2023/06/17 by Giorgio Fusco, Carlos Frederico Duarte Rocha, Fusco, Giorgio +1 · 1 citation
Computer Science · Engineering · Mathematics · #35 #37 #39 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2306.10403
openalex publication_date 2023/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C0[0,1] or on C1[0,1] which admit discrete Lyapunov functions related to the zero number. We show that, if the space is C1[0,1], the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is C0[0,1], the corresponding equations are generalizations of monotone feedback delay differential equations.