Evolution equations for nonlinear degenerate parabolic PDE

We define a convex function on H-1(Ω)H-1(Ω) and characterize its subdifferential, by which we introduce an evolution equation as a weak formulation of a class of nonlinear, degenerate, singular and noncoercive parabolic PDEs associated with an arbitrary maximal monotone graph and with the Dirichlet boundary condition.