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Giuseppe Viglialoro

Publications and source records attributed to Giuseppe Viglialoro.

At least 19 recordsLinked to original sources
arXiv2026

Global boundedness of a two-species attraction-attraction chemotaxis model with bilinear boundary influx

Since its introduction, the Keller--Segel model has become a cornerstone in the mathematical theory of chemotaxis and it has generated extensive analytical activity. Most studies consider homogeneous Neumann boundary conditions, which ensure mass conservation and simplify the qualitative analysis of solutions. To the best of the authors' knowledge, at present chemotaxis models incorporating boundary conditions that generate inward fluxes have only been studied in two recent papers, and we believe that this topic deserves and it may attract further mathematical attention. In this sense, in the present paper we investigate a two-species chemotaxis system with positive total flux. The model consists of two interacting populations, $u$ and $w$, coupled through elliptic/parabolic chemical signals $v$ and $z$, and subject to Robin-type boundary conditions allowing inward fluxes that depend on the product of the cellular and chemical densities. Unlike the classical conservative setting, the total mass is not preserved and it exhibits quadratic growth in time, exactly in line with one of the investigations above mentioned and dealing with a single-species taxis model. We show that, within the considered framework, standard logistic damping is not sufficient to compensate for the mass increase induced by the positive boundary flux. To restore control of the dynamics, stronger dissipative mechanisms involving gradient-dependent damping terms are required. Under suitable assumptions, we establish the global existence and boundedness of classical solutions in the presence of logistic-gradient damping.

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arXiv2026

Uniform boundedness for the two-dimensional Keller-Segel system with Gompertz growth

It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping mechanisms that can still stabilize the system. Previous work showed that, under suitable assumptions on the initial cell distribution, even weaker growth terms than the classical logistic one can prevent aggregation. In this paper, we study the effect of a Gompertz-type growth term in a minimal two-dimensional chemotaxis model. This term provides a weaker damping effect than those previously considered. We analyze how it influences the system and identify conditions that guarantee that solutions exist for all time and remain bounded.

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arXiv2026

Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

This paper investigates a {three-component} chemotaxis system involving both attraction and repulsion effects, as well as a nonlocal logistic-type source term. Mathematically, if $u=u(x,t)$, $v = v(x,t)$ and $w = w(x,t)$ denote the cell distribution, and the attractive and the repulsive chemical signals, the model is then described by \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot (u \nabla v) + ξ\nabla \cdot (u \nabla w) + a u^α- b u^α\int_Ωu^β, & x \in Ω, \ t > 0, τv_t = Δv - v + f(u), & x \in Ω, \ t > 0, τw_t = Δw - w + g(u), & x \in Ω, \ t > 0. \end{cases} \end{equation*} Here, $Ω\subset \mathbb{R}^n$ ($n \geq 1$) is a bounded smooth domain, $τ\in\{0,1\}$, $a,b,α,β,χ,ξ>0$, the production functions $f(u)$ and $g(u)$ are assumed to satisfy algebraic growth conditions of order $\ell$ and $ρ$, generalizing prototypes of the form $u^\ell$ and $u^ρ$, $\ell,ρ>0$. The work is devoted to proving the global existence and boundedness of classical solutions under a suitable balance between the signal production exponents $\ell, ρ$ and the nonlocal damping exponents $α, β$, for regular enough initial data and zero-flux boundary restrictions. In this regard, two main theorems are established for the cases where the chemical signals satisfy either elliptic ($τ=0$) or parabolic ($τ=1$) partial differential equations, highlighting how sufficiently strong nonlocal damping prevents the formation of singularities in time. We extend the results obtained in [Chiyo et al., Appl. Math. Optim. 89:9 (2024)], where the fully parabolic ($τ=1$) and only attraction version is studied. In our context, we establish well-posedness of the system and the long-time behavior of solutions.

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arXiv2025

Global dynamics of chemotaxis-consumption systems with oppositely acting nonlocal terms

This paper studies a chemotaxis system where cells move in response to a chemical signal within a confined habitat. The model includes external source terms that combine local and nonlocal growth with dampening effects. The main focus is on conditions under which solutions exist for all time and remain uniformly bounded, preventing cell aggregation. Two types of source terms are considered. In the first case, the structure of the source term ensures that the total cell mass remains controlled over time. In the second case, this mass control is not guaranteed, which can lead to different dynamic behaviors. The results extend previous studies that examined similar systems but with more specific source terms and slightly different chemical dynamics. This work highlights how variations in the reaction terms influence the long-term behavior of the system.

