The Mesa Problem for the Fractional Porous Medium Equation

Original · EN

We investigate the behaviour of the solutions uₘ(x,t) of the fractional porous medium equation uₜ+(-Δ)ˢ (uᵐ)=0, x∈ Rⁿ, t>0. with initial data u(x,0)≥ 0, x∈ Rⁿ, in the limit as m→∞ with fixed s∈ (0,1). We first identify the limit of the Barenblatt solutions as the solution of a fractional obstacle problem, and we observe that, contrary to the case s=1, the limit is not compactly supported but exhibits a typical fractional tail with power-like decay. In other words, we do not get a plain mesa in the limit, but a mesa with tails. We then study the limit for a class of nonnegative initial data and derive counterexamples to expected propagation and comparison properties based on symmetrization.