Long-time dynamics for the energy critical heat equation in R5

We investigate the long-time behavior of global solutions to the energy critical heat equation in R-5 {partial derivative(t)u = Delta u + vertical bar u vertical bar(4/3)u in R(5)x(t(0),infinity), u(center dot, t(0)) = u(0) in R-5. For t(0) sufficiently large, we show the existence of positive solutions for a class of initial value u(0)(x) similar to vertical bar x vertical bar(-gamma) as vertical bar x vertical bar -> infinity with gamma > 3/2 such that the global solutions behave asymptotically parallel to u(center dot, t)parallel to(L infinity(R5)) similar to {t(-3(2-gamma)/2) if 3/2 < gamma < 2 (ln t)(-3) if gamma = 2 for t > t(0), 1 if gamma > 2 which is slower than the self-similar time decay t(-3/4). These rates are inspired by Fila and King (2012, Conjecture 1.1).