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∂tu=Δu+|x|σup,(x,t)∈RN×(0,T), are obtained, in the range of exponents p>1 , σ≥−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→∞ 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.