  This paper is concerned with the uniqueness, existence, comparison principle
and long-time behavior of solutions to the initial-boundary value problem for a
unidirectional diffusion equation. The unidirectional evolution often appears
in Damage
  Mechanics due to the strong irreversibility of crack propagation or damage
evolution. The existence of solutions is proved in an L^2-framework by
introducing a peculiar discretization of the unidirectional diffusion equation
by means of variational inequities of obstacle type and by developing a
regularity theory for such variational inequalities. The novel discretization
argument will be also applied to prove the comparison principle as well as to
investigate the long-time behavior of solutions.
