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

[2607.13944] Propagation dynamics of an acid-mediated invasion model with degenerate tumor diffusion Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2607.13944 (math) [Submitted on 15 Jul 2026 ( v1 ), last revised 2 Nov 2026 (this version, v3)] Title: Propagation dynamics of an acid-mediated invasion model with degenerate tumor diffusion Authors: Xinyue Cao , Quentin Griette , Xiong Li , Yang Wang View PDF HTML (experimental) Abstract: 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 $\theta\ge 2\sqrt{rD(0)}$. 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. Comments: 28 pages Subjects: Analysis of PDEs (math.AP) MSC classes: 35K57, 35C07, 35B40, 92C17 Cite as: arXiv:2607.13944 [math.AP] (or arXiv:2607.13944v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2607.13944 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xinyue Cao [ view email ] [v1] Wed, 15 Jul 2026 15:29:37 UTC (21 KB) [v2] Fri, 28 Aug 2026 02:37:36 UTC (21 KB) [v3] Mon, 2 Nov 2026 15:03:33 UTC (22 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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