On the paper ON THE CAUCHY PROBLEM FOR REACTION-DIFFUSION EQUATIONS Wang studies the Hardy-Hénon equation

\begin{equation} \left\{ \begin{array}{rll} 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{array} \right. \end{equation} 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?