Removable singularities of semilinear parabolic equations

Abstract

We prove that the parabolic equation ft = Δf + F(x,f,∇f,t), in (Rm\ {0})×(0,T), m ≥ 3, has removable singularities at {0}×(0,T) if ||f||L∞(Rm\{0}×(0,T)) < ∞ and ||∇f||L∞(Rm\{0}×(0,T)) < ∞. We also prove that the solution u of the heat equation in (ω\ {0}) × (0,T) has removable singularities at {0}×(0,T), ω ⊂ Rm, m ≥ 3, if and only if for any 0 < t1 < t2 < T and δ ∈ (0,1) there exists BR0(0)⊂ω depending on t1, t2 and δ, such that |u(x,t)| ≤ δ|x|2-m for any 0 < |x| ≤ R0 and t1≤t≤t2.