Asymptotic behavior for the critical nonhomogeneous porous medium\n equation in low dimensions

2015/11/24 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Ariel Sánchez +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1511.07806

openalex publication_date 2015/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We deal with the large time behavior for a porous medium equation posed in\nnonhomogeneous media with singular critical density \n|x|^-2
partial_tu(x,t)=
Delta u^m(x,t),
quad (x,t)
in\n
real^N
times(0,
infty),
m
geq1, posed in dimensions N=1 and N=2,\nwhich are also interesting in applied models according to works by Kamin and\nRosenau. We deal with the Cauchy problem with bounded and continuous initial\ndata u₀. We show that in dimension N=2, the asymptotic profiles are\nself-similar solutions that vary depending on whether u₀(0)=0 or\nu₀(0)=K\∈(0,\∞). In dimension N=1, things are strikingly different,\nand we find new asymptotic profiles of an unusual mixture between self-similar\nand traveling wave forms. We thus complete the study performed in previous\nrecent works for the bigger dimensions N\≥3.\n

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