Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains

This paper deals with the quasilinear fully parabolic Keller–Segel system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂RNΩ⊂RN with smooth boundary, N∈NN∈N. The diffusivity D(u)D(u) is assumed to satisfy some further technical conditions such as algebraic growth and D(0)⩾0D(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+ε)αS(u)/D(u)⩽K(u+ε)α for u>0u>0 with α<2/Nα<2/N, K>0K>0 and ε⩾0ε⩾0. When D(0)>0D(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=RNΩ=RN and D(0)⩾0D(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.