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This paper deals with the quasilinear fully parabolic Keller–Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R N with smooth boundary, N ∈ N . The diffusivity D ( u ) is assumed to satisfy some further technical conditions such as algebraic growth and D ( 0 ) ⩾ 0 , which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that S ( u ) / D ( u ) ⩽ K ( u + e ) α for u > 0 with α 2 / N , K > 0 and e ⩾ 0 . When D ( 0 ) > 0 , this paper represents an improvement of Tao and Winkler [17] , because the domain does not necessarily need to be convex in this paper. In the case Ω = R N and D ( 0 ) ⩾ 0 , uniform-in-time boundedness is an open problem left in a previous paper [7] . This paper also gives an answer to it in bounded domains. |
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