Global weak solution and boundedness in a three-dimensional competing chemotaxis

We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model \begin{document}$ \left\{ \begin{array}{l}u_t = Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w),&x∈ Ω,&t>0,\\v_t = Δ v-β v+α u,&x∈Ω,&t>0,\\0 = Δ w-δ w+γ u,&x∈Ω,&t>0\\\end{array} \right. $ \end{document} in a bounded domain $Ω\subset \mathbb{R}^3$ with positive parameters $χ, ξ, α, β, γ$ and $δ$.It is firstly proved that if the repulsion dominates in the sense that $ξγ>χα$, then for any choice of sufficiently smooth initial data $(u_0, v_0)$ the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when $ξγ>χα$, and extends the results in Lin et al. (2016) [19] and Jin and Xiang (2017) [14] to more general case.Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that $ξγ$ is suitable large as related to $χα$, then the classical solutions to the above system are uniformly-in-time bounded.

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