On the well-posedness for Keller-Segel system with fractional diffusion

Communicated by M. Costabel 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 B˙q2−2αRn×B˙q2−2αRn with q ∈ [2, ∞], where 1 < α ≤ 2. Making...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 34; no. 14; pp. 1739 - 1750
Main Authors: Wu, Gang, Zheng, Xiaoxin
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 30.09.2011
Wiley
Subjects:
Exact sciences and technology
fractional diffusion
Functional analysis
Global analysis, analysis on manifolds
Ill-posedness
Keller-Segel model
Mathematical analysis
Mathematics
nonlinear evolution equations
nonlinear parabolic equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Partial differential equations
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Well-posedness
Keller-Segel model
Numerical linear algebra
Fourier analysis
Evolution equation
Well posed problem
Littlewood Paley theory
Mathematical method
Cauchy problem
Non linear equation
Regularization method
Parabolic equation
Numerical analysis
fractional diffusion
Linear equation
III-posedness
Applied mathematics
nonlinear parabolic equations
Ill posed problem
Localization
nonlinear evolution equations
Well-posedness
ISSN:0170-4214, 1099-1476
Online Access:Get full text
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Summary:Communicated by M. Costabel 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 B˙q2−2αRn×B˙q2−2αRn 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.
Bibliography:Chinese Postdoctoral Science Foundation - No. 20090460553
istex:DAAB3592B02342C1BD3920DA905A7BAACEA5AF7A
ArticleID:MMA1480
ark:/67375/WNG-BQ34G498-8
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.1480