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Type II blow-up in the 5-dimensional energy critical heat equation. (English) Zbl 1417.35074

Summary: We consider the Cauchy problem for the energy critical heat equation \[\begin{cases}u_t =\Delta u+|u|^{\frac{4}{n-2}}u\;\;\;\text{in}\;\mathbb{R}^n\times(0,T)\\u(\cdot,0)=u_0\;\;\;\text{in}\;\mathbb{R}^n\end{cases}\] in dimension \(n =5\). More precisely 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 (0.1) blows-up at exactly those \(k\) points with rates type II, namely with absolute size \(\sim (T - t)^{-\alpha}\) for \(\alpha > \frac{3}{4}\). The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

MSC:

35K58 Semilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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