  This work is concerned with the development of an adaptive numerical method
for semilinear heat flow models featuring a general (possibly) nonlinear
reaction term that may cause the solution to blow up in finite time. The fully
discrete scheme consists of a high order discontinuous Galerkin (dG) time
stepping method and a conforming finite element discretisation (cG) in space.
The proposed adaptive procedure is based on rigorously devised conditional a
posteriori error bounds in the $L^{\infty}(L^{\infty})$ norm. Numerical
experiments complement the theoretical results.
