We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d\geq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.
We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v_1) + ξ\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = Δv_1 - βv_1 + αu \\ \partial_t v_2 = Δv_2 - δv_2 + γu, \end{cases} \end{equation*} posed with homogeneous Neumann boundary conditions in bounded domains $Ω=B_R \subset \mathbb{R}^3$, $R>0$, admits radially symmetric solutions which blow-up in finite time if it is attraction-dominated in the sense that $χα-ξγ>0$.
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $χ, ξ>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the …
In this paper we study a zero-flux attraction-repulsion che\-mo\-taxis-system. We show that despite any mutual interplay between the repulsive and attractive coefficients from the corresponding chemo-sensitivities, even less any restriction on their own sizes, if the production rate of that chemical signal responsible of the cellular coalescence is sublinear, then any initial data emanate a unique global classical solution, which is as well bounded. Further, in a remark of the manuscript, we also address an open question given in \cite{Viglialoro2019RepulsionAttraction}.
This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = Δu - χ\nabla\cdot(u \nabla v) + ξ\nabla\cdot (u \nabla w) + λu - μu^k, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δv + αu - βv, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δw + γu - δw, \quad &x \in Ω,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $Ω\subset \mathbb{R}^n$ ($n \ge 3$), with constant parameters $λ\in \mathbb{R}$, $k>1$, $μ, χ, ξ, α, β, γ, δ>0$. Blow-up phenomena in the system have been well investigated in the c…
In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} u_{ t}=Δu -χ\nabla \cdot (u\nabla v)+ξ\nabla \cdot (u\nabla w) & \textrm{in }Ω\times (0,t^*), \\ 0=Δv+αu-βv & \textrm{in } Ω\times (0,t^*),\\ 0=Δw+γu-δw & \textrm{in } Ω\times (0,t^*),\\ \end{cases} \end{equation} where $Ω$ is a smooth and bounded domain of $\mathbb{R}^2$, $t^*$ is the blow--up time and $α,β,γ,δ,χ,ξ$ are positive real numbers. From the literature it is known that under a proper interplay between the above p…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0…
In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. M.P. Brenner et al.1999, Nonlinearity, 12, 1071-1098); one type is self-similar, and may be viewed as a trade-off between diffusion and attraction, while in the other type the attraction prevails over the diffusion and a non-self-simil…
For the parabolic-elliptic Keller-Segel system in R^2 it has been proved that if the initial mass is less than 8π/χ global solution exist and in the case that the initial mass is larger than 8π/χ blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the case of two especies case that allows us to determine if the system has blow-up or has a global in time solution.
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d \geq 3$. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a set of critical values: there is a critical value of a parameter pair in the system of equations for which there is a global-in-time energy solution and there exist blowing-up free-energy solutions under a criticality …
This work considers a chemotaxis system for multi-species that includes birth or death rate terms, which implies no mass preservation of the populations. We aim to show the convergence to a $L^{\infty}\ -\ $weak solutions, that is local in time, of the JKO - scheme arising from the Optimal Transport Theory, in the spirit of [35,14]. Currently, $L^{\infty}$ solutions have shown to be important in order to get uniqueness. Since death rate case does not ensure global solutions, for arbitrary initial data, in this framework, it could be interest to analyze the Blowing-up phenomenon of this system.…
In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,\infty}(1\leq j\leq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one spec…
Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner- and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighbouring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum …
This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a two-species population in interaction appearing for instance in bacterial chemotaxis. Blow-up of classical solutions occurs in finite time. This raises the question to define measure-valued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniquenes…
Refined blow-up criteria and global solutions for triangular cross-diffusion systems
Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Département de Mathématiques et Applications (DMA), Ecole Normale Supérieure (ENS-PSL)（索邦大学、法国国家科学研究中心、巴黎西岱大学、雅克-路易·利翁斯实验室、数学与应用系、巴黎高等师范学院）
This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1- u^{κ_1-1} - a_1 v^{λ_1-1}), &\quad x \in Ω,\ t>0,\\ % v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1- a_2 u^{λ_2-1} - v^{κ_2-1}), &\quad x \in Ω,\ t>0,\\ % 0 = d_3 Δw + αu + βv - h(u,v,w), &\quad x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=γw$ or $h=\frac{1}{|Ω|}\int_Ω(αu+ βv)\,dx$. In the case that $κ_1=λ_1=κ_2=λ_2=2$ and $h=γw$, it is known that sm…
We study the collapse of a many-body system which is used to model two-component Bose-Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for $N$ identical bosons in $\mathbb R^2$, interacting with potentials rescaled in the mean-field manner $-N^{2β-1}w^{(σ)}(N^βx)$ with $\int_{\mathbb R^{2}}w^{(σ)}(x){\rm d}x=1$. Assuming that $0<β<1/2$, we first show that the leading order of the quantum energy is captured correctly by the Gross-Pitaevskii energy. Secondly, we…
The aim of this paper is to study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inhere…
This paper investigates the global well-posedness of a class of reaction-advection-diffusion models with nonlinear diffusion and Lotka-Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic-elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka-Volterra competition systems.
We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant $-α$, with $α>0$. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behaviour, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The …
A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Technical University Berlin（柏林工业大学）；German Aerospace Center (DLR)（德国航空航天中心）
We analyze a reaction-diffusion system describing the growth of microbial species in a model of flocculation type that arises in biology. Existence of global classical positive solutions is proved under general growth assumptions, with flocculation and deflocculation rates polynomially bounded above, that guarantee uniform sup norm bounds for all time t obtained by an $L^{p}-$energy functional estimate. We also show finite time blow up can occur when the yield coefficients are large enough. Also, using arguments relying on the spectral and fixed theory, we show persistence and existence of non…
In this study, we consider the following extended attraction chemotaxis system of two species parabolic-parabolic-elliptic type with nonlocal terms \[ \begin{cases} u_t=d_1Δu-χ_1\nabla (u\cdot \nabla w)+u\left(a_0-a_1u-a_2v-a_3\int_Ωu-a_4\int_Ωv\right),\quad x\in Ω\quad\cr v_t=d_2Δv-χ_2\nabla (v\cdot \nabla w)+v\left(b_0-b_1u-b_2v-b_3\int_Ωu-b_4\int_Ωv\right),\quad x\in Ω\quad\cr 0=d_3Δw+k u+lv-λw,\quad x\in Ω\quad\cr \end{cases} \] under homogeneous Neuman boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n\ge1)$ with smooth boundary, where $a_0,b_0,\, \,a_1,$ and $ b_2$ are posi…
This article is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the inter-specific competitions are strong, while if the inter-specifi…
