Monotone solutions to quasilinear parabolic equations (Q1335904)

General quasilinear parabolic equations on a bounded domain in \(\mathbb{R}^ N\) under linear boundary conditions are considered. Due to the maximum principle, the solution flows of such equations belong to the class of strongly monotone (order preserving) dynamical systems. The main result of the present contribution states that if the nonlinearities in the equation are real analytic and some growth and dissipative conditions are satisfied then most solutions (that is, solutions emanating from an open and dense set of initial conditions) become eventually monotone in time.