Infinite-time blow-up for the 3-dimensional energy-critical heat equation

Abstract

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3

$$u_t=\mathrm{\Delta }u+u^5\text{ in }{ℝ}^3\times \left(0,\infty \right),u\left(x,0\right)=u_0\left(x\right)\text{ in }{ℝ}^3.$$

For each $\gamma >1$ we find initial data (not necessarily radially symmetric) with $\underset{\left|x\right|\to \infty }{lim}|x{|}^{\gamma }{u}_0\left(x\right)>0$ such that as $t\to \infty$

$$\parallel u\left(\cdot ,t\right){\parallel }_{\infty }\sim t^{\gamma -\frac{1}{2}}\text{ if }1<\gamma <2,\parallel u\left(\cdot ,t\right){\parallel }_{\infty }\sim \sqrt{t}\text{ if }\gamma >2,\parallel u\left(\cdot ,t\right){\parallel }_{\infty }\sim \sqrt{t}{\left(lnt\right)}^{-1}\text{ if }\gamma =2.$$

Furthermore we show that this infinite-time blow-up is codimensional-1 stable. The existence of such solutions was conjectured by Fila and King (Netw. Heterog. Media 7:4 (2012), 661–671).