Convergence in smooth strongly monotone flows defined by semilinear parabolic equations (Q1825353)

After the experimental discovery of oscillatory evolutions in processes described by parabolic systems, two approaches have been used to determine asymptotic properties of the corresponding flows: linearization near known equilibria, and imposition of `suitable' smoothness conditions on the form of equations, allowing a comparison argument in a `suitable' space. The author uses the latter approach to establish an order relation for bounded trajectories, but cannot dispense with linearizations. Although the number of hypotheses and restrictions is quite large, the author claims that they apply to models of experimental processes. This claim is not substantiated by a comparison of `suitable' and realistic spaces.