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 $L^1$.

对于具有单调增加源项f的半线性热方程，在Lebesgue空间中获得非负解的局部存在的必要和充分条件的问题直到最近才得以解决（Laister等人（2016））。在那里，对于更困难的初始数据${\mathrm{大号}}^{\mathrm{1个}}$，出现了f的充要条件。在此条件下，我们考虑具有${\mathrm{大号}}^{\mathrm{1个}}$不确定符号的初始数据，即：唯一性，规则性，连续依赖性和比较性。我们还为全球范围内的小型初始数据的解决方案的及时全球延续建立了充分的条件。${\mathrm{大号}}^{\mathrm{1个}}$。