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

We deal with the large time behavior for a porous medium equation posed in nonhomogeneous media with singular critical density|x|−2∂tu(x,t)=Δum(x,t),(x,t)∈RN×(0,∞),m≥1, posed in dimensions N=1N=1 and N=2N=2, which are also interesting in applied models according to works by Kamin and Rosenau. We deal with the Cauchy problem with bounded and continuous initial data u0u0. We show that in dimension N=2N=2, the asymptotic profiles are self-similar solutions that vary depending on whether u0(0)=0u0(0)=0 or u0(0)=K∈(0,∞)u0(0)=K∈(0,∞). In dimension N=1N=1, things are strikingly different, and we find new asymptotic profiles of an unusual mixture between self-similar and traveling wave forms. We thus complete the study performed in previous recent works for the higher dimensions N≥3N≥3.