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1.
Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity 下载免费PDF全文
Kentarou Fujie Michael Winkler Tomomi Yokota 《Mathematical Methods in the Applied Sciences》2015,38(6):1212-1224
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. 相似文献
2.
Ke Lin Chunlai Mu Liangchen Wang 《Mathematical Methods in the Applied Sciences》2015,38(18):5085-5096
This paper is concerned with the following coupled chemotaxis system with homogeneous Neumann boundary conditions in a bounded domain Ω?Rn(n≥2) with smooth boundary, where λ, χ1, χ2, μ1, μ2, a1, a2, b1, and b2 are supposed to be positive and τ = 0,1. In the case τ = 0, based on some energy estimates for both u and v, it is shown that for any parameters, the system possesses a unique globally bounded solution if n = 2. Moreover, when τ = 1, relying on a comparison principle, for a range of parameters, the existence of a unique global bounded classical solution of problem is established for any n≥2 if Ω is convex. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
3.
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. 相似文献
4.
《Mathematische Nachrichten》2018,291(14-15):2318-2333
In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough. 相似文献
5.
The asymptotic behavior of the attraction–repulsion Keller–Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction–repulsion Keller–Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time‐dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform‐in‐time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
6.
We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type ut = ? ? (?(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, ? ≥ c1sp 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. 相似文献
7.
8.
This paper deals with positive solutions of the fully parabolic system under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain with positive parameters and nonnegative smooth initial data .Global existence and boundedness of solutions were shown if in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying . This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has -dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in . 相似文献
9.
This paper deals with the Keller–Segel system where Ω is a bounded domain in with smooth boundary , ; χ is a nonnegative function satisfying for some and . In the case that and , Fujie 2 established global existence of bounded solutions under the condition . On the other hand, when , Winkler 14 asserted global existence of bounded solutions for arbitrary . However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary . Moreover, the condition for K when cannot connect with the condition when . The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases and . 相似文献
10.
Hans-Christoph Kaiser Hagen Neidhardt Joachim Rehberg 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(3):287-310
Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients
including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space
of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions
enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have
in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system. 相似文献
11.
Gabriela Li?canu Cristian Morales-Rodrigo 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):77-98
In this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two dimensional Euclidean space. This system appears as a mathematical model for some biological processes. Global existence and uniqueness of a nonnegative classical Hölder continuous solution are proved. The last part of the paper is devoted to the study of the asymptotic behavior of the solutions. 相似文献
12.
Liangchen Wang Chunlai Mu Xuegang Hu Pan Zheng 《Journal of Differential Equations》2018,264(5):3369-3401
This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractantunder homogeneous Neumann boundary conditions in a bounded domain () with smooth boundary, where the initial data and are non-negative and the parameters , , and . The chemotactic function () is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for ,(i) and(ii) .Moreover, we prove asymptotic stabilization of solutions in the sense that:? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution algebraically converge to as . 相似文献
13.
Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source 下载免费PDF全文
This paper deals with a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source under homogeneous Neumann boundary conditions in a smooth bounded domain . The functions χ and f are assumed to generalize the chemotactic sensitivity function and logistic source respectively. Under some conditions, we obtain that the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded. 相似文献
14.
Qingshan Zhang 《Mathematische Nachrichten》2016,289(17-18):2323-2334
We consider the chemotaxis system with rotation under no‐flux boundary conditions in the bounded domain , . Here the matrix‐valued function fulfills () for all with some nondecreasing function S0 and is a nonnegative function with for all . Moreover, f satisfies for all with nondecreasing function f0. It is shown that for the nonnegative initial data and with , if at least one of the following assumptions holds:
- ,
- , and ,
- ,
15.
Sachiko Ishida Takashi Ono Tomomi Yokota 《Mathematical Methods in the Applied Sciences》2013,36(7):745-760
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. 相似文献
16.
Michael Winkler 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(2):1044-1064
We consider the elliptic-parabolic PDE system
17.
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. 相似文献
18.
Flávio Dickstein Miguel Loayza 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(1):1-23
We consider the Cauchy problem for the weakly coupled parabolic system ∂
t
w
λ−Δ w
λ = F(w
λ) in R
N
, where λ > 0, w
λ = (u
λ, v
λ), F(w
λ) = (v
λ
p
, u
λ
q
) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ1, φ2
C
0(R
N
). If (p, q) is sub-critical or either φ1 or φ2 has slow decay at ∞, w
λ blows up for all λ > 0. Under these conditions, we study the blowup of w
λ for λ small.
相似文献
19.
E. Nakaguchi 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(1):286-297
We study the global existence of solutions to a parabolic-parabolic system for chemotaxis with a logistic source in a two-dimensional domain, where the degradation order of the logistic source is weaker than quadratic. We introduce nonlinear production of a chemoattractant, and show the global existence of solutions under certain relations between the degradation and production orders. 相似文献
20.
Vasilii V. Kurta 《Archiv der Mathematik》2005,85(6):563-571
We obtain a new comparison principle for weak solutions of the Cauchy problem for a wide class of quasilinear parabolic inequalities.
This is a nonlinear result with no analogue in linear theory.
Received: 13 January 2005 相似文献