The convergence rate of solutions for a class of three-dimensional Keller–Segel–Stokes systems with indirect signal production mechanism

This paper is concerned with a class of three-dimensional Keller–Segel–Stokes systems with indirect signal production mechanism. By Lyapunov functionals, it is shown that the global bounded solution of the model with some boundary conditions converges to its constant steady state as time tends to infinity under the assumption that χ, r > 0, α∈(53,2) and μ > 0 is sufficiently large. More precisely, the convergence rate is exponential as time tends to infinity for such kind of models.