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 $${\partial }_{t}u+(-\Delta {)}^{\sigma /2}\phi (u)=0,$$ posed for $x\in {\mathbb{R}}^N$, $t>0$, with $0<\sigma <2$, $N\ge 1$. If the nonlinearity satisfies some not very restrictive conditions: $\phi \in C^{1,\gamma }(\mathbb{R})$, $1+\gamma >\sigma$, and ${\phi }^{\prime }(u)>0$ 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 $\phi (u)=|u{|}^{m-1}u$, $m>0$, are also considered, and the existence of classical solutions in the power case is proved.

28 pages, 1 figure