The initial-value problem is studied for evolution equations in Hilbert space of the general form d se(u)+N(u) l:, dt where and are maximal monotone operators. Existence of a solution is proved when1 is a subgradient and either is strongly monotone or 9 is coercive; existence is established also in the case where 1 is strongly monotone and is subgradient. Uniqueness is provedwhen one of or is continuous self-adjoint and thesum is strictlymonotone; examples of nonuniqueness are given. Applications are indicated for various classes of degenerate nonlinear partial differential equations or systems of mixed elliptic-parabolic-pseudo- parabolic types and problems with nonlocal nonlinearity.