Classical solutions and higher regularity for nonlinear fractional diffusion equations

Abstract

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-Δ)^{σ/2}φ(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<σ<2$, $N\ge1$. If the nonlinearity satisfies some not very restrictive conditions: $φ\in C^{1,γ}(\mathbb{R})$, $1+γ>σ$, and $φ'(u)>0$ for every $u\in\mathbb{R}$, 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 $C^\infty$ regularity. Degenerate and singular cases, including the power nonlinearity $φ(u)=|u|^{m-1}u$, $m>0$, are also considered, and the existence of classical solutions in the power case is proved.