Classical solutions and higher regularity for nonlinear fractional diffusion equations

By Juan Luis V'azquez, A. de Pablo, F. Quir'os, A. Rodr'iguez. 2013

85 citations recorded by the source corpus.

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0 \sigma$, and $\varphi'(u)>0$ 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 $\varphi(u)=|u|^{m-1}u$, $m>0$, are also considered, and the existence of classical solutions in the power case is proved.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/67271dc5d98481e18442fff77e7f92a2d84a49d8

Citations in the corpus

• Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, unknown): “…problem with a dynamical boundary condition Lσw = 0 for x ∈ Ω, t > 0, ∂w ∂yσ − ∂w 1/m ∂t = 0 for x ∈ Γ, t > 0, w(x, 0, 0) = fm(x) for x ∈ Γ. (19) This problem has been considered by Athanasopoulos and Caffarelli [11], who prove that any bounded weak solution is Hölder continuous if m > 1.”