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

Abstract

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+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 $\Vert u{\Vert }_{L^{\infty }}\sim (T-t{)}^{-k}$, 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.

43 pages; comments welcome

References in corpus (1)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

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