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 $$|x{|}^{-2}{\partial }_{t}u=\hat{I}”u+|x{|}^{\ddot{I}}ƒu^p,(x,t)\in {\mathbb{R}}^N\times (0,T),$$ are obtained, in the range of exponents $p>1$, $\ddot{I}ƒ\ge -2$. 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 $\ddot{I}ƒ=-2$ 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.