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E. Keller, L. Segel (1971)
Traveling bands of chemotactic bacteria: a theoretical analysis.Journal of theoretical biology, 30 2
Zhian Wang (2012)
Mathematics of traveling waves in chemotaxis --Review paper--Discrete and Continuous Dynamical Systems-series B, 18
P. Biler (1999)
Global solutions to some parabolic-elliptic systems of chemotaxisAdv. Math. Sci. Appl., 9
Xinru Cao, J. Lankeit (2016)
Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivitiesCalculus of Variations and Partial Differential Equations, 55
E. Keller, L. Segel (1970)
Initiation of slime mold aggregation viewed as an instability.Journal of theoretical biology, 26 3
J. Lankeit (2016)
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusionarXiv: Analysis of PDEs
M. Winkler (2018)
Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorptionJournal of Differential Equations, 264
P. Biler (1999)
347Adv. Math. Sci. Appl., 9
J. Lankeit, M. Winkler (2017)
A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial dataNonlinear Differential Equations and Applications NoDEA, 24
Dirk Horstmann
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences
Xiangdong Zhao, Sining Zheng (2018)
Global existence and asymptotic behavior to a chemotaxis–consumption system with singular sensitivity and logistic sourceNonlinear Analysis: Real World Applications
M. Winkler (2016)
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation propertiesMathematical Models and Methods in Applied Sciences, 26
(1998)
Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis
J. Lankeit (2015)
A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivityMathematical Methods in the Applied Sciences, 39
N. Bellomo, N. Bellomo, A. Bellouquid, Youshan Tao, M. Winkler (2015)
Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissuesMathematical Models and Methods in Applied Sciences, 25
Yilong Wang (2016)
Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivityBoundary Value Problems, 2016
Ann Pellegrini (2011)
MovementMaterial Religion, 7
Sachiko Ishida, Kiyotaka Seki, T. Yokota (2014)
Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domainsJournal of Differential Equations, 256
Elisa Lankeit, J. Lankeit (2018)
Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorptionNonlinear Analysis: Real World Applications
Kentarou Fujie, T. Senba (2018)
A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis systemNonlinearity, 31
Renjun Duan, A. Lorz, P. Markowich (2010)
Global Solutions to the Coupled Chemotaxis-Fluid EquationsCommunications in Partial Differential Equations, 35
M. Winkler (2010)
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel modelJournal of Differential Equations, 248
C. Weibull (1960)
The Bacteria, vol. I
M. Marin, Andreas Öchsner (2018)
Elliptic Partial Differential EquationsNumerical Methods for Engineers and Scientists
Jian‐Guo Liu, A. Lorz (2011)
A coupled chemotaxis-fluid model: Global existenceAnnales De L Institut Henri Poincare-analyse Non Lineaire, 28
Dongmei Liu (2018)
Global classical solution to a chemotaxis consumption model with singular sensitivityNonlinear Analysis: Real World Applications
Kentarou Fujie (2015)
Boundedness in a fully parabolic chemotaxis system with singular sensitivityJournal of Mathematical Analysis and Applications, 424
Tobias Black (2017)
Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2DJournal of Differential Equations
D. Horstmann (2003)
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences IJahresber. Dtsch. Math.-Ver., 105
Zhian Wang, Zhaoyin Xiang, Pei Yu (2016)
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesisJournal of Differential Equations, 260
In this paper we study the zero-flux chemotaxis-system{ut=Δu−χ∇⋅(uv∇v)vt=Δv−f(u)v\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \textstyle\begin{cases} u_{t}=\Delta u -\chi\nabla\cdot(\frac{u}{v} \nabla v) \\ v_{t}=\Delta v-f(u)v \end{cases} $$\end{document} in a smooth and bounded domain Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\varOmega$\end{document} of R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{R}^{2}$\end{document}, with χ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi >0$\end{document} and f∈C1(R)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$f\in C^{1}(\mathbb{R})$\end{document} essentially behaving like uβ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u^{\beta}$\end{document}, 0<β<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$0<\beta<1$\end{document}. Precisely for χ<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi<1$\end{document} and any sufficiently regular initial data u(x,0)≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u(x,0)\geq0$\end{document} and v(x,0)>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$v(x,0)>0$\end{document} on Ω¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\bar{\varOmega}$\end{document}, we show the existence of global classical solutions. Moreover, if additionally m:=∫Ωu(x,0)dx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m:=\int _{\varOmega}u(x,0) dx$\end{document} is sufficiently small, then also their boundedness is achieved.
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 3, 2020
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