Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production

This work is concerned with a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production: ut=∇⋅((1+u)−α∇u)−∇⋅(u(1+u)β−1∇v)+f(u),vt=Δv−v+uγ, with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain, where α,β∈R and γ>0. The logistic type source term f(u) satisfies that either f(u)≡0 or f(u)=ru−μuk with r∈R, μ>0 and k>1. The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameters conditions, which improves the known results.