Asymptotic behavior for the heat equation in nonhomogeneous media with critical density

We study the long-time behavior of solutions to the heat equation in nonhomogeneous media with critical singular density |x|−2∂tu=Δu,in  RN×(0,∞) in dimensions N≥3. The asymptotic behavior proves to have some interesting and quite striking properties. We show that there are two completely different asymptotic profiles depending on whether the initial data u0 vanishes at x=0 or not. Moreover, in the former the results are true only for radially symmetric solutions, and we provide counterexamples to convergence to symmetric profiles in the general case.