  In this paper, we provide a new PDE proof for the celebrated
Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection
between the Borell--Brascamp--Lieb inequality and properties of diffusion
equations of porous medium type pertaining to the large time asymptotics and
preservation of a generalized concavity of the solutions. We also recover the
equality condition in the special case of the Pr\'ekopa--Leindler inequality by
further exploiting known properties of the heat equation including the eventual
log-concavity and backward uniqueness of solutions.
