Strong maximum principle for nonlocal diffusion in measure spaces: new characterization and population dynamics applications
Abstract
We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form a0(x)u(x)−∫Ωk(x,y)(u(y)−u(x)) dμ(y)=F(x,u), u≥ 0, on a general measure space (Ω,M,μ), where the kernel k is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.
Related
- Characterizing the breakdown of the maximum principle in a class of nonlocal elliptic problems
- Simultaneous recovery of multiple parameters in nonlocal diffusion equations from internal measurements
- Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range
- Turing bifurcation, criticality, and multistability in a nonlocal advection-reaction-diffusion equation
- Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules
BibTeX
@misc{casadosanchez2026strong,
title = {{Strong maximum principle for nonlocal diffusion in measure spaces: new characterization and population dynamics applications}},
author = {Ana Casado-Sanchez and Monica Molina-Becerra and Antonio Suarez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07191},
eprint = {2609.07191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07191},
}
RIS
TY - UNPB
TI - Strong maximum principle for nonlocal diffusion in measure spaces: new characterization and population dynamics applications
AU - Ana Casado-Sanchez
AU - Monica Molina-Becerra
AU - Antonio Suarez
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07191
UR - https://arxiv.org/abs/2609.07191
AB - We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form a_0(x)u(x)−∫_Ωk(x,y)(u(y)−u(x)) dμ(y)=F(x,u), u≥ 0, on a general measure space (Ω,M,μ), where the kernel k is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.
ER -