On the profile of solutions with time-dependent singularities for the heat equation

Let N ≥ 2, T ∈ (0,∞] and ξ ∈ C(0,T; RN). Under some regularity condition for ξ, it is known that the heat equation uₜ − Δu = 0, x ∈ RN \ξ(t)\, t ∈ (0,T) has a solution behaving like the fundamental solution of the Laplace equation as x → ξ(t) for any fixed t. In this paper we construct a singular solution whose behavior near x = ξ(t) suddenly changes from that of the fundamental solution of the Laplace equation at some t.