Large densities in a competitive two-species chemotaxis system in the non-symmetric case

This paper deals with the two-species chemotaxis system with Lotka–Volterra competitive kinetics ( Formula Presented), under homogeneous Neumann boundary conditions and suitable initial conditions, where Ω ⊂ Rⁿ (n ∈ N) is a bounded domain with smooth boundary, d1, d2, d3, χ1, χ2, µ1, µ2 > 0, a1, a2 ≥ 0 and α, β, γ > 0. Under largeness conditions on χ1 and χ2, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends previous results of [38] to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic–hyperbolic–elliptic limit system with d1 = d2 = 0.