Global boundedness in a quasilinear chemotaxis system with general density-signal governed sensitivity

Abstract

In this paper we study the global boundedness of solutions to the quasilinear parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇φ(v)), 0=Δv−v+u, subject to homogeneous Neumann boundary conditions and the initial data u0 in a bounded and smooth domain Ω⊂Rn (n≥2), where the diffusivity D(u) is supposed to satisfy D(u)≥a0(u+1)−α with a0>0 and α∈R, while the density-signal governed sensitivity fulfills 0≤S(u)≤b0(u+1)β and 0<φ′(v)≤χvk for b0,χ>0 and β,k∈R. It is shown that the solution is globally bounded if α+β<(1−2n)k+2n with n≥3 and k<1, or α+β<1 for k≥1. This implies that the large k benefits the global boundedness of solutions due to the weaker chemotactic migration of the signal-dependent sensitivity at high signal concentrations. Moreover, when α+β arrives at the critical value, we establish the global boundedness of solutions for the coefficient χ properly small. It should be emphasized that the smallness of χ under k>1 is positively related to the total cellular mass ∫Ωu0dx, which is attributed to the stronger singularity of φ(v) at v=0 for k>1 and the fact that v can be estimated from below by a multiple of ∫Ωu0dx. In addition, distinctive phenomena concerning this model are observed by comparison with the known results.