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Boundedness and large time behavior of solutions to a prey-taxis system accounting in liquid surrounding. (English) Zbl 1455.35192

Summary: In this paper, we are concerned with the following prey-taxis system with liquid surrounding describing by the incompressible Navier-Stokes equations \[\begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot (\chi n \nabla c) + \gamma n F (c) - \theta n - \alpha n^2, \quad & x \in \Omega,\ t > 0, \\ c_t + u \cdot \nabla c = D \Delta c - n F (c) + f (c), & x \in \Omega,\ t > 0, \\ u_t + u \cdot \nabla u = \Delta u + \nabla P + n \nabla \phi, \nabla \cdot u = 0, & x \in \Omega,\ t > 0, \\ \frac{\partial n}{\partial \nu} = \frac{\partial c}{\partial \nu} = u = 0, & x \in \partial \Omega,\ t > 0, \\ n (x, 0) = n_0 (x), c (x, 0) = c_0 (x), u (x, 0) = u_0 (x), & x \in \Omega, \end{cases}\] where \(\Omega \subset \mathbb{R}^2\) is a bounded domain with smooth boundary. Using the \(L^p\)-energy estimate, we obtain the global existence of solutions with uniform-in-time bound. Moreover, by constructing some Lyapunov functionals, we also establish the large time behavior of solutions.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76Z99 Biological fluid mechanics
92D25 Population dynamics (general)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
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