Type II blow-up in the 5-dimensional energy critical heat equation

Abstract

We consider the Cauchy problem for the energy critical heat equation $$u_t=\Delta u+|u{|}^{\frac{4}{n-2}}uin{\mathbb{R}}^n\times (0,T),u(\cdot ,0)=u_{0}in{\mathbb{R}}^n$$ in dimension $n=5$. More precisely we find that for given points $q_1,q_2,\dots ,q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem 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.

15 pages

References in corpus (3)

• Singularity formation for the two-dimensional harmonic map flow into $S^2$
• Geometry driven Type II higher dimensional blow-up for the critical heat equation
• On strongly anisotropic type II blow up

Cited by in corpus (2)

• The blow-up rate for a non-scaling invariant semilinear heat equation
• Existence and stability of infinite time bubble towers in the energy critical heat equation