Existence of Weak solution of a PDE with singular term

Question

On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation $$ \begin{cases} u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T), \\ {u}(0) = {u}_{0}& \mbox{ in } \mathbb{R}^n \\ \end{cases}. $$ The notion of a weak solution is often mentioned, and it is established that Mild solutions are weak solutions. I would like to know if it is possible to show directly for this equation the existence of a weak solution, in the sense $$ \int u\phi dx\left|_{0}^{T}\right. = \int_{0}^{T} \int[u (\Delta\phi + \partial_t \phi) + |\cdot|^{l}u^{p}\phi] dxds$$ Is there any way to show this existence, even with this singular term? Can you reference any papers for reading? I even think of Galerkin's method, but that singular term seems to get in the way.