{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T21:55:18Z","timestamp":1782597318667,"version":"3.54.5"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T00:00:00Z","timestamp":1754956800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T00:00:00Z","timestamp":1754956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12171349"],"award-info":[{"award-number":["12171349"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12271389"],"award-info":[{"award-number":["12271389"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s00332-025-10198-9","type":"journal-article","created":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T16:14:37Z","timestamp":1755015277000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analysis of a Nonlinear Free Boundary Problem Modeling the Radial Growth of Two-Layer Tumors"],"prefix":"10.1007","volume":"35","author":[{"given":"Junde","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuehong","family":"Zhuang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,8,12]]},"reference":[{"key":"10198_CR1","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1137\/060654815","volume":"68","author":"H Bueno","year":"2008","unstructured":"Bueno, H., Ercole, G., Zumpano, A.: Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases. SIAM J. Appl. Math. 68, 1004\u20131025 (2008)","journal-title":"SIAM J. Appl. Math."},{"key":"10198_CR2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/0025-5564(96)00023-5","volume":"135","author":"H Byrne","year":"1996","unstructured":"Byrne, H., Chaplain, M.: Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187\u2013216 (1996)","journal-title":"Math. Biosci."},{"key":"10198_CR3","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1017\/S0956792597003264","volume":"8","author":"H Byrne","year":"1997","unstructured":"Byrne, H., Chaplain, M.: Free boundary problems associated with the growth and development of multicellular spheroids. Euro. J. Appl. Math. 8, 639\u2013658 (1997)","journal-title":"Euro. J. Appl. Math."},{"key":"10198_CR4","doi-asserted-by":"publisher","first-page":"1071","DOI":"10.1007\/s10114-004-0483-3","volume":"21","author":"S Cui","year":"2005","unstructured":"Cui, S.: Analysis of a free boundary problem modeling tumor growth. Acta Math. Sin. English Ser. 21, 1071\u20131082 (2005)","journal-title":"Acta Math. Sin. English Ser."},{"key":"10198_CR5","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1016\/S0252-9602(06)60104-5","volume":"26","author":"S Cui","year":"2006","unstructured":"Cui, S.: Formation of necrotic cores in the growth of tumors: analytic results. Acta Math. Scientia 26, 781\u2013796 (2006)","journal-title":"Acta Math. Scientia"},{"key":"10198_CR6","doi-asserted-by":"publisher","first-page":"1845","DOI":"10.1016\/j.jde.2008.10.014","volume":"246","author":"S Cui","year":"2009","unstructured":"Cui, S.: Lie group action and stability analysis of stationary solutions for a free boundary problem modeling tumor growth. J. Differ. Equ. 246, 1845\u20131882 (2009)","journal-title":"J. Differ. Equ."},{"key":"10198_CR7","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1137\/060657509","volume":"39","author":"S Cui","year":"2007","unstructured":"Cui, S., Escher, J.: Bifurcation analysis of an elliptic free boundary problem modeling stationary growth of avascular tumors. SIAM J. Math. Anal. 39, 210\u2013235 (2007)","journal-title":"SIAM J. Math. Anal."},{"key":"10198_CR8","doi-asserted-by":"publisher","first-page":"636","DOI":"10.1006\/jmaa.2000.7306","volume":"255","author":"S Cui","year":"2001","unstructured":"Cui, S., Friedman, A.: Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. Appl. 255, 636\u2013677 (2001)","journal-title":"J. Math. Anal. Appl."},{"key":"10198_CR9","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1016\/j.ejca.2008.10.026","volume":"45","author":"EA Eisenhauera","year":"2009","unstructured":"Eisenhauera, E.A., Therasseb, P., et al.: New response evaluation criteria in solid tumours: revised RECIST guideline (version 1.1). Eur. J. Cancers 45, 228\u2013247 (2009)","journal-title":"Eur. J. Cancers"},{"key":"10198_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-009-0037-6","volume":"17","author":"J Escher","year":"2010","unstructured":"Escher, J., Matioc, A.