Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production

This paper investigates a class of chemotaxis systems modeling lethal interactions in a smooth, bounded domain $\Omega \subset \mathbb{R}^n$ with homogeneous Neumann boundary conditions. We examine two distinct cases: (i) a fully parabolic system where both equations exhibit parabolic dynamics, and (ii) a parabolic-elliptic system featuring a parabolic first equation coupled with an elliptic second equation. Under appropriate parameter constraints, we establish the existence of unique globally bounded classical solutions for arbitrary spatial dimensions $n \geq 1$. Additionally, we employ carefully constructed Lyapunov functionals to analyze the long-term behavior of solutions, obtaining rigorous asymptotic stability results.