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 $ \chi \le \chi^*(\xi, \mu, A) $ and wave speeds $ c \ge c^*(\chi, \xi, \mu, A) $. Notably, when $ \xi $ is sufficiently small and chemotaxis vanishes ($ \chi \to 0 $), the speed threshold $ c^* $ approaches the classical Fisher-KPP critical speed $ 2\sqrt{\mu} $. 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.