6533b86dfe1ef96bd12c9e23

RESEARCH PRODUCT

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

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Abstract We study the large time behavior of solutions to the Cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u m , in R N × ( 0 , ∞ ) , where m > 1 and N ≥ 3 , with nonnegative initial condition u ( x , 0 ) = u 0 ( x ) ≥ 0 . 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 u 0 vanishes at x = 0 or not. Moreover, when u 0 ( 0 ) = 0 , we show the convergence towards a peak-type profile presenting a jump discontinuity, coming from an interesting asymptotic simplification to a conservation law, while when u 0 ( 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.