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

2012/06/06 by Razvan Gabriel Iagar, Razvan Iagar, Iagar, Razvan +2
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1206.1167

arxiv created 2013/02/22 · arxiv updated 2013/02/26

Abstract

We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density |x|^-2∂_tu=Δu, \hboxin \real^N×(0,∞). 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.

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