On the structure of the infinitesimal generators of semigroups with discrete Lyapunov functionals

Dynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $C^0[0,1]$ or on $C^1[0,1]$ which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $C^1[0,1]$, the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $C^0[0,1]$, the corresponding equations are generalizations of monotone feedback delay differential equations.