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Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

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Abstract

We consider the Cauchy problem for the energy critical heat equation

$$\begin{cases}u_t ={\Delta}u+|u|^\frac{4}{n-2}u\;\;\;\text{in}\;\mathbb{R}^n\times(0,T)\\u(\centerdot,0)=u_0\;\;\;\text{in}\;\mathbb{R}^n\end{cases}$$

in dimension n = 5. More precisely we find that for given points q1,q2,...,qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t)-α for α > \(\frac{3}{4}\). The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Manuel del Pino & Monica Musso

  2. Departamento de Ingeniería Matemática-CMM, Universidad de Chile, Santiago, 837-0456, Chile

    Manuel del Pino

  3. Departamento de Matemáticas, Universidad Católica de Chile, Macul, 782-0436, Chile

    Monica Musso

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

    Jun Cheng Wei

Authors
  1. Manuel del Pino
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  2. Monica Musso
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  3. Jun Cheng Wei
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Correspondence to Manuel del Pino.

Additional information

Dedicated to Carlos Kenig on the occasion of his 65th birthday

M. del Pino has been supported by a UK Royal Society Research Professorship and Fondo Basal CMM-Chile.

M. Musso has been partly supported by grants Fondecyt 1160135, Chile. The research of J. Wei is partially supported by NSERC of Canada

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del Pino, M., Musso, M. & Wei, J.C. Type II Blow-up in the 5-dimensional Energy Critical Heat Equation. Acta. Math. Sin.-English Ser. 35, 1027–1042 (2019). https://doi.org/10.1007/s10114-019-8341-5

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  • DOI: https://doi.org/10.1007/s10114-019-8341-5

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