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Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

Loïc Constantin, Carlos Alberto Santos, Guillaume Warnault

TL;DR

The paper addresses logistic-type equations driven by the $p$-fractional Laplacian on bounded domains, establishing existence and uniqueness of weak solutions for the elliptic problem and analyzing their dependence on the parameter $\lambda$. It then develops a parabolic theory, proving local and global existence, uniqueness, energy dissipation, and long-time behaviors such as stabilization to stationary states, extinction, or blow-up, highlighting the distinct effects of nonlocal operators. The analysis combines Mountain Pass techniques, Picone-type inequalities, comparison principles, energy methods, and Sattinger-type stability analysis to obtain a comprehensive view of both elliptic and parabolic fractional $p$-Laplacian logistic dynamics. Overall, the work extends known local results to the nonlocal setting, revealing how nonlocality influences blow-up, extinction, and asymptotic behavior across the domain.

Abstract

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-Δ)_p^s u_λ=λu_λ^q - b(x)u_λ^r \quad \text{in} \;Ω,\\ u_λ=0 \quad \text{in} \; ( \mathbb{R}^d \backslash Ω), \\ u_λ>0 \text{ in} \; Ω. \end{cases}\ \end{equation*} We also prove behavior of $u_λ$ with respect to $λ,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.

Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

TL;DR

The paper addresses logistic-type equations driven by the -fractional Laplacian on bounded domains, establishing existence and uniqueness of weak solutions for the elliptic problem and analyzing their dependence on the parameter . It then develops a parabolic theory, proving local and global existence, uniqueness, energy dissipation, and long-time behaviors such as stabilization to stationary states, extinction, or blow-up, highlighting the distinct effects of nonlocal operators. The analysis combines Mountain Pass techniques, Picone-type inequalities, comparison principles, energy methods, and Sattinger-type stability analysis to obtain a comprehensive view of both elliptic and parabolic fractional -Laplacian logistic dynamics. Overall, the work extends known local results to the nonlocal setting, revealing how nonlocality influences blow-up, extinction, and asymptotic behavior across the domain.

Abstract

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-Δ)_p^s u_λ=λu_λ^q - b(x)u_λ^r \quad \text{in} \;Ω,\\ u_λ=0 \quad \text{in} \; ( \mathbb{R}^d \backslash Ω), \\ u_λ>0 \text{ in} \; Ω. \end{cases}\ \end{equation*} We also prove behavior of with respect to underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.

Paper Structure

This paper contains 12 sections, 26 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. The elliptic problem
  4. Asymptotic behavior
  5. As lambda to sup Lambda q
  6. As lambda to inf Lambda q
  7. For For q>p-1
  8. The parabolic problem
  9. Existence and properties of solutions
  10. Qualitative behavior
  11. Case q in (p-1,p*-1]
  12. Appendix

Key Result

Theorem 1.2

Problem PbLE admits a unique weak solution $u_\lambda$iff$\lambda\in \Lambda_q$. Furthermore, for any $\lambda$, $\mu \in \Lambda_q$, $u_\lambda$ belongs to $C^s({\mathbb{R}^d})$ satisfying $u_\lambda \geq cd(\cdot,\Omega^c)^s$ in $\Omega$ and for $\lambda<\mu$, we have $u_{\lambda}\leq u_{\mu}$ in

Theorems & Definitions (62)

  • Definition 1.1: Weak solution
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 1.6: Weak solution
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 52 more