  We consider the Cauchy problem for semi-linear heat equations with
exponential nonlinearity. The main purpose of this paper is to prove the
existence of solutions lying on the borderline between global existence and
blow-up infinite time. The existence has been shown for semi-linear heat
equations with power type nonlinearity. We explain the main strategy to prove
the existence. By using the definition of exponential function, we approximate
the solution to exponential type equation by that of power type equation. Then
we can use directly the knowledge for power type equation.
