  In this paper we establish the existence and uniqueness of solutions for
nonlinear evolution equations on Banach space with locally monotone operators,
which is a generalization of the classical result by J.L. Lions for monotone
operators. In particular, we show that local monotonicity implies the
pseudo-monotonicity. The main result is applied to various types of PDE such as
reaction-diffusion equations, generalized Burgers equation, Navier-Stokes
equation, 3D Leray-$\alpha$ model and $p$-Laplace equation with non-monotone
perturbations.
