Global boundedness of solutions for a quasilinear chemotaxis‐Stokes system with nonlinear production

Abstract In this paper we study the global boundedness of solutions to the quasilinear chemotaxis‐Stokes system with nonlinear production subject to no‐flux/no‐flux/Dirichlet boundary conditions in a smoothly bounded domain with , where , , and generalize the prototypes and for all with and . It is shown that if for , or for , then for any reasonably regular initial datum, the corresponding initial‐boundary value problem of () possesses a unique globally bounded classical solution. Compared to the global boundedness condition for in the relevant fluid‐free system [J. Differ. Equ. 268 6729–6777 (2020)], the present result indicates that the slow fluid motion described by the Stokes equation might possibly be adverse to the global solvability in the case that with . What is more, this paper provides for the quasilinear problems with general signal production a universal approach capable of deriving the boundedness of solutions in the optimal range of parameters.