JournalsjemsVol. 19, No. 7pp. 1949–1975

Classical solutions and higher regularity for nonlinear fractional diffusion equations

  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
  • Arturo de Pablo

    Universidad Carlos III de Madrid, Leganes, Spain
  • Fernando Quirós

    Universidad Autónoma de Madrid, Spain
  • Ana Rodríguez

    Universidad Politécnica de Madrid, Spain
Classical solutions and higher regularity for nonlinear fractional diffusion equations cover
Download PDF

A subscription is required to access this article.

Abstract

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion

tu+(Δ)σ/2φ(u)=0,\partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0,

posed for xRNx\in \mathbb{R}^N, t>0t>0, with 0<σ<20<\sigma<2, N1N\ge1. If the nonlinearity satisfies some not very restrictive conditions: φC1,γ(R)\varphi\in C^{1,\gamma}(\mathbb{R}), 1+γ>σ1+\gamma>\sigma, and φ(u)>0\varphi'(u)>0 for every uRu\in\mathbb{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 CC^\infty regularity. Degenerate and singular cases, including the power nonlinearity φ(u)=um1u\varphi(u)=|u|^{m-1}u, m>0m>0, are also considered, and the existence of positive classical solutions until the possible extinction time if m<NσN,N>σm < \frac{N-\sigma}{N}, N > \sigma, and for all times otherwise, is proved.

Cite this article

Juan Luis Vázquez, Arturo de Pablo, Fernando Quirós, Ana Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 1949–1975

DOI 10.4171/JEMS/710