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Logistic diffusion equations governed by the superposition of operators of mixed fractional order

Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci

TL;DR

The paper studies stationary solutions to logistic diffusion models of Fisher-KPP type driven by a superposition of fractional Laplacians $L_\mu$ under Dirichlet hostilities in bounded domains. It develops a rigorous spectral framework around the principal eigenvalue $\lambda_\mu(\Omega)$ and uses a variational energy $E$ to prove existence of nontrivial, nonnegative weak solutions when resources overcome the spectral threshold, while showing extinction when resources are insufficient. It reveals that the negative component of the measure $\mu$ can promote survival by concentrating mass away from lethal regions, and that nonlocal diffusion can enable persistence even when isolated subregions cannot. The results further demonstrate that combining disconnected safe zones via Lévy-type dispersal can sustain populations and that the relative advantage of diffusion exponents depends on domain size. Overall, the work provides a mathematical foundation for how mixed-order, possibly signed, nonlocal diffusion interacts with environmental resources to determine persistence in hostile environments.

Abstract

We discuss the existence of stationary solutions for logistic diffusion equations of Fisher-Kolmogoroff-Petrovski-Piskunov type driven by the superposition of fractional operators in a bounded region with "hostile" environmental conditions, modeled by homogeneous external Dirichlet data. We provide a range of results on the existence and nonexistence of solutions tied to the spectral properties of the ambient space, corresponding to either survival or extinction of the population. We also discuss how the possible presence of nonlocal phenomena of concentration and diffusion affect the endurance or disappearance of the population. In particular, we give examples in which both classical and anomalous diffusion leads to the extinction of the species, while the presence of an arbitrarily small concentration pattern enables survival.

Logistic diffusion equations governed by the superposition of operators of mixed fractional order

TL;DR

The paper studies stationary solutions to logistic diffusion models of Fisher-KPP type driven by a superposition of fractional Laplacians under Dirichlet hostilities in bounded domains. It develops a rigorous spectral framework around the principal eigenvalue and uses a variational energy to prove existence of nontrivial, nonnegative weak solutions when resources overcome the spectral threshold, while showing extinction when resources are insufficient. It reveals that the negative component of the measure can promote survival by concentrating mass away from lethal regions, and that nonlocal diffusion can enable persistence even when isolated subregions cannot. The results further demonstrate that combining disconnected safe zones via Lévy-type dispersal can sustain populations and that the relative advantage of diffusion exponents depends on domain size. Overall, the work provides a mathematical foundation for how mixed-order, possibly signed, nonlocal diffusion interacts with environmental resources to determine persistence in hostile environments.

Abstract

We discuss the existence of stationary solutions for logistic diffusion equations of Fisher-Kolmogoroff-Petrovski-Piskunov type driven by the superposition of fractional operators in a bounded region with "hostile" environmental conditions, modeled by homogeneous external Dirichlet data. We provide a range of results on the existence and nonexistence of solutions tied to the spectral properties of the ambient space, corresponding to either survival or extinction of the population. We also discuss how the possible presence of nonlocal phenomena of concentration and diffusion affect the endurance or disappearance of the population. In particular, we give examples in which both classical and anomalous diffusion leads to the extinction of the species, while the presence of an arbitrarily small concentration pattern enables survival.
Paper Structure (12 sections, 21 theorems, 180 equations)

This paper contains 12 sections, 21 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. The eigenvalue problem
  3. Logistic equations of Fisher-Kolmogoroff-Petrovski-Piskunov type
  4. Organization of the paper
  5. Preliminary results
  6. The eigenvalue problem driven by the operator $L_\mu$
  7. The logistic diffusion equation governed by the operator $L_\mu$
  8. Proofs of Theorems \ref{['thmlogistica1']}, \ref{['thmlogistica2']} and \ref{['thmlogistica3']}
  9. Proof of Theorem \ref{['thmlogistica4']}
  10. Proofs of Proposition \ref{['proplogistica5']} and Theorem \ref{['thmduemisure']}
  11. Conclusion
  12. On the assumptions \ref{['mu2']} and \ref{['mu2forte']}

Key Result

Theorem 1.1

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^N$ with Lipschitz boundary. Then, there exists $\gamma_0>0$, depending only on $N$ and $\Omega$, such that if $\gamma\in[0,\gamma_0]$ the following statements hold true. Let $\mu$ satisfy mu0, mu1 and mu2. Let $s_\sharp$ be as in scritico. Pr An eigenfunction $e_\mu$ corresponding to the eigenvalue $\lambda_\mu(\Omega)$ attains the minimum

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • ...and 29 more