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Diffusive logistic equation with a non Lipschitz nonlinear boundary condition arising from coastal fishery harvesting: the resonant case

Kenichiro Umezu

TL;DR

This work analyzes a diffusive logistic equation with a sublinear boundary harvesting term in a bounded domain, focusing on the resonant case $\beta_{\Omega}=1$ where the boundary nonlinearity is not right-differentiable at zero. The authors develop a nonstandard bifurcation-from-zero framework built on an energy approach, a regularized problem, and Whyburn topological arguments to map the positive solution set. They establish both bounded and unbounded continua of nonnegative/positive solutions depending on the product $pq$, showing that for $pq\le1$ a bounded subcontinuum connects $(0,0)$ to $(0,1)$, while for $pq>1$ an unbounded continuum exists with detailed asymptotics indicating boundary concentration along the principal eigenfunction. The results advance bifurcation theory for nonlinear Neumann boundary conditions with non-Lipschitz terms and have implications for coastal fishery harvesting models under resonance.

Abstract

For bifurcation analysis, we study the positive solution set for a semilinear elliptic equation of the logistic type, equipped with a sublinear boundary condition modeling coastal fishery harvesting. This work is a continuation of the author's previous studies, where certain results were obtained in a non resonant case, including the existence, uniqueness, multiplicity, and strong positivity for positive solutions. In this paper, we consider the delicate resonant case and develop a sort of non standard bifurcation technique at zero to evaluate the positive solution set depending on a parameter. The nonlinear boundary condition is not right-differentiable at zero.

Diffusive logistic equation with a non Lipschitz nonlinear boundary condition arising from coastal fishery harvesting: the resonant case

TL;DR

This work analyzes a diffusive logistic equation with a sublinear boundary harvesting term in a bounded domain, focusing on the resonant case where the boundary nonlinearity is not right-differentiable at zero. The authors develop a nonstandard bifurcation-from-zero framework built on an energy approach, a regularized problem, and Whyburn topological arguments to map the positive solution set. They establish both bounded and unbounded continua of nonnegative/positive solutions depending on the product , showing that for a bounded subcontinuum connects to , while for an unbounded continuum exists with detailed asymptotics indicating boundary concentration along the principal eigenfunction. The results advance bifurcation theory for nonlinear Neumann boundary conditions with non-Lipschitz terms and have implications for coastal fishery harvesting models under resonance.

Abstract

For bifurcation analysis, we study the positive solution set for a semilinear elliptic equation of the logistic type, equipped with a sublinear boundary condition modeling coastal fishery harvesting. This work is a continuation of the author's previous studies, where certain results were obtained in a non resonant case, including the existence, uniqueness, multiplicity, and strong positivity for positive solutions. In this paper, we consider the delicate resonant case and develop a sort of non standard bifurcation technique at zero to evaluate the positive solution set depending on a parameter. The nonlinear boundary condition is not right-differentiable at zero.
Paper Structure (9 sections, 16 theorems, 125 equations, 4 figures)

This paper contains 9 sections, 16 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Energy approach
  3. A priori upper bounds of $\lambda$ for positive solutions
  4. Limiting behavior of positive solutions
  5. Limiting behavior as $\lambda \to 0^{+}$
  6. Limiting behavior as $\| u \|_{C(\overline{\Omega})}\to 0$
  7. Bounded subcontinua of nonnegative solutions: case $\beta_\Omega=1$ and $pq\leq1$
  8. Unbounded positive solution sets: case $\beta_\Omega = 1$ and $pq>1$
  9. Asymptotic property for Dirichlet logistic problem

Key Result

Theorem 0

(I)$u<1$ in $\overline{\Omega}$ and $u>0$ on $\Gamma$ with $|\Gamma|>0$ for some $\Gamma \subset \partial\Omega$ for a positive solution $u$ of p, provided that $\lambda>0$. (II) Assume that $\beta_\Omega < 1$. Then, $u>0$ in $\overline{\Omega}$ for a positive solution $u$ of p. Conversely, problem (III) Assume that $\beta_\Omega > 1$. Then, problem p has at least two positive solutions $U_{1}$

Figures (4)

  • Figure 1: Case with either $\beta_\Omega < 1$ or $\beta_\Omega =1$ and $pq>1$.
  • Figure 2: Case with either $\beta_\Omega=1$ and $pq\leq 1$ or $\beta_\Omega > 1$.
  • Figure 3: Admissible case for $\mathcal{C}_{\alpha, \beta}$.
  • Figure 4: Admissible case for $\mathcal{C}_{\beta}$.

Theorems & Definitions (35)

  • Theorem 0
  • Remark 0
  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 25 more