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arXiv · 2002.05765

Type II Finite time blow-up for the three dimensional energy critical heat equation

Manuel del Pino·Monica Musso·Juncheng Wei·Qidi Zhang·Yifu Zhang
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Abstract

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3. \end{cases} \end{equation*} We construct type II finite time blow-up solution $u(x,t)$ with the blow-up rates $ \| u\|_{L^\infty} \sim (T-t)^{-k}$, where $ k=1,2,... $. This gives a rigorous proof of the formal computations by Filippas, Herrero and Velazquez \cite{fhv}. This is the first instance of type II finite time blow-up for three dimensional energy critical heat equation.

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Manuel del Pino, Monica Musso, Juncheng Wei, Qidi Zhang, Yifu Zhang. 2020-02-13. Type II Finite time blow-up for the three dimensional energy critical heat equation. https://arxiv.org/abs/2002.05765

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