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

The second author is partially supported by NSFC (Grant No. 11771062 and 11571062), the Fundamental Research Funds for the Central Universities (Grant No. 10611CDJXZ238826) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007), and the third author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059) and Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006)
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  • We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model

    $ \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. $

    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.

    Mathematics Subject Classification: Primary: 92C17; Secondary: 35K51, 35A01, 35B65.


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