Large positive solutions for a class of 1-D diffusive logistic problems with general boundary conditions

TL;DR

This work addresses a one-dimensional singular boundary value problem with general boundary conditions, proving existence of positive large solutions for all bifurcation parameters $\lambda$ and providing a sharp uniqueness result in the constant-weight case. The authors employ phase-plane analysis, shooting methods, and comparison principles to establish existence, monotonicity in $\lambda$, and asymptotic behavior as $\lambda\to\pm\infty$, with distinctions determined by Dirichlet, Neumann, and Robin boundary types. For general positive weight $a(x)$, they construct minimal and maximal positive solutions $L_{\lambda,a}^{\min}$ and $L_{\lambda,a}^{\max}$, showing that any positive solution is trapped between these envelopes, and they prove a weight-monotonicity-based uniqueness when $a$ is nonincreasing and $\lambda\ge0$. The results yield detailed bifurcation and blow-up rate descriptions, including uniform convergence statements on compact subsets and asymptotic regimes across the boundary conditions, enhancing understanding of logistic-type diffusive models under general boundary constraints.

Abstract

The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or non-classical; in the sense that, as soon as $\mathcal{B}u(0)=-u'(0)+βu(0)$, the coefficient $β$ can take any real value, not necessarily $β\geq 0$ as in the classical Sturm--Liouville theory. Since the function $f(u):=au^p -λu$, $u\geq 0$, is not increasing if $λ>0$, the uniqueness of the positive solution of (1.1) is far from obvious, in general, even for the simplest case when $a(x)$ is a positive constant. The second goal of this paper is to establish the uniqueness of the positive solution of (1.1) in that case. At a later stage, denoting by $L_λ$ the unique positive solution of (1.1) when $a(x)$ is a positive constant, we will characterize the point-wise behavior of $L_λ$ as $λ\to \pm \infty$. It turns out that any positive solution of (1.1) mimics the behavior of $L_λ$ as $λ\to \pm\infty$. Finally, we will establish the uniqueness of the positive solution of (1.1) when $a(x)$ is non-increasing in $[0,R]$, $λ\geq 0$, and $β<0$ if $-u'(0)+βu(0)=0$.

Figures (16)

• Figure 1: Phase diagram of \ref{['eq:3.1']} for $\lambda>0$. The green trajectory corresponds to a positive solution of \ref{['eq:3.1']} satisfying $u(0)=0$.
• Figure 2: Phase diagrams of \ref{['eq:3.1']} for $\lambda <0$ (left) and $\lambda=0$ (right). The green trajectory corresponds to a positive solution of \ref{['eq:3.1']} satisfying $u(0)=0$.
• Figure 3: Graph of the blow-up time $\mathcal{T}_{\mathcal{D}}(v_0)$ defined in \ref{['3.6']} for $\lambda > 0$ (left) and $\lambda\leq 0$ (right).
• Figure 4: Phase diagram of the equation \ref{['eq:3.1']} for $\lambda>0$. The green trajectory corresponds to a positive solution satisfying $u'(0)=v(0)=0$.
• Figure 5: Phase diagram of the equation \ref{['eq:3.1']} for $\lambda <0$ (left) and $\lambda=0$ (right). The green trajectory corresponds to a positive solution satisfying $u'(0)=0$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4
• proof