-V.: Radially symmetric growth of nonnecrotic tumors. Nonlinear Differ. Equ. Appl. 17, 1\u201320 (2010)","journal-title":"Nonlinear Differ. Equ. Appl."},{"key":"10198_CR11","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1016\/j.jde.2005.09.008","volume":"227","author":"A Friedman","year":"2006","unstructured":"Friedman, A., Hu, B.: Asymptotic stability for a free boundary problem arising in a tumor model. J. Differ. Equ. 227, 598\u2013639 (2006)","journal-title":"J. Differ. Equ."},{"key":"10198_CR12","doi-asserted-by":"publisher","first-page":"5291","DOI":"10.1090\/S0002-9947-08-04468-1","volume":"360","author":"A Friedman","year":"2008","unstructured":"Friedman, A., Hu, B.: Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model. Trans. Amer. Math. Soc. 360, 5291\u20135342 (2008)","journal-title":"Trans. Amer. Math. Soc."},{"key":"10198_CR13","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/s002850050149","volume":"38","author":"A Friedman","year":"1999","unstructured":"Friedman, A., Reitich, F.: Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262\u2013284 (1999)","journal-title":"J. Math. Biol."},{"key":"10198_CR14","doi-asserted-by":"publisher","first-page":"4238","DOI":"10.1137\/21M1437494","volume":"54","author":"W He","year":"2022","unstructured":"He, W., Xing, R., Hu, B.: Linear stability analysis for a free boundary problem modeling multilayer tumor growth with time delay. SIAM J. Math. Anal. 54, 4238\u20134276 (2022)","journal-title":"SIAM J. Math. Anal."},{"key":"10198_CR15","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1016\/j.jde.2024.03.028","volume":"399","author":"Y Huang","year":"2024","unstructured":"Huang, Y., Hu, B.: Periodic solution for a free-boundary tumor model with small diffusion-to-growth ratio. J. Differ. Equ. 399, 252\u2013280 (2024)","journal-title":"J. Differ. Equ."},{"key":"10198_CR16","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1002\/mma.5412","volume":"42","author":"Y Huang","year":"2019","unstructured":"Huang, Y., Zhang, Z., Hu, B.: Linear stability for a free boundary tumor model with a periodic supply of external nutrients. Math. Methods Appl. Sci. 42, 1039\u20131054 (2019)","journal-title":"Math. Methods Appl. Sci."},{"key":"10198_CR17","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1016\/j.nonrwa.2011.08.010","volume":"13","author":"W Hao","year":"2012","unstructured":"Hao, W., Hauenstein, J.D., Hu, B., et al.: Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core. Nonlinear Anal. Real World Appl. 13, 694\u2013709 (2012)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"10198_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2023.104004","volume":"76","author":"Y Liu","year":"2024","unstructured":"Liu, Y., Zhuang, Y.: Analytic results of a double-layered radial tumor model with different consumption rates. Nonlinear Anal. Real World Appl. 76, 104004 (2024)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"10198_CR19","doi-asserted-by":"publisher","first-page":"2655","DOI":"10.1002\/mma.10456","volume":"48","author":"Y Liu","year":"2025","unstructured":"Liu, Y., Zhuang, Y.: Time delays in a double-layered radial tumor model with different living cells. Math. Methods Appl. Sci. 48, 2655\u20132664 (2025)","journal-title":"Math. Methods Appl. Sci."},{"key":"10198_CR20","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s00285-022-01862-9","volume":"86","author":"M Lu","year":"2023","unstructured":"Lu, M., Hao, W., Hu, B., Li, S.: Bifurcation analysis of a free boundary model of vascular tumor growth with a necrotic core and chemotaxis. J. Math. Biol. 86, 19 (2023)","journal-title":"J. Math. Biol."},{"key":"10198_CR21","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1186\/1754-1611-6-4","volume":"6","author":"M Soltani","year":"2012","unstructured":"Soltani, M., Chen, P.: Effect of tumor shape and size on drug delivery to solid tumors. J. Biol. Eng. 6, 4 (2012)","journal-title":"J. Biol. Eng."