Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotationu t = ∇ · ( D ( u ) ∇ u − uS ( u , v , x ) ∇ v ) ,x ∈ Ω ,t > 0 ,v t = Δ v − uf ( v ) ,x ∈ Ω ,t > 0 ,∇ v · ν = 0 ,( D ( u ) ∇ u − uS ( u , v , x ) ∇ v ) · ν = 0 ,x ∈ ∂ Ω , t > 0 ,where Ω ⊂ R n ( n ≥ 2 ) is a bounded domain with smooth boundary, D ( u ) is supposed to be sufficiently smooth and satisfies D ( u )≥ D 0 u m − 1 ( m ≥1) and D ( u )≤ D 1 ( u + 1) K − m u m − 1 ( K ≥1) for all u ≥0 with some positive constants D 0 and D 1 , and f ( u ) is assumed to be smooth enough and non‐negative for all u ≥0 and f (0) = 0, while S ( u , v , x ) = ( s i j ) n × n is a matrix withs ij ∈ C 2 ( [ 0 , ∞ ) × [ 0 , ∞ ) × Ω ¯ ) and | S ( u , v , x ) | ≤ u l − 2S ~ ( v ) with l ≥2, whereS ~ ( v ) is nondecreasing on [0, ∞ ). It is proved that when m > l − 2 n , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.