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Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $

  • * Corresponding author: Manuel del Pino

    * Corresponding author: Manuel del Pino 

The first author has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. The second author has been partly supported by Fondecyt grant 1160135, Chile. The research of the third author is partially supported by NSERC of Canada

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  • We consider the Cauchy problem for the energy critical heat equation

    $ \begin{equation} \left\{ \begin{aligned} u_t & = \Delta u + u^3 {\quad\hbox{in } }\ \mathbb R^4 \times (0, T), \\ u(\cdot, 0) & = u_0 {\quad\hbox{in } } \mathbb R^4. \end{aligned}\right. ~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation} $

    We find that for given points $ q_1, q_2, \ldots, q_k $ and any sufficiently small $ T>0 $ there is an initial condition $ u_0 $ such that the solution $ u(x, t) $ of (1) blows up at exactly those $ k $ points with a type Ⅱ rate, namely larger than $ (T-t)^{-\frac 12} $. In fact $ \|u(\cdot, t)\|_\infty \sim (T-t)^{-1}\log^2(T-t) $. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

    Mathematics Subject Classification: Primary: 35K58; Secondary: 35B40.


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