AI 中文总结
研究具有密度依赖退化扩散的酸介导肿瘤侵袭模型的行波前沿，通过引入非线性变量变换等方法，证明了特定波速下行波前沿存在，连接肿瘤主导与健康状态，并建立了相关估计。
AI 中文摘要
我们研究了具有密度依赖退化扩散的酸介导肿瘤侵袭模型的行波前沿。该模型是Gatenby - Gawlinski型的部分扩散PDE - ODE系统，其中肿瘤扩散系数\(D(U)\)是满足\(D(1)=0\)的一般递减函数。这种退化导致肿瘤分量的行波方程在健康状态附近失去一致椭圆性，标准的非退化反应扩散系统论证不适用。为克服此困难，我们引入非线性变量变换消除肿瘤方程最高阶项的退化。对于每个固定的可允许肿瘤轮廓，酸轮廓由格林核表示，健康组织轮廓通过显式积分公式获得。然后将变换后的肿瘤轮廓构造为一致抛物辅助问题的平稳极限。通过结合比较原理、局部绍德尔估计、精心选择的上解和下解以及绍德尔不动点定理，我们证明了对于每个波速\(\theta\geq2\sqrt{rD(0)}\)行波前沿的存在性。所得波在\(z = -\infty\)处连接肿瘤主导状态\((0,1,1)\)到\(z = +\infty\)处的健康状态\((1,0,0)\)。我们进一步建立了严格的逐点界、所有波分量的单调性以及在变换变量和原始行波变量中的单边指数渐近估计。
英文摘要
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 the Schauder fixed-point theorem, we prove the existence of traveling wave fronts for every wave speed $θ\ge 2\sqrt{rD(0)}$. 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.
Comments27 pages
