  This work is concerned with the development of a space-time adaptive
numerical method, based on a rigorous a posteriori error bound, for the
semilinear heat equation with a general local Lipschitz reaction term whose
solution may blow-up in finite time. More specifically, conditional a
posteriori error bounds are derived in the $L^{\infty}L^{\infty}$ norm for a
first order in time, implicit-explicit (IMEX), conforming finite element method
in space discretization of the problem. Numerical experiments applied to both
blow-up and non blow-up cases highlight the generality of our approach and
complement the theoretical results.
