Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation arts (-Delta)(sigma/2)u = f and its elliptic counterpart hv + (-Delta)(sigma/2)v = f, h > 0, using the concept of comparison of concentrations. The results extend to the nonlinear version, partial derivative(t)u+ (-Delta)(sigma/2)A(u) = f, but only when the nondecreasing function A : R+ -> R+ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(-Delta)(sigma/2)v = f when B(v) is a convex nonnegative function for v > 0 with B(0) = 0, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B is concave. Such counterexamples do not exist in the standard diffusion case sigma = 2. (C) 2013 Elsevier Masson SAS. All rights reserved.