Classical solutions and higher regularity for nonlinear fractional diffusion equations

2013/11/28 by Vázquez, Juan Luis, de Pablo, Arturo, Quirós, Fernando +1 · 2 citations
#35B65 #35K55 #35R11 #35S10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1311.7427

Abstract

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion ∂_tu+(-Δ)^σ/2φ(u)=0, posed for x∈ ℝ^N, t>0, with 0σ, and φ'(u)>0 for every u∈ℝ, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even C^∞ regularity. Degenerate and singular cases, including the power nonlinearity φ(u)=|u|^m-1u, m>0, are also considered, and the existence of classical solutions in the power case is proved.

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