Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density

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We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density |x|⁻²∂ₜu=Δuᵐ, in ⁿ×(0,∞), where m>1 and N≥3. The asymptotic behavior proves to have some interesting and striking properties. We show that there are different asymptotic profiles for the solutions, depending on whether the continuous initial data u₀ vanishes at x=0 or not. Moreover, when u₀(0)=0, we show the convergence towards a profile presenting a discontinuity in form of a shockwave, coming from an unexpected asymptotic simplification to a conservation law, while when u₀(0)>0, the limit profile remains continuous. These phenomena illustrate the strong effect of the singularity at x=0. We improve the time scale of the convergence in sets avoiding the singularity. On the way, we also study the large-time behavior for a porous medium equation with convection which is interesting for itself.