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

Abstract: We establish symmetrization results for the solutions of the linear fractional diffusion equation and its elliptic counterpart , , using the concept of comparison of concentrations. The results extend to the nonlinear version, , but only when the nondecreasing function is concave. In the elliptic case, complete symmetrization results are proved for when is a convex nonnegative function for with , 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 . [Copyright &y& Elsevier]