A critical non-homogeneous heat equation with weighted source

Abstract Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source \begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*} are obtained, in the range of exponents $p\gt 1$ , $\sigma \ge -2$ . More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to \infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $\sigma =-2$ , we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher–KPP equation is derived and employed in order to deduce these properties.