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

Abstract

AbstractIn this paper, we study the global solvability and stabilization to a chemotaxis‐Stokes model with porous medium diffusion 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: (i) and (ii) . For case (i), 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.