},{"key":"10198_CR22","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1186\/1754-1611-8-12","volume":"8","author":"M Sefidgar","year":"2014","unstructured":"Sefidgar, M., Soltani, M., et al.: Effect of tumor shape, size, and tissue transport properties on drug delivery to solid tumors. J. Biol. Eng. 8, 12 (2014)","journal-title":"J. Biol. Eng."},{"key":"10198_CR23","doi-asserted-by":"publisher","first-page":"5875","DOI":"10.1016\/j.jde.2015.12.023","volume":"260","author":"J Wu","year":"2016","unstructured":"Wu, J.: Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation. J. Differ. Equ. 260, 5875\u20135893 (2016)","journal-title":"J. Differ. Equ."},{"key":"10198_CR24","doi-asserted-by":"publisher","first-page":"2955","DOI":"10.1088\/1361-6544\/ab15a8","volume":"32","author":"J Wu","year":"2019","unstructured":"Wu, J.: Asymptotic stability of a free boundary problem for the growth of multi-layer tumours in the necrotic phase. Nonlinearity 32, 2955\u20132974 (2019)","journal-title":"Nonlinearity"},{"key":"10198_CR25","doi-asserted-by":"publisher","first-page":"3399","DOI":"10.3934\/dcds.2019140","volume":"39","author":"J Wu","year":"2019","unstructured":"Wu, J.: Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete Contin. Dyn. Syst. 39, 3399\u20133411 (2019)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"10198_CR26","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/s10884-019-09817-3","volume":"33","author":"J Wu","year":"2021","unstructured":"Wu, J.: Analysis of a nonlinear necrotic tumor model with two free boundaries. J. Dynam. Diff. Equ. 33, 511\u2013524 (2021)","journal-title":"J. Dynam. Diff. Equ."},{"key":"10198_CR27","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.nonrwa.2019.04.012","volume":"50","author":"J Wu","year":"2019","unstructured":"Wu, J., Wang, C.: Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors. Nonlinear Anal. Real World Appl. 50, 25\u201333 (2019)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"7","key":"10198_CR28","first-page":"2453","volume":"25","author":"J Wu","year":"2020","unstructured":"Wu, J., Xu, S.: Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply. Discrete Contin. Dyn. Syst. Ser. B 25(7), 2453\u20132460 (2020)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"10198_CR29","doi-asserted-by":"publisher","first-page":"977","DOI":"10.1080\/00036811.2021.1969013","volume":"102","author":"S Xu","year":"2023","unstructured":"Xu, S., Zhang, F., Zhou, Q.: A free boundary problem for necrotic tumor growth with angiogenesis. Appl. Anal. 102, 977\u2013987 (2023)","journal-title":"Appl. Anal."},{"key":"10198_CR30","doi-asserted-by":"publisher","first-page":"1958","DOI":"10.3934\/dcds.2024016","volume":"44","author":"J Zheng","year":"2024","unstructured":"Zheng, J., Li, R., Zhuang, Y.: Analysis of the growth of a radial tumor with triple-layered structure. Discrete Contin. Dyn. Syst. 44, 1958\u20131981 (2024)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"10198_CR31","doi-asserted-by":"publisher","first-page":"1829","DOI":"10.1016\/j.jde.2020.01.022","volume":"269","author":"X Zhao","year":"2020","unstructured":"Zhao, X., Hu, B.: Symmetry-breaking bifurcation for a free-boundary tumor model with time delay. J. Differ. Equ. 269, 1829\u20131862 (2020)","journal-title":"J. Differ. Equ."},{"key":"10198_CR32","doi-asserted-by":"publisher","first-page":"620","DOI":"10.1016\/j.jde.2018.03.005","volume":"265","author":"Y Zhuang","year":"2018","unstructured":"Zhuang, Y., Cui, S.: Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis. J. Differ. Equ. 265, 620\u2013644 (2018)","journal-title":"J. Differ. Equ."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10198-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10198-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10198-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T16:57:55Z","timestamp":1757523475000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10198-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,12]]},"references-count":32,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["10198"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10198-9","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,12]]},"assertion":[{"value":"23 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"98"}}