  The problem of obtaining necessary and sufficient conditions for local
existence of non-negative solutions in Lebesgue spaces for semilinear heat
equations having monotonically increasing source term $f$ has only recently
been resolved (Laister et al. (2016)). There, for the more difficult case of
initial data in $L^1$, a necessary and sufficient integral condition on $f$
emerged. Here, subject to this integral condition, we consider other
fundamental properties of solutions with $L^1$ initial data of indefinite sign,
namely: uniqueness, regularity, continuous dependence and comparison. We also
establish sufficient conditions for the global-in-time continuation of
solutions for small initial data in $L^1$.
