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

Abstract

Let N ≥ 2, T ∈ (0,∞] and ξ ∈ C(0,T; R^N). Under some regularity condition for ξ, it is known that the heat equation
u_t − Δu = 0, x ∈ R^N \ {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.