC of Canada, Y. Zheng is partially supported by NSF of China (11301374) and China Scholarship Council (CSC).
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Communicated by Y. Giga.
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Musso, M., Sire, Y., Wei, J. et al. Infinite time blow-up for the fractional heat equation with critical exponent. Math. Ann. 375, 361–424 (2019). https://doi.org/10.1007/s00208-018-1784-7
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DOI: https://doi.org/10.1007/s00208-018-1784-7