2013/09/27 by Razvan Gabriel Iagar, Iagar, Razvan, Ariel Sánchez Valdés +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1309.7291
openalex publication_date 2013/09/27 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28
We study the large time behavior of solutions to the porous medium equation\nin nonhomogeneous media with critical singular density \n|x|-2
partialtu=
Delta um,
quad
hboxin
realN
times(0,
infty),\nwhere m>1 and N\≥3. The asymptotic behavior proves to have some\ninteresting and striking properties. We show that there are different\nasymptotic profiles for the solutions, depending on whether the continuous\ninitial data u0 vanishes at x=0 or not. Moreover, when u0(0)=0, we show\nthe convergence towards a profile presenting a discontinuity in form of a\nshockwave, coming from an unexpected asymptotic simplification to a\nconservation law, while when u0(0)>0, the limit profile remains continuous.\nThese phenomena illustrate the strong effect of the singularity at x=0. We\nimprove the time scale of the convergence in sets avoiding the singularity. On\nthe way, we also study the large-time behavior for a porous medium equation\nwith convection which is interesting for itself.\n