On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation $$ \begin{cases} u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T), \\ {u}(0) = {u}_{0}& \mbox{ in } \mathbb{R}^n \\ \end{cases}. $$ On page 554 he says that "when $l<0$ it is generally impossible to obtain a classical solution for the equation". I was curious if there really isn't any classical solution, or if there aren't mathematical methods yet to determine whether or not there is a classical solution. In the case $l<0$ it deals only with mild solutions. Has there been any progress in the literature in this direction?