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Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators. (English) Zbl 1227.35184
Summary: We establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDEs of various types such as porous medium equations, reaction-diffusion equations, the generalized Burgers equation, the Navier-Stokes equation, the 3D Leray-$$\alpha$$ model and the $$p$$-Laplace equation with non-monotone perturbations.

MSC:
 35K90 Abstract parabolic equations 35Q30 Navier-Stokes equations 47H05 Monotone operators and generalizations 35K57 Reaction-diffusion equations 35K92 Quasilinear parabolic equations with $$p$$-Laplacian
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