On the Borel summability of divergent solutions of the heat equation

In recent years, the theory of Borel summability or multisummability of div ergent power series of one variable has been established and it has been pr oved that every formal solution of an ordinary differential equation with i rregular singular point is multisummable. For partial differential equation s the summability problem for divergent solutions has not been studied so w ell, and in this paper we shall try to develop the Borel summability of div ergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet dive regent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its gr owth condition of Cauchy data to infinity along a stripe domain, and the Bo rel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel s um by taking; a special Cauchy data.