Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u_F:=e^tΔu₀; we prove that there exist 2 positive constants σ₀ and C₀ such that if the gravitational potential and the initial data (u₀,n₀,c₀) satisfy for some p,q with and , then the global solutions can be established in critical Besov spaces.     

A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model in a smooth, bounded, convex two‐dimensional domain and show global existence and boundedness of solutions for χ∈(0,χ₀) for some χ₀>1, thereby proving that the value χ = 1 is not critical in this regard. Our main tool is consideration of the energy functional for a > 0, b≥0, where using nonzero values of b appears to be new in this context. Copyright © 2015 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.     

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.     

Strong solutions to the Keller‐Segel system with the weak Ln /2 initial data and its application to the blow‐up rate

We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u₀ in L^{n /2}_w (?ⁿ), the weak L^{n /2}‐space on ?ⁿ. If ‖u₀‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L^{n /2}_w (?ⁿ) stems from obtaining a self‐similar solution which does not belong to any usual L^p(?ⁿ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖u (t)‖ for n /2 < p < ∞ as t → T_max, where T_max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.     

Global solutions in a fully parabolic chemotaxis system with singular sensitivity

The Neumann boundary value problem for the chemotaxis system is considered in a smooth bounded domain Ω??ⁿ, n?2, with initial data and v₀∈W^{1, ∞}(Ω) satisfying u₀?0 and v₀>0 in . It is shown that if then for any such data there exists a global‐in‐time classical solution, generalizing a previous result which asserts the same for n=2 only. Furthermore, it is seen that the range of admissible χ can be enlarged upon relaxing the solution concept. More precisely, global existence of weak solutions is established whenever . Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

We consider a chemotaxis consumption system with singular sensitivity , v_t=εΔv−uv in a bounded domain with χ,α,ε>0. The global existence of classical solutions is obtained with n=1. Moreover, for any global classical solution (u,v) to the case of n,α≥1, it is shown that v converges to 0 in the L^∞‐norm as t→∞ with the decay rate established whenever ε∈(ε₀,1) with .     

Boundedness in a higher‐dimensional chemotaxis system with porous medium diffusion and general sensitivity

This paper deals with the following chemotaxis system: in a bounded domain with smooth boundary under no‐flux boundary conditions, where satisfies for all with l ?2 and some nondecreasing function on [0,∞ ). Here, f (v )∈C ¹([0,∞ )) is nonnegative for all v ?0. It is proved that when , the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 39 : 1159–1175) which shows global existence and boundedness of weak solutions under the condition . Copyright © 2017 John Wiley & Sons, Ltd.     

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

This paper is concerned with the two‐species chemotaxis‐competition system where Ω is a bounded domain in with smooth boundary ∂Ω, n≥2; χ_i and μ_i are constants satisfying some conditions. The above system was studied in the cases that a₁,a₂∈(0,1) and a₁>1>a₂, and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a₁,a₂, and have not been obtained in the case that a₁,a₂≥1. Moreover, convergence rates in the cases that a₁,a₂∈(0,1) and a₁>1>a₂ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a₁,a₂>0 which covers the case that a₁,a₂≥1, and lead to convergence rates for solutions of the above system in the cases that a₁,a₂∈(0,1) and a₁≥1>a₂.     

Global existence and blow up of solutions of quasilinear chemotaxis system

In this paper, we study the global existence of solution for the quasilinear chemotaxis system with Dirichlet boundary conditions, and further we show that the blow up properties of the solution depend only on the first eigenvalue. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis

In this paper we study the global in-time and blow-up solutionsfor the simplified Keller–Segel system modelling chemotaxis.We prove that there is a critical number which determines theoccurrence of blowup in the two-dimensional case for 1 <p < 2. In three- or higher-dimensional cases, we show thatthe radial symmetrical solution will blow up if 1 < p <N/N–2 (N 3) for non-negative initial value.     

Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D₀u^{m ? 1}(m≥1) and D(u)≤D₁(u + 1)^{K ? m}u^{m ? 1}(K≥1) for all u≥0 with some positive constants D₀ and D₁, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (s_ij)_{n × n} is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 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.     

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.