Classical solutions and higher regularity for nonlinear fractional diffusion equations

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion, partial derivative(t)u + (-Delta)(sigma/2)phi(u) = 0, for x is an element of R-N and t > 0, with 0 < sigma < 2 and N >= 1. If the nonlinearity satisfies some not very restrictive conditions: phi is an element of C-1,C- gamma (R) with 1 + gamma > sigma, and phi '(u) > 0 for every u is an element of R, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the nonlinearity to obtain higher regularity for the solutions, even C-infinity regularity. Degenerate and singular cases, including the power nonlinearity phi(u) = |u|(m-1)u, m > 0, are also considered, and the existence of positive classical solutions until the possible extinction time if m < N-sigma/N, N > sigma, and for all times otherwise, is proved.

nonlinear fractional diffusion; nonlocal diffusion operators; classical solutions; optimal regularity; porous-medium equation; parabolic equations

Classical solutions and higher regularity for nonlinear fractional diffusion equations Articles