  Studying the behavior of the K\"ahler-Ricci flow on mildly singular
varieties, one is naturally lead to study weak solutions of degenerate
parabolic complex Monge-Amp\`ere equations.
  In this article, the third of a series on this subject, we study the long
term behavior of the normalized K\"ahler-Ricci flow on mildly singular
varieties of positive Kodaira dimension, generalizing results of Song and Tian
who dealt with smooth minimal models.
