On the Borel summability of divergent solutions of the heat equation

Abstract

In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel 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 sum by taking a special Cauchy data.