On a chemotaxis-type Solow-Swan model for economic growth with capital-induced labor migration

Abstract

We study the Neumann initial-boundary value problem of the parabolic chemotaxis system(⋆){ut=Δu−χ∇⋅(u∇v)+μu(1−u),vt=Δv−v+κu1−αvα with parameters χ,κ>0, μ≥0 and α∈(0,1), in a bounded domain Ω⊂Rn with n≥1 and smooth boundary. The system (⋆) was originally proposed in [15] as a spatial Solow-Swan model for economic growth with the capital-induced labor migration. Our result asserts that for any suitably regular initial data, whenever either μ>0 or 1−α<2n, the corresponding Neumann initial-boundary value problem possesses a global classical solution, which is uniformly bounded.