Borel summability of the heat equation with variable coefficients

We prove Borel summability in nonsingular directions of solutions of partial differential equations of the form ut=a(z)uzz where a(z) is a quartic polynomial and the initial condition is analytic. In the special case a(z)=z we obtain the detailed resurgent structure; even in such a simple case, the structure of the singular manifolds is quite intricate.