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Semi-linear evolution equations via positive semigroups

Wolfgang Arendt, Daniel Daners

TL;DR

This work develops a sub- and super-solution framework for semi-linear evolution equations where the linear part is the generator of a positive $C_0$-semigroup and the nonlinearity is quasi-increasing. By constructing a fixed-point map $G$ and monotone iterates, the authors obtain minimal and maximal mild solutions and establish their convergence to equilibria under either order-continuous norm or compactness assumptions, with a detailed comparison principle. The theory is then applied to concrete PDE models including diffusion-augmented logistic growth, Lotka–Volterra competition, and the Fisher equation, illustrating existence, uniqueness under Lipschitz regimes, and asymptotic convergence to equilibria. Overall, the paper provides a robust, minimal-regularity approach to existence, comparison, and long-time behavior of semi-linear evolutions in ordered Banach spaces, with broad applicability to reaction-diffusion systems.

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.

Semi-linear evolution equations via positive semigroups

TL;DR

This work develops a sub- and super-solution framework for semi-linear evolution equations where the linear part is the generator of a positive -semigroup and the nonlinearity is quasi-increasing. By constructing a fixed-point map and monotone iterates, the authors obtain minimal and maximal mild solutions and establish their convergence to equilibria under either order-continuous norm or compactness assumptions, with a detailed comparison principle. The theory is then applied to concrete PDE models including diffusion-augmented logistic growth, Lotka–Volterra competition, and the Fisher equation, illustrating existence, uniqueness under Lipschitz regimes, and asymptotic convergence to equilibria. Overall, the paper provides a robust, minimal-regularity approach to existence, comparison, and long-time behavior of semi-linear evolutions in ordered Banach spaces, with broad applicability to reaction-diffusion systems.

Abstract

We study semi-linear evolutionary problems where the linear part is the generator of a positive -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 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.
Paper Structure (12 sections, 28 theorems, 172 equations, 1 figure)

This paper contains 12 sections, 28 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The fixed point map and mild solutions
  3. Monotone iterations and convergence
  4. Existence and comparison of mild solutions
  5. Asymptotic behaviour: Convergence to equilibria
  6. Uniqueness of solutions
  7. Admissible Operators
  8. Applications
  9. The logistic equation
  10. A Lotka-Volterra competition model
  11. The Fisher Equation
  12. Appendix: Some facts on ordered Banach spaces

Key Result

Theorem 1.2

Suppose that $E$ is an ordered Banach space with normal cone and let $-A$ be the generator of a positive $C_0$-semigroup $S((t))_{t\geq 0}$ on $E$. Further assume that $\underline u,\overline u\in E$ is an ordered pair of weak sub- and super-solutions of eq:stationary-equation and let $F\in C([\unde

Figures (1)

  • Figure 6.1: Maximal and minimal solutions (shown in red) of \ref{['eq:scalar-non-unique']} on $[\![-M,M]\!]$.

Theorems & Definitions (66)

  • Definition 1.1: sub/super-solution
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Translation of mild solution
  • ...and 56 more