  The interplay between Mott and Anderson routes to localization in disordered
interacting systems gives rise to different transitions and transport regimes.
Here, we investigate the phase diagram at finite temperatures using dynamical
mean field theory combined with typical medium theory, which is an effective
theory of the Mott-Anderson metal-insulator transition. We mainly focus on the
properties of the coexistence region associated with the Mott phase transition.
For weak disorder, the coexistence region is found to be similar as in the
clean case. However, as we increase disorder Anderson localization effects are
responsible for shrinking the coexistence region and at sufficiently strong
disorder (approximately equal to twice the bare bandwidth) it drastically
narrows, the critical temperature abruptly goes to zero, and we observe a phase
transition in the absence of a coexistence of the metallic and insulating
phases. In this regime, the effects of interaction and disorder are found to be
of comparable importance for charge localization.
