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) = Delta u + u(5) in R-3 x (0, infinity), u(x, 0) = u(0)(x) in R-3. For each gamma > 1 we find initial data (not necessarily radially symmetric) with lim(vertical bar x vertical bar ->infinity) vertical bar x vertical bar(gamma) u(0)(x) > 0 such that as t -> infinity parallel to u(., t)parallel to(infinity) similar to t(gamma-1/2) if 1 < gamma < 2, parallel to u(., t)parallel to(infinity) similar to root t if gamma < 2, parallel to u(., t)parallel to(infinity) similar to root t(lnt)(-1) 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).