A simple characterization of positivity preserving semi-linear parabolic systems

TL;DR

This paper provides a straightforward proof for characterizing positivity-preserving semi-flows in ordinary differential systems and extends the approach to reaction-diffusion systems with boundary conditions, offering optimal results for invariant rectangles.

Contribution

It introduces a simple, direct proof method for positivity preservation in semi-flows and applies it to reaction-diffusion systems with boundary conditions, unifying the conditions with or without diffusion.

Findings

Characterization of positivity-preserving semi-flows for ODEs.

Extension of the characterization to reaction-diffusion systems.

Optimal results for invariant rectangles under Neumann conditions.

Abstract

We give a simple and direct proof of the characterization of positivity preserving semi-flows for ordinary differential systems. The same method provides an abstract result on a class of evolution systems containing reaction-diffusion systems in a bounded domain of $ \mathbb{R}^n$ with either Neumann or Dirichlet homogeneous boundary conditions. The conditions are exactly the same with or without diffusion. A similar approach gives the optimal result for invariant rectangles in the case of Neumann conditions.