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 (Formula Presented) in a bounded domain Ω ⊂ R³ 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 (u0, v0) 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.