Leibenson's equation on graphs

[2608.15168] Leibenson's equation on graphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2608.15168 (math) [Submitted on 15 Aug 2026] Title: Leibenson's equation on graphs Authors: Philipp Sürig , Xinrong Zhao View a PDF of the paper titled Leibenson's equation on graphs, by Philipp S\"urig and Xinrong Zhao View PDF HTML (experimental) Abstract: In this paper we study on infinite graphs the Leibenson equation $$ \partial_t u = \Delta_p u^q, $$ where $p>1$, $q>0$ and $\Delta_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber--Krahn inequality, we obtain sharp $\ell^1$-$\ell^\infty$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an \textit{isoperimetric inequality}. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the $\ell^\infty$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions. Comments: 32 pages, no figures Subjects: Analysis of PDEs (math.AP) MSC classes: 35R02, 35K55, 39A12 Cite as: arXiv:2608.15168 [math.AP] (or arXiv:2608.15168v1 [math.AP] for this version) https://doi.org/10.48550/arXiv.2608.15168 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xinrong Zhao [ view email ] [v1] Sat, 15 Aug 2026 11:11:20 UTC (34 KB) Full-text links: Access Paper: View a PDF of the paper titled Leibenson's equation on graphs, by Philipp S\"urig and Xinrong Zhao View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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