Boundedness and large time behavior of solutions to a prey-taxis system accounting in liquid surrounding. (February 2021) - Feng, Zefu; Zhang, Mei
Abstract: In this paper, we are concerned with the following prey-taxis system with liquid surrounding describing by the incompressible Navier–Stokes equations n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( χ n ∇ c ) + γ n F ( c ) − θ n − α n 2, x ∈ Ω, t > 0, c t + u ⋅ ∇ c = D Δ c − n F ( c ) + f ( c ), x ∈ Ω, t > 0, u t + u ⋅ ∇ u = Δ u + ∇ P + n ∇ ϕ, ∇ ⋅ u = 0, x ∈ Ω, t > 0, ∂ n ∂ ν = ∂ c ∂ ν = u = 0, x ∈ ∂ Ω, t > 0, n ( x, 0 ) = n 0 ( x ), c ( x, 0 ) = c 0 ( x ), u ( x, 0 ) = u 0 ( x ), x ∈ Ω, where Ω ⊂ 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.