  In this paper we study the heat transfer with a general fractional diffusion
term of an incompressible fluid in a porous medium governed by Darcy's law. We
show formation of singularities with infinite energy and for finite energy we
obtain existence and uniqueness results of strong solutions for the
sub-critical and critical cases. We prove global existence of weak solutions
for different cases. Moreover, we obtain the decay of the solution in $L^p$,
for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the
existence of an attractor in a weak sense and, for the sub-critical dissipative
case with $\alpha\in (1,2]$, we obtain the existence of the global attractor
for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$.
