On the profile of solutions with time-dependent singularities for the heat equation (Q487278)

Let \(N\geq 2,\) \(T\in (0,\infty]\) and \(\xi\in C(0,T; \mathbb R^N).\) Under suitable regularity assumptions on \(\xi,\) it is known that the heat equation \[ u_t -\Delta u = 0,\quad x \in \mathbb R^N\setminus \{\xi(t)\},\, t \in (0,T), \] has a solution behaving as the fundamental solution of the Laplace equation as \(x\to \xi(t)\) for any fixed \(t.\) In the paper under review, the authors construct a singular solution whose behavior near \(x =\xi(t)\) suddenly changes from that of the fundamental solution of the Laplace equation at some \(t\).