Global boundedness in a fully parabolic attraction–repulsion chemotaxis model

This paper deals with a fully parabolic attraction–repulsion chemotaxis system in two‐dimensional smoothly bounded domains. It is shown that the system admits global bounded classical solutions whenever the repulsion is dominated. The proof is based on an entropy‐like inequality and coupled estimate techniques. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

In this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in 3 .     

Boundedness in a two‐dimensional attraction–repulsion system with nonlinear diffusion

This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion: where χ > 0, ζ > 0, α_i>0, β_i>0 (i = 1,2) and f(s)≤κ ? μs^τ. In two‐space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ > 0. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundedness in a four‐dimensional attraction‐repulsion chemotaxis system with logistic source

In this paper, we study the attraction‐repulsion chemotaxis system with logistic source: u_t = Δu−χ∇·(u∇v)+ξ∇·(u∇w)+f(u), 0 = Δv−βv+αu, 0 = Δw−δw+γu, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain , where χ,α,ξ,γ,β, and δ are positive constants, and is a smooth function satisfying f(s) ≤ a−bs^3/2 for all s ≥ 0 with a ≥ 0 and b > 0. It is proved that when the repulsion cancels the attraction (ie, ξγ=χα), for any nonnegative initial data , the solution is globally bounded. This result corresponds to the one in the classical 2‐dimensional Keller‐Segel model with logistic source bearing quadric growth restrictions.     

Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

In this paper, we are interested in a model derived from the 1‐D Keller‐Segel model on the half line x >  as follows: where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l < 0. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.     

A stochastic Keller–Segel model of chemotaxis

We introduce stochastic models of chemotaxis generalizing the deterministic Keller–Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean’s approach, we derive the exact kinetic equation satisfied by the density distribution of cells. In the mean field limit where statistical correlations between cells are neglected, we recover the Keller–Segel model governing the smooth density field. We also consider hydrodynamic and kinetic models of chemotaxis that take into account the inertia of the particles and lead to a delay in the adjustment of the velocity of cells with the chemotactic gradient. We make the connection with the Cattaneo model of chemotaxis and the telegraph equation.     

Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

This paper deals with the attraction‐repulsion chemotaxis system with nonlinear diffusion u _t =?·(D (u )?u )??·(u χ (v )?v )+?·(u ^γ ξ (w )?w ), τ ₁v _t =Δv ?α ₁v +β ₁u , τ ₂w _t =Δw ?α ₂w +β ₂u , subject to the homogenous Neumann boundary conditions, in a smooth bounded domain , where the coefficients α _i , β _i , and τ _i ∈{0,1}(i =1,2) are positive. The function D fulfills D (u )?C _D u ^{m ?1} for all u >0 with certain C _D >0 and m >1. For the parabolic‐elliptic‐elliptic case in the sense that τ ₁=τ ₂=0 and γ =1, we obtain that for any and all sufficiently smooth initial data u ₀, the model possesses at least one global weak solution under suitable conditions on the functions χ and ξ . Under the assumption , it is also proved that for the parabolic‐parabolic‐elliptic case in the sense that τ ₁=1, τ ₂=0, and γ ?2, the system possesses at least one global weak solution under different assumptions on the functions χ and ξ .     

Global solutions of a two‐dimensional chemotaxis system with attraction and repulsion rotational flux terms

We consider the attraction–repulsion chemotaxis system with rotational flux terms where is a bounded domain with smooth boundary. Here, S₁ and S₂ are given parameter functions on [0,∞)²×Ω with values in . It is shown that for any choice of suitably regular initial data (u₀,v₀,w₀) fulfilling a smallness condition on the norm of v₀,w₀ in L^∞(Ω), the corresponding initial‐boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness for Keller–Segel system with fractional diffusion

In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier‐Herz spaces with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.     

Possibility of the existence of blow‐up solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear ‘degenerate’ Keller–Segel system of parabolic–parabolic type under the super‐critical condition. In the ‘non‐degenerate’ case, Winkler (Math. Methods Appl. Sci. 2010; 33:12–24) constructed the initial data such that the solution blows up in either finite or infinite time. However, the blow‐up under the super‐critical condition is left as an open question in the ‘degenerate’ case. In this paper, we try to give an answer to the question under assuming the existence of local solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Local solvability of the Keller–Segel system with Cauchy data in the Besov spaces

The Cauchy problem for the Keller–Segel system of parabolic elliptic type is considered for initial data in the Besov spaces with p < ∞ , and a sufficient condition is given on the existence and the uniqueness of local solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Global classical solutions of viscous liquid–gas two‐phase flow model

In this paper, we consider the global existence and uniqueness of the classical solutions for the three‐dimensional where the existence of global classical solutions to the compressible Navier–Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model

In this article, a new numerical scheme for a degenerate Keller–Segel model with heterogeneous anisotropic tensors is treated. It is well‐known that standard finite volume scheme not permit to handle anisotropic diffusion without any restrictions on meshes. Therefore, a combined finite volume‐nonconforming finite element scheme is introduced, developed, and studied. The unknowns of this scheme are the values at the center of cell edges. Convergence of the approximate solution to the continuous solution is proved only supposing the shape regularity condition for the primal mesh. This scheme ensures the validity of the discrete maximum principle under the classical condition that all transmissibilities coefficients are positive. Therefore, a nonlinear technique is presented, as a correction of the diffusive flux, to provide a monotone scheme for general tensors. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1030–1065, 2014     

Stationary solutions to the drift–diffusion model in the whole spaces

We study the stationary problem in the whole space ?ⁿ for the drift–diffusion model arising in semiconductor device simulation and plasma physics. We prove the existence and uniqueness of stationary solutions in the weighted L^p spaces. The proof is based on a fixed point theorem of the Leray–Schauder type. Copyright © 2008 John Wiley & Sons, Ltd.