We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3 ut = 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.
- Critical exponents
- Nonlinear parabolic equations
ASJC Scopus subject areas
- Numerical Analysis
- Applied Mathematics