We investigate  quantitative properties of nonnegative solutions
$u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u +
\mathcal{L} (u^m)=0$, posed in a bounded domain, $x\in\Omega\subset \RR^N$ for $t>0$
and $m>1$. As $\mathcal{L}$ we can take the most common definitions of the
fractional Laplacian $(-\Delta)^s$, $0<s<1$,  in a bounded domain with zero
Dirichlet boundary conditions, as well as more general classes of operators. We
consider a class of very weak solutions for the equation at hand, that we call weak
dual solutions, and we obtain a priori estimates in the form of smoothing effects,
absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate
the boundary behaviour and we obtain sharp estimates from above and below.  The
standard Laplacian case $s=1$ or the linear case $m=1$ are recovered as limits. The
method is quite general, suitable to be applied to a number of similar problems that
will be briefly discussed as examples. 
  As a consequence, we can prove existence and uniqueness of minimal weak dual
solutions with data in $L^1_{\Phi_1}$\,, where $\Phi_1$ is the first eigenfunction
of $\mathcal{L}$. We also briefly show existence and uniqueness of $H^{-s}$
solutions with a different approach.   As a byproduct, we derive similar estimates
for the elliptic semilinear equation $\mathcal{L}S^m=S$ and we prove existence and
uniqueness of $H^{-s}(\Omega)$ solutions via parabolic techniques. Solutions to
this elliptic problem represents the asymptotic profiles of the rescaled
solutions, namely the stationary states of the rescaled equation $\partial_t v =
-\mathcal{L} (v^m)+ v$. 
Finally, we will study the asymptotic behaviour. We will prove sharp rates of decay
of the rescaled solution to the unique stationary profile S and also for the
relative error $v/S-1$. The sharp rates of convergence can be obtained with two
different methods: one is based on the above estimates, that guarantee existence of
the "friendly giant". Another approach is given by a new entropy method, based on
the so-called Caffarelli-Silvestre extension. 
This is a joint work with J. L. Vazquez (UAM, Madrid, Spain) and Y. Sire (Univ.
Marseille, France). 

References:
[BV1] M.  B., J. L. Vazquez,  A Priori Estimates  for Fractional Nonlinear 
Degenerate Diffusion Equations  on bounded domains. To appear in  Arch. Rat. Mech.
Anal (2015) http://arxiv.org/abs/1311.6997 <http://arxiv.org/abs/1311.6997>

[BSV] M. B.,  Y. Sire, J. L. Vazquez,  Existence, Uniqueness and Asymptotic
behaviour for fractional porous medium equations on bounded domains. To appear in
Discr. Cont. Dyn. Sys. (2015) http://arxiv.org/abs/1404.6195
<http://arxiv.org/abs/1404.6195>

[BV2] M.  B., J. L. Vazquez,  Nonlinear Degenerate Diffusion Equations on bounded
domains with Restricted Fractional Laplacian. In Preparation (2015).