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We study the asymptotic behavior of solutions to the heat equation innonhomogeneous media with critical singular density $$|x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$The asymptotic behavior proves to have some interesting and quite strikingproperties. We show that there are two completely different asymptotic profilesdepending on whether the initial data $u_0$ vanishes at $x=0$ or not. Moreover,in the former the results are true only for radially symmetric solutions, andwe provide counterexamples to convergence to symmetric profiles in the generalcase.

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