Global Solutions in the Species Competitive Chemotaxis System with Inequal Diffusion Rates

Abstract EN

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates u 1 t = Δ u 1 - ∇ · u 1 χ 1 v ∇ v + μ 1 u 1 1 - u 1 - a 1 u 2 , x ∈ Ω , t > 0 , u 2 t = Δ u 2 - ∇ · u 2 χ 2 v ∇ v + μ 2 u 2 1 - a 2 u 1 - u 2 , x ∈ Ω , t > 0 , v t = τ Δ v - γ v + u 1 + u 2 , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded and regular domain Ω ⊂ R n ( n ≥ 1 ) .

If the nonnegative initial date ( u 10 , u 20 , v 0 ) ∈ ( C 1 ( Ω ¯ ) ) 3 and v 0 ∈ ( v _ , v ¯ ) where the constants v ¯ > v _ ≥ 0 , the system possesses a unique global solution that is uniformly bounded under some suitable assumptions on the chemotaxis sensitivity functions χ 1 ( v ) , χ 2 ( v ) and linear chemical production function - γ v + u 1 + u 2 .