Work

On the existence of a positive solution to a nonlocal logistic system with nonlinear advection terms

Year: 2025

Type: article

Abstract: In this paper, we study a nonlocal logistic system with nonlinear advection terms$ \begin{equation*} \left\{ \begin{array}{lcl} -\Delta u+\vec{\alpha}(x)\cdot \nabla (|u|^{p-1}u)& = &\left(a-\int_{\Omega}K_1(x, y)f(u, v)dy \right)u+bv\text{ in }\Omega, \\ -\Delta v+\vec{\beta}(x)\cdot \nabla (|v|^{q-1}v)& = &\left(d-\int_{\Omega}K_2(x, y)g(u, v)dy \right)v+cu\text{ in }\Omega, \\ \qquad \qquad \qquad \qquad u = v& = &0\text{ on }\partial\Omega, \end{array} \right. \end{equation*} $where $ \Omega\subset \mathbb{R}^N $, $ N\geq1 $, is a bounded domain with a smooth boundary, $ \vec{\alpha}(x) = (\alpha_1(x), \cdots, \alpha_N(x)) $ and $ \vec{\beta}(x) = (\beta_1(x), \cdots, \beta_N(x)) $ are flows satisfying suitable conditions, $ p, q\geq1 $, $ a, b, c, d>0 $ and $ K_1, K_2:\Omega\times\Omega\rightarrow \mathbb{R} $ are nonnegative functions, with their specific conditions detailed below. The functions $ f $ and $ g $ satisfy some assumptions which allow us to use bifurcation theory to prove the existence of solution to this problem. It is important to highlight that the inclusion of the integral nonlocal term on the right-hand side makes the problem more representative of real-world situations.

Source: Communications on Pure &amp Applied Analysis

DOI: 10.3934/cpaa.2025104

Authors: Willian Cintra, Romildo Lima, Mayra Soares

Institutions: Universidade de Brasília, Universidade Federal de Campina Grande

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação de Apoio à Pesquisa do Distrito Federal

Awards: G6756993358, G7493349729, G7918167441