A chemotaxis‐Stokes system with nonhomogeneous boundary condition: Global large‐datum solution and asymptotic behavior

Abstract

In this paper, we study the global solvability and stabilization to a chemotaxis-Stokes model with porous medium diffusion Δ𝑢 𝑚 and mixed nonhomogeneous boundary value conditions in three-dimensional space. When 𝑚 is slightly bigger than 1, we can get a solution with strong regularity, but when 𝑚 is close to 1, the regularity of the solution becomes weak. Specifically, our results are divided into two cases:we obtain a global bounded weak existence with good regularity for any initial datum, and for decay incoming oxygen, we also prove that the bounded solution will converge to a constant steady state. But for case (ii), it is hard to obtain the boundedness of solutions, and a global "very" weak solution is obtained.