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On the well‐posedness for Keller–Segel system with fractional diffusion
Author(s) -
Wu Gang,
Zheng Xiaoxin
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1480
Subject(s) - mathematics , dissipative system , fourier transform , initial value problem , norm (philosophy) , space (punctuation) , mathematical analysis , diffusion , cauchy distribution , computer science , law , physics , thermodynamics , quantum mechanics , political science , operating system
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 ( u 0 , v 0 ) in critical Fourier‐Herz spacesB ˙ q 2 − 2 αR n × B ˙ q 2 − 2 αR nwith 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.