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arXiv2025

Chemotaxis models with mixed mechanisms: boundedness in growth-dominated regimes

We study a chemotaxis-growth system with nonlinear local and nonlocal reactions and gradient-dependent damping. Under suitable conditions on the system parameters and spatial dimension, we prove that solutions exist globally in time and remain uniformly bounded. Unlike classical cases, when local growth dominates, mass control is not automatic. To address this, we use a two-step approach: first ensuring bounded total mass, then establishing full uniform boundedness. The results highlight how chemotaxis, damping, and nonlocal effects interact to prevent blow-up in structured models.

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arXiv2025

To what extent does the consideration of positive total flux influence the dynamics of Keller-Segel-type models?

Since the introduction of the Keller-Segel model in 1970 to describe chemotaxis (the interactions between cell distributions u and chemical distributions v), there has been a significant proliferation of research articles exploring various extensions and modifications of this model within the scientific community. From a technical standpoint, the totality of results concerning these variants are characterized by the assumption that the total flux, involving both distributions, of the model under consideration is zero. This research aims to present a novel perspective by focusing on models with a positive total flux. Specifically, by employing Robin-type boundary conditions for u and v, we seek to gain insights into the interactions between cells and their environment, uncovering important dynamics such as how variations in boundary conditions influence chemotactic behavior. In particular, the choice of the boundary conditions is motivated by real-world phenomena and by the fact that the related analysis reveals some interesting properties of the system.

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arXiv2025

Boundedness in a nonlinear chemotaxis-consumption model with gradient terms

We study a chemotaxis-consumption mechanism, in which some chemical signal and cells density interact each other. In order to control the concentration of such a population, sources involving gradient nonlinearities, which introduce a dampening effect on the model, are considered. Moreover, the system is characterized by nonlinear diffusion and sensitivity terms. We derive conditions on some data of the problem so to ensure the boundedness of related solutions.

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arXiv2024

Dissipation through combinations of nonlocal and gradient nonlinearities in chemotaxis models

This work concerns with a class of chemotaxis models in which external sources, comprising nonlocal and gradient-dependent damping reactions, influence the motion of a cell density attracted by a chemical signal. The mechanism of the two densities is studied in bounded and impenetrable regions. In particular, it is seen that no gathering effect for the cells can appear in time provided that the damping impacts are sufficiently strong.

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arXiv2023

Boundedness through nonlocal dampening effects in a fully parabolic chemotaxis model with sub and superquadratic growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough.

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arXiv2023

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

We introduce a novel gradient-based damping term into a Keller-Segel type taxis model with motivation from ecology and consider the following system equipped with homogeneous Neumann-boundary conditions: \begin{equation} \begin{cases} u_t= Δu - χ\nabla \cdot (u \nabla v)+a u^α-b u^β-c|\nabla u|^γ,\\ τv_t=Δv-v+u .\\ \end{cases} \end{equation} The problem is formulated in a bounded and smooth domain $Ω$ of $\mathbb{R}^N$, with $N\geq 2$, for some positive numbers $a,b,c,χ>0$, $τ\in \{0,1\}$, $γ\geq 1$, $β>α\geq 1$. As far as we know, Keller-Segel models with gradient-dependent sources are new in the literature and, accordingly, beyond giving a reasonable ecological interpretation the objective of the paper is twofold: 1.) to provide a rigorous analysis concerning the local existence and exensibility criterion for a class of models generalizing the above problem, obtained by replacing $a u^α-b u^β-c|\nabla u|^γ$ with $f(u)-g(\nabla u)$; 2.) to establish sufficient conditions on the data of the problem itself, such that it admits a unique classical solution $(u,v)$, for $T_{max}=\infty$ and with both $u$ and $v$ bounded. We handle 1.) whenever appropriately regular initial distributions $u(x,0)=u_0(x)\geq 0$, $τv(x,0)=τv_0(x)\geq 0$ are considered and $f$ and $g$ obey some regularity properties and, moreover, some growth restrictions. Further, as to 2.), for the same initial data considered in the previous case, global boundedness of solutions is proven for any $τ\in \{0,1\}$, provided that $\frac{2N}{N+1}<γ\leq 2$.

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arXiv2023

Properties of given and detected unbounded solutions to a class of chemotaxis models

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction-repulsion model, with nonlinear productions, diffusion, sensitivities and logistic term, we detect Lebesgue spaces where given unbounded solutions blow-up also in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved.

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arXiv2022

Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production and consumption

This paper is framed in a series of studies on attraction-repulsion chemotaxis models combining different effects: nonlinear diffusion and sensitivities and logistic sources, for the dynamics of the cell density, and consumption and/or production impacts, for those of the chemicals. In particular, herein we focus on the situation where the signal responsible of gathering tendencies for the particles' distribution is produced, while the opposite counterpart is consumed. In such a sense, this research complements two recent results, where the chemicals evolve according to different laws.

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arXiv2021

Improvements and generalizations of results concerning attraction-repulsion chemotaxis models

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.

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