{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:18:37Z","timestamp":1760059117777,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T00:00:00Z","timestamp":1747785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"],"award-info":[{"award-number":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Projects at the Academy Level of China Fire and Rescue Institute","award":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"],"award-info":[{"award-number":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"]}]},{"name":"Teaching Reform Projects at the Academy Level of China Fire and Rescue Institute","award":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"],"award-info":[{"award-number":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"]}]},{"name":"Research Projects at the Academy Level of China Fire and Rescue Institute","award":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"],"award-info":[{"award-number":["2024YFB3411303","XFKYY202510","2025RGZN01Z","XFKYY202513"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form \u2202tu\u2212\u2211i=12nXiAi(Xu)=K(x,t,u,Xu), where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For 2\u2264p\u22644, we establish the local Lipschitz regularity (u\u2208Cloc0,1), with the horizontal gradient satisfying Xu\u2208Lloc\u221e; (ii) For 2\u2264p&lt;3, we establish the local second-order horizontal Sobolev regularity (u\u2208HWloc2,2), with the second-order horizontal derivative satisfying XXu\u2208Lloc2. These results solve an open problem proposed by Capogna et al.<\/jats:p>","DOI":"10.3390\/sym17050799","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T07:44:57Z","timestamp":1747813497000},"page":"799","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group"],"prefix":"10.3390","volume":"17","author":[{"given":"Huiying","family":"Wang","sequence":"first","affiliation":[{"name":"China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9915-5739","authenticated-orcid":false,"given":"Chengwei","family":"Yu","sequence":"additional","affiliation":[{"name":"China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"China Fire and Rescue Institue, 4 Nanyan Road, Changping District, Beijing 102202, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yue","family":"Zeng","sequence":"additional","affiliation":[{"name":"Beihang University, Haidian District, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"ref_1","first-page":"184","article-title":"Degenerate quasilinear elliptic systems","volume":"7","year":"1968","journal-title":"Zap. 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