Boundedness in a fully parabolic chemotaxis-type Solow-Swan model

In this paper, we investigate the following fully parabolic chemotaxis-type Solow-Swan model for economic growth with capital-induced labor migration$ {eqnarray*} {1a} \{ {aligned} &uₜ = Δ u-χ ∇ · (uθ∇ v)+μ₁ u(1-uκ),&(x,t)∈ Ω× (0,∞),\\ &vₜ = Δ v-v+μ ₂u1-αvα,&(x,t)∈ Ω× (0,∞),\\ {aligned} . {eqnarray*} $under homogeneous Neumann boundary conditions in a smoothly bounded domain $ Ω ⊂ Rⁿ $ $ (n≥ 1) $, where $ χ>0 $, $ μ ₁≥0 $, $ μ₂>0 $, $ θ,κ>1 $, and $ α∈(0,1) $. When $ μ₁ = 0 $, we prove that this problem possesses a global classical solution that is uniformly bounded in time under the conditions of $ 1-α≤1/n $, $ θ<1+1/n $ and $ n≥2 $. When $ μ₁>0 $, we also show that this problem has a globally uniformly bounded classical solution if $ θ≤κ $.