Global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity

In this paper we study the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇φ(v)), vt=Δv−v+u, where bounded domain Ω⊂Rn (n≥2) subject to the non-flux boundary conditions, the diffusivity fulfills D(u)=a0(u+1)−α with a0>0 and α≥0, while the density-signal governed sensitivity satisfies 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 provided α+β<1 and k≤1. This result demonstrates the effect of signal-dependent sensitivity on the blow-up prevention.