6533b833fe1ef96bd129c047

RESEARCH PRODUCT

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

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Abstract We study the long-time behavior of solutions to the heat equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u , in  R N × ( 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 u 0 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.