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

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3¶ν=Δν+ν5<mspace class="quad" width="1em"/> in ℝ3×(0,∞),<mspace class="qquad" width="2em"/>ν(,0)=ν0()<mspace class="quad" width="1em"/> in ℝ3.¶ For each [math] we find initial data (not necessarily radially symmetric) with [math] such that as [math]¶∥ν(⋅,)∥∞∼γ−12<mspace class="quad" width="1em"/> if 1<γ<2,<mspace class="qquad" width="2em"/>∥ν(⋅,)∥∞∼<msqrt></msqrt><mspace class="quad" width="1em"/> if γ>2,<mspace class="qquad" width="2em"/>∥ν(⋅,)∥∞∼<msqrt></msqrt>()−1<mspace class="quad" width="1em"/> if γ=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).