Journal ArticleDOI
Global solutions in a fully parabolic chemotaxis system with singular sensitivity
TLDR
In this article, the Neumann boundary value problem for the chemotaxis system is considered in a smooth bounded domain and 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.Abstract:
The Neumann boundary value problem for the chemotaxis system
is considered in a smooth bounded domain Ω⊂ℝn, n⩾2, with initial data and v0∈W1, ∞(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.read more
Citations
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Journal ArticleDOI
Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues
TL;DR: In this article, a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions.
Journal ArticleDOI
Boundedness in a fully parabolic chemotaxis system with singular sensitivity
TL;DR: In this article, a fully parabolic chemotaxis system with singular sensitivity χ v ( χ > 0 ) on a bounded domain Ω ⊂ R n, n ≥ 2 was studied and the main result solved the open problem of uniform-in-time boundedness of solutions for χ 2 n.
Journal ArticleDOI
Global weak solutions in a chemotaxis system with large singular sensitivity
TL;DR: In this article, a generalized solution concept is introduced for the Neumann problem associated with (⋆), and within this concept global-in-time solutions are shown to exist regardless of the size of χ>0.
Journal ArticleDOI
Mathematics of traveling waves in chemotaxis --Review paper--
TL;DR: In this paper, the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of biological or medical phenomena including bacterial chemotactic motion, initiation of angiogenesis and reinforced random walks, are surveyed.
Journal ArticleDOI
A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data
Johannes Lankeit,Michael Winkler +1 more
TL;DR: In this paper, a generalized solution framework for the chemotaxis system is introduced within which an extension of previously known ranges for the key parameter with regard to global solvability is achieved.
References
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Book
Navier-Stokes Equations: Theory and Numerical Analysis
TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
MonographDOI
Linear and Quasi-linear Equations of Parabolic Type
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.
Journal ArticleDOI
Initiation of slime mold aggregation viewed as an instability.
Evelyn Fox Keller,Lee A. Segel +1 more
TL;DR: A mathematical formulation of the general interaction of amoebae, as mediated by acrasin is presented, and a detailed analysis of the aggregation process is provided.
Journal ArticleDOI
A user’s guide to PDE models for chemotaxis
Thomas Hillen,Kevin J. Painter +1 more
TL;DR: This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.
Journal ArticleDOI
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
TL;DR: In this article, the authors considered the classical parabolic-parabolic Keller-Segel system with homogeneous Neumann boundary conditions in a smooth bounded domain and proved that for each q > n 2 and p > n one can find e 0 > 0 such that if the initial data ( u 0, v 0 ) satisfy L q ( Ω ) e and ∇ v 0 ‖ L p (Ω) e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic
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