Global boundedness in a quasilinear chemotaxis system with signal-dependent sensitivity

This paper deals with a quasilinear parabolic-elliptic chemotaxis system with signal-dependent sensitivity{ut=∇⋅(φ(u)∇u)−∇⋅(uχ(v)∇v),(x,t)∈Ω×(0,∞),0=Δv−v+u,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥2), with nonnegative initial data 0≢u0∈C0(Ω¯), where the given functions φ(u) and χ(v) are the nonlinear diffusion and chemotactic sensitivity function, respectively. Firstly, under the case of non-degenerate diffusion φ(u), it is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in Ω×(0,∞). Moreover, under the case of degenerate diffusion φ(u), we prove that the corresponding problem asserts at least one nonnegative global-in-time bounded weak solution.