The mesa problem for the fractional porous medium equation
Juan Luis Vázquez · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2015
We investigate the behaviour of the solutions u_m(x,t) of the fractional porous medium equation u_t+(-\Delta)^s (u^m)=0, \quad x\in \mathbb R^N, \ t>0 with initial data u(x,0)\ge 0 , x\in \mathbb R^N , in the limit m\to\infty with fixed s\in (0,1) . We first identify the limit F_\infty of the Barenblatt solutions U_m(x,t) ormalcolor as the solution of a stationary 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 a tail. This is not the whole story since the limit of V_m(x,t)= mt\, U_m^m(x,t) exists and is compactly supported (in x ). We then study the limit m\to\infty for a wide class of solutions with nonnegative initial data, and show also in this setting the phenomenon of initial discontinuity, whereby the solution does not take on the prescribed initial data. Finally, we derive counterexamples to expected propagation and comparison properties based on symmetrization and pose a related open problem.