Eventual smoothness in a two-species chemotaxis system with Lotka–Volterra competitive kinetics

This study investigates a two-species chemotaxis system incorporating Lotka–Volterra competitive kinetics within a bounded domain with smooth boundaries. Unlike prior works, we relax the regularity assumptions on the initial data to (u0, v0, w0, z0) ∈ L2(Ω) × W1,2(Ω) × L2(Ω) × W1,2(Ω). We establish the existence of global weak solutions for the system in arbitrary dimensions. Additionally, in three-dimensional settings, we prove that these weak solutions evolve into classical solutions after some finite waiting time, provided that the condition riminμi,μi32+ω<ξ with some ξ = ξ(ω) > 0, whenever ω > 0. This result demonstrates that the eventual smoothness of weak solutions is achieved when either intrinsic growth rates ri is sufficiently small or the self-limitation effects μi is sufficiently large. To the best of our knowledge, our findings not only significantly generalize existing results on global weak solutions for such systems but also provide the first known result on the eventual smoothness of solutions in three dimensions.