Global boundedness of a chemotaxis model with logistic growth and general indirect signal production

We consider the nonnegative solutions of the following chemotaxis-growth system{ut=Δu−∇⋅(u∇v1)+μ(u−uα),x∈Ω,t>0,v1,t=Δv1−v1+v2,x∈Ω,t>0,v2,t=Δv2−v2+v3,x∈Ω,t>0,⋅⋅⋅vk,t=Δvk−vk+u,x∈Ω,t>0, in a smooth bounded domain Ω⊂Rn (n⩾2) with nonnegative initial data and null Neumann boundary condition, where μ>0,α>0. This model can be regard as a chemotaxis type model with general indirect signal. We reveal the influence of space dimension n and the number of equations k≥2 by showing that the solutions are globally bounded whenever α>min⁡{2,max⁡{1,n+22k}}.