Semi-linear evolution equations via positive semigroups

TL;DR

This paper investigates semi-linear evolution equations with positive semigroup generators, establishing existence, convergence, and applications to biological models like logistic and Lotka-Volterra equations.

Contribution

It introduces a method for solving semi-linear evolution problems with quasi-increasing nonlinearities using sub- and super-solutions, and analyzes long-term behavior.

Findings

Existence of solutions between sub- and super-solutions

Convergence of solutions as time approaches infinity

Application to biological models such as logistic and Lotka-Volterra equations

Abstract

We study semi-linear evolutionary problems where the linear part is the generator of a positive $C_0$-semigroup. The non-linear part is assumed to be quasi-increasing. Given an initial value in between a sub- and a super-solution of the stationary problem we find a solution of the semi-linear evolutionary problem. Convergence as $t\to\infty$ is also studied for the solutions. Our results are applied to the logistic equation with diffusion, to a Lotka-Volterra competition model and the Fisher equation from population genetics.