Traveling waves in a tumor invasion model with chemotaxis and haptotaxis effects

This paper investigates the existence and asymptotic properties of traveling wave solutions for a cross-diffusion tumor invasion model incorporating chemotaxis, haptotaxis, and logistic tumor growth. We establish the existence of smooth traveling waves connecting biologically relevant equilibria for chemotactic sensitivities χ ≤ χ^*(ξ, μ, A) and wave speeds c ≥ c^*(χ, ξ, μ, A). Notably, when ξ is sufficiently small and chemotaxis vanishes (χ → 0), the speed threshold c^* approaches the classical Fisher-KPP critical speed 2√μ. Furthermore, we derive the optimal exponential decay rates of the wave profiles near both equilibrium states, providing a precise description of the invasion front's shape. A central result of this work is the demonstration that logistic growth is indispensable: we prove that in its absence, no smooth, non-negative traveling wave exists, highlighting the essential role of tumor cell proliferation in supporting invasive fronts. The analysis overcomes the mathematical challenges arising from the strongly coupled, non-quasimonotone structure of the system through a layered decoupling strategy, parabolic regularization, and a fixed-point argument.