Propagation dynamics of an acid-mediated invasion model with degenerate tumor diffusion

We investigate traveling wave fronts for an acid-mediated tumor invasion model with density-dependent degenerate diffusion.

The model is a partially diffusive PDE--ODE system of Gatenby--Gawlinski type, in which the tumor diffusion coefficient $D(U)$ is allowed to be a general decreasing function satisfying $D(1)=0$.

This degeneracy causes the traveling wave equation for the tumor component to lose uniform ellipticity near the healthy state, and hence standard arguments for nondegenerate reaction diffusion systems are not directly applicable.

To overcome this difficulty, we introduce a nonlinear change of variables which removes the degeneracy from the highest-order term of the tumor equation.

For each fixed admissible tumor profile, the acid profile is represented by a Green kernel, while the healthy-tissue profile is obtained from an explicit integral formula.

The transformed tumor profile is then constructed as the stationary limit of a uniformly parabolic auxiliary problem.

By combining comparison principles, local Schauder estimates, carefully chosen super- and sub-solutions, and Schauder's fixed point theorem, we prove the existence of traveling wave fronts for every wave speed $\theta\ge 2\sqrt{rD(0)}$.

The resulting wave connects the tumor-dominant state $(0,1,1)$ at $z=-\infty$ to the healthy state $(1,0,0)$ at $z=+\infty$.

We further establish strict pointwise bounds, monotonicity of all wave components, and one-sided exponential asymptotic estimates in both the transformed variable and the original traveling-wave variable.