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A critical non-homogeneous heat equation with weighted source

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $σ\ge-2$. 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 $σ=-2$ 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.