We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-\Delta)^s$, $0 < s < 1$, in a bounded domain with zero Dirichlet boundary conditions, but it also includes many other examples, since our theory only needs some basic properties that are typical of ``linear heat semigroups''. The nonlinearity $F$ is assumed to be increasing and is allowed to be degenerate, the prototype being $F(u)=|u|^{m-1}u$, with $m>1$.
We will present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions. We then show sharper lower and upper estimates up to the boundary, which fairly combine into various forms of Harnack type inequalities. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit in most of the results.
When the nonlinearity is of the form $F(u)=|u|^{m-1}u$, with $m>1$, global Harnack estimates are the key tool to understand the sharp asymptotic behaviour of the solutions.
We finally show that solutions are classical, and even $C^{\infty}$ in space in the interior of the domain, when the operator $\mathcal{L}$ is the (restricted) fractional Laplacian.
The above results are contained on a series of recent papers with J. L. Vazquez, and also with A. Figalli, Y. Sire and X. Ros-Oton.
