A new result for global solvability in a singular chemotaxis-growth system with indirect signal production

In this paper, we study a chemotaxis system with singular sensitivity, indirect signal production and logistic source: ut=Δu−χ∇⋅(uv−1∇v)+ru−μu2; τvt=Δv−α1v+β1w; τwt=Δw−α2w+β2u, x∈Ω, t>0 in a bounded and smooth domain Ω⊂RN (N≥1) with no-flux boundary conditions, where α1,α2,β1,β2,χ,λ,μ are positive constants, τ=0. Compared with previous work K. Fujie, M. Winkler, T. Yokota (2014) [7], the novelty here is that logistic source is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in any space dimensional setting are not only in the space two dimensional setting. For any appropriately regular initial data, we prove that the system has a unique globally defined classical solution in any space dimensional setting even for arbitrarily large χ, and our result significantly improves and extends the result obtained in previously known ones.