This paper investigates a coupled chemotaxis-Stokes system with logistic source terms, \( {\left\{ \begin{array}{ll} \begin{aligned} n_t + u \cdot \nabla n & = \Delta n - \nabla \cdot (n \nabla c) + a n - b n^2, \\ c_t + u \cdot \nabla c & = \Delta c - n c, \\ u_t & = \Delta u + \nabla P + n \nabla \phi , \\ \nabla \cdot u & = 0, \end{aligned} \end{array}\right. } \) posed in a three-dimensional bounded domain \( \Omega \) , subject to regular initial data and the boundary conditions \(\begin{aligned} \frac{\partial n}{\partial \nu } - n \frac{\partial c}{\partial \nu } = 0,\quad c = c^*(x, t),\quad u = 0, \qquad x \in \partial \Omega , \,\, t > 0, \end{aligned}\) where \( c^* \) is a prescribed nonnegative function that is not assumed to be constant or spatially homogeneous, in contrast to assumptions in several recent works. Such Dirichlet boundary conditions for the chemical signal \( c \) are more biologically realistic in certain settings involving chemotaxis-fluid interactions. By identifying an explicit threshold \( b_0 \ge 0 \) , depending on the initial data \( c(x, 0) \) and the boundary value \( c^* \) , we establish that for all logistic damping parameters \( b > b_0 \) , the system admits a global classical solution which remains uniformly bounded in time. Moreover, under additional integrability conditions on \( c^* \) , the solution exhibits stabilization in the sense that \(\begin{aligned}\Big \Vert n(\cdot , t) - \frac{a_+}{b}\Big \Vert _{W^{1,\infty }(\Omega )} + \big \Vert c(\cdot , t) - c^*(\cdot , t)\big \Vert _{W^{1,\infty }(\Omega )} + \Vert u(\cdot , t)\Vert _{W^{1,\infty }(\Omega )} \rightarrow 0 \quad \,\,\, \text {as} \,\,\, t \rightarrow \infty , \end{aligned}\) where \( a_+ := \max \big \{a, \, 0\big \} \) . Furthermore, if \( c^* \) decays exponentially in time, the convergence is exponential for \( a \ne 0 \) , and algebraic for \( a = 0 \) . To the best of our knowledge, this stabilization result and the corresponding convergence rates are the first to address inhomogeneous Dirichlet boundary conditions for the chemical signal, even in the fluid-free setting. All these results remain valid for arbitrary \( b > 0 \) in the two-dimensional case, even when the Navier-Stokes fluid coupling is considered.
