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

This paper is devoted to studying the two‐species competitive chemotaxis system with signal‐dependent chemotactic sensitivities and inequal diffusion rates u1t = Δu1 − ∇·(u1χ1(v)∇v) + μ1u1(1 − u1 − a1u2), x ∈ Ω, t > 0, u2t = Δu2 − ∇·(u2χ2(v)∇v) + μ2u2(1 − a2u1 − u2), x ∈ Ω, t > 0, vt = τΔv − γv + u1 + u2, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded and regular domain Ω⊂Rn (n≥1). If the nonnegative initial date (u10,u20,v0)∈(C1(Ω¯)) 3 and v0∈(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 χ1(v), χ2(v) and linear chemical production function −γv + u1 + u2.