Time-variable singularities for solutions of the heat equation

Abstract:

A solution $u(x,t)$ of the two-dimensional heat equation ${u_{xx}} = {u_t}$ may have the representation \[ u(x,t) = \int _{ - \infty }^\infty {k(x - y,t)\;d\alpha (y)} \] where $k(x,t) = {(4\pi t)^{ - 1/2}}\exp [ - {x^2}/(4t)]$, valid in some strip $0 < t < c$ of the x, t-plane. If so, $u({x_0},t)$ is known to be an analytic function of the complex variable t in the disc $\operatorname {Re} (1/t) > 1/c$, for each fixed real ${x_0}$. It is shown here that if $\alpha (y)$ is nondecreasing and not absolutely continuous then $u({x_0},t)$ must have a singularity at $t = 0$. Examples show that both restrictions on $\alpha (y)$ are necessary for that conclusion. It is shown further under the same hypothesis on $\alpha (y)$, that for each fixed positive ${t_0} < c,u(x,{t_0})$ is an entire function of x of order 2 and of type $1/(4{t_0})$. Compare the function $k(x,t)$ itself for a check on both conclusions.