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

2018/08/31 by del Pino, Manuel, Musso, Monica, Wei, Juncheng · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.10637

Abstract

We consider the Cauchy problem for the energy critical heat equation u_t = Δu + |u|^\frac 4n-2u \hboxin \mathbb Rⁿ × (0, T), u(⋅,0) =u₀ \hboxin \mathbb Rⁿ in dimension n=5. More precisely we find that for given points q₁, q₂,…, q_k and any sufficiently small T>0 there is an initial condition u₀ 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 ∼ (T-t)^-α for α> \frac 34 . The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

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