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 u_t = Δ u + u⁵ in R³ × (0, ∞), u(x, 0) = u₀(x) in R³. For each γ > 1 we find initial data (not necessarily radially symmetric) with lim |x| → ∞ |x| γ u₀ (x) > 0 such that as t → ∞ ||u (·, t) || ∞ ~ tγ-1/2 if 1 < γ < 2, ||u (·, t) || ∞ ~ √t if γ > 2, ||u (·, t) || ∞ ~ √t(Int) ^-1 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.