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Infinite time blow-up for the fractional heat equation with critical exponent

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Abstract

We consider positive solutions for the fractional heat equation with critical exponent

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t = -(-\Delta )^{s}u + u^{\frac{n+2s}{n-2s}}&{}\text {in } \Omega \times (0, \infty ),\\ u = 0&{}\text {on } ({\mathbb {R}}^n{\setminus } \Omega )\times (0, \infty ),\\ u(\cdot , 0) = u_0&{}\text {in }{\mathbb {R}}^n, \end{array}\right. } \end{aligned}$$

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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Authors and Affiliations

  1. University of Bath, North Rd., Bath, BA2 7AY, UK

    Monica Musso

  2. Department of Mathematics, Johns Hopkins University, Krieger Hall, Baltimore, MD, 21218, USA

    Yannick Sire

  3. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Juncheng Wei & Yifu Zhou

  4. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Youquan Zheng

Authors
  1. Monica Musso
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  2. Yannick Sire
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Correspondence to Yannick Sire.

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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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