On the existence of a positive solution to a nonlocal logistic system with nonlinear advection terms
Willian Cintra, Romildo Lima, Mayra Soares
TL;DR
The paper studies a two-species nonlocal logistic system with nonlinear advection on a bounded domain, incorporating nonlocal reaction terms via kernels and homogeneous Dirichlet boundary conditions. The authors recast the problem through linear solution operators and nonlinear maps, then apply Rabinowitz global bifurcation theory to obtain existence and nonexistence results for positive classical solutions, with spectral thresholds involving the largest eigenvalue of the coupling matrix $A=[a\ b; c\ d]$ and the principal eigenvalue of the advection-diffusion operator. For the linear advection case ($p=q=1$) with $\operatorname{div}\vec{α}=0$, positivity occurs iff $\lambda_A>\lambda_1^{\alpha}$; for the nonlinear advection case ($p,q>1$) with $\gamma>\max\{p,q\}$ and $\operatorname{div}\vec{α}=\operatorname{div}\vec{β}=0$, positivity requires $\lambda_A>\lambda_1$. The paper also develops a local bifurcation analysis via Crandall–Rabinowitz to describe a local branch of positive solutions near the bifurcation point, including conditions under which positive solutions exist when $\lambda_A$ is slightly below $\lambda_1$. These results advance understanding of nonlocal logistic systems with advection, linking spectral properties to the emergence of coexistence states in population dynamics models.
Abstract
In this paper, we study a nonlocal logistic system with nonlinear advection terms \begin{equation*} \left\{ \begin{array}{lcl} -Δu+\vecα(x)\cdot \nabla (|u|^{p-1}u)&=&\left(a-\int_ΩK_1(x,y)f(u,v)dy \right)u+bv\mbox{ in }Ω,\\ -Δv+\vecβ(x)\cdot \nabla (|v|^{q-1}v)&=&\left(d-\int_ΩK_2(x,y)g(u,v)dy \right)v+cu\mbox{ in }Ω,\\ \qquad \qquad \qquad \qquad u=v&=&0\mbox{ on }\partialΩ, \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$, $N\geq1$, is a bounded domain with a smooth boundary, $\vecα(x)=(α_1(x),\cdots,α_N(x))$ and $\vecβ(x)=(β_1(x),\cdots,β_N(x))$ are flows satisfying suitable conditions, $p,q\geq1$, $a,b,c,d>0$ and $K_1,K_2:Ω\timesΩ\rightarrow\mathbb{R}$ 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 problem $(P)$. 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.