  In this paper, we introduce a method for computing rigorous local inclusions
of solutions of Cauchy problems for nonlinear heat equations for complex time
values. Using a solution map operator, we construct a simplified Newton
operator and show that it has a unique fixed point. The fixed point together
with its rigorous bounds provides the local inclusion of the solution of the
Cauchy problem. The local inclusion technique is then applied iteratively to
compute solutions over long time intervals. This technique is used to prove the
existence of a branching singularity in the nonlinear heat equation. Finally,
we introduce an approach based on the Lyapunov-Perron method to calculate part
of a center-stable manifold and prove that an open set of solutions of the
Cauchy problem converge to zero, hence yielding the global existence of the
solutions in the complex plane of time.
