2013/03/12 by Juan Luis Vázquez, Vázquez, Juan Luis, Bruno Volzone +1 · 1 citation
Computer Science · Mathematics · #35J60 #35K55 #35K65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35J60 #msc:35K55 #msc:35K65
paper · pdf · doi:10.48550/arxiv.1303.2970
42 pages, 1 figure
arxiv created 2013/03/12 · openalex publication_date 2013/03/12 · arxiv updated 2013/03/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We establish symmetrization results for the solutions of the linear fractional diffusion equation ∂t u +(-Δ)σ/2u=f and itselliptic counterpart h v +(-Δ)σ/2v=f, h>0, using the concept of comparison of concentrations. The results extend to the nonlinear version, ∂t u+(-Δ)σ/2A(u)=f, but only when A:\re+→\re+ is a concave function. In the elliptic case, complete symmetrization results are proved for B(v)+(-Δ)σ/2v=f when B(v) is a convex nonnegative function for v>0 with B(0)=0, and partial results 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 σ=2.