The work deals with a study of a nonlinear parabolic equation with hysteresis, containing a nonlinear monotone operator in the diffusion term. The well-posedness of the model equation is addressed by using an implicit time discretization scheme in conjunction with the piecewise monotonicity of the hysteresis operator, and a fundamental inequality due to M. Hilpert. A characterization of the $\omega$-limit set of the solution is then given through the study of the long-time behaviour of the solution of the equation in which we investigate the convergence of trajectories to limit points.
"Well-posedness and Long-time Behaviour for a Nonlinear Parabolic Equation with Hysteresis." Commun. Math. Anal. 23 (1) 38 - 62, 2020.