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| In this paper, the following model is considered
ut = ∆u − ∇ · (χu∇v) + ∇ · (ξu∇w), x ∈ Ω, t > 0,
τvt = ∆v + αu − βv, x ∈ Ω, t > 0,τwt = ∆w + γu − δw, x ∈ Ω, t > 0,under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn(n ≥ 2). It
is proved that, when τ = 1, in the space dimension n ≥ 3, for each q > n2 and p > n one can find ε0 > 0 such that if the initial data (u0, v0, w0) satisfies ∥u0∥Lq(Ω) < ϵ, ∥∇v0∥Lp(Ω) < ϵ and∥∇w0∥Lp(Ω) < ϵ for any ϵ < ϵ0, then the solution of the above system is globally bounded.Moreover, this solution converges to the stationary solution (m, m, m) as t → ∞, where
m := ∫Ω u0. When τ = 0, the global solvability of the system for appropriately small initial data is also established under the assumptions χα − ξγ > 0, δ > β and n = 2. |
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| Keywords:Applied mathematic; Chemotaxis; Attraction-repulsion; Global existence; Boundedness; Steady state |
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