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

Abstract

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