  This work is concerned with the development of a space-time adaptive
numerical method, based on a rigorous a posteriori error bound, for a
semilinear convection-diffusion problem which may exhibit blow-up in finite
time. More specifically, a posteriori error bounds are derived in the
$L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time
implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) in space
discretization of the problem, although the theory presented is directly
applicable to the case of conforming finite element approximations in space.
The choice of the discretization in time is made based on a careful analysis of
adaptive time stepping methods for ODEs that exhibit finite time blow-up. The
new adaptive algorithm is shown to accurately estimate the blow-up time of a
number of problems, including one which exhibits regional blow-up.
