  The aim of this paper is to develop the theory of distributions, not
necessarily of compact support, in a topos model of Synthetic Differential
Geometry, the so-called "Cahiers Topos". As an application, we study the
evolution through time of a heat distribution. In the first part of the paper
we study distributions in the sense of Schwartz in the context of convenient
vector spaces. In particular, we study smoothness (with respect to time) of the
fundamental distribution solution of the heat equation. In the second part, we
show how the results of the first part may be used to internalize these notions
to prove the existence of the fundamental distribution solution of the heat
equation in the Cahiers topos.}
