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Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data
Author(s) -
Yokota Tomomi,
Yoshino Noriaki
Publication year - 2016
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.3780
Subject(s) - lipschitz continuity , mathematics , degenerate energy levels , smoothness , nonlinear system , mathematical analysis , pure mathematics , diffusion , physics , quantum mechanics , thermodynamics
This paper is concerned with local and global existence of solutions to the parabolic‐elliptic chemotaxis system∂b ∂t − Δ D ( b ) + ∇ · ( K ( b , c ) b ∇ c ) = 0 , − Δ c + c = b . Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear m ‐accretive operators to giving existence of local solutions to this system when 0 < D 0 ≤ D ′( r )≤ D ∞ < ∞ and ( r 1 , r 2 )↦ K ( r 1 , r 2 ) r 1 is Lipschitz continuous onR 2 , provided that the initial data is assumed to be small. The smallness assumption on the initial data was recently removed (J. Math. Anal. Appl. 2014; 419:756–774). However the case of non‐Lipschitz and degenerate diffusion, such as D ( r ) = r m ( m > 1), is left incomplete. This paper presents the local and global solvability of the system with non‐Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley & Sons, Ltd.