{"id":"https://openalex.org/W2999185166","doi":"https://doi.org/10.1137/19m125827x","title":"Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion","display_name":"Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2999185166","doi":"https://doi.org/10.1137/19m125827x","mag":"2999185166"},"language":"en","primary_location":{"id":"doi:10.1137/19m125827x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m125827x","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022412709","display_name":"Jos\u00e9 A. Carrillo","orcid":"https://orcid.org/0000-0001-8819-4660"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Jose A. Carrillo","raw_affiliation_strings":["Department of Mathematics, Imperial College London, London SW7 2AZ"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Imperial College London, London SW7 2AZ","institution_ids":["https://openalex.org/I47508984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087971771","display_name":"Xinfu Chen","orcid":null},"institutions":[{"id":"https://openalex.org/I170201317","display_name":"University of Pittsburgh","ror":"https://ror.org/01an3r305","country_code":"US","type":"education","lineage":["https://openalex.org/I170201317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xinfu Chen","raw_affiliation_strings":["Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260","institution_ids":["https://openalex.org/I170201317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100644737","display_name":"Qi Wang","orcid":"https://orcid.org/0000-0003-2612-8224"},"institutions":[{"id":"https://openalex.org/I204831749","display_name":"Southwestern University of Finance and Economics","ror":"https://ror.org/04ewct822","country_code":"CN","type":"education","lineage":["https://openalex.org/I204831749"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qi Wang","raw_affiliation_strings":["Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China","institution_ids":["https://openalex.org/I204831749"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031464759","display_name":"Zhi\u2010An Wang","orcid":"https://orcid.org/0000-0003-2945-5810"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Zhian Wang","raw_affiliation_strings":["Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100388564","display_name":"Lu Zhang","orcid":"https://orcid.org/0000-0001-6829-703X"},"institutions":[{"id":"https://openalex.org/I178169726","display_name":"Southern Methodist University","ror":"https://ror.org/042tdr378","country_code":"US","type":"education","lineage":["https://openalex.org/I178169726"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lu Zhang","raw_affiliation_strings":["Department of Mathematics, Southern Methodist University, 6425 Boaz Lane, Dallas TX 75205, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Southern Methodist University, 6425 Boaz Lane, Dallas TX 75205, USA","institution_ids":["https://openalex.org/I178169726"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5022412709"],"corresponding_institution_ids":["https://openalex.org/I47508984"],"apc_list":null,"apc_paid":null,"fwci":0.5167,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.72345074,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"80","issue":"1","first_page":"232","last_page":"261"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9868999719619751,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/metastability","display_name":"Metastability","score":0.6888712048530579},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.678623616695404},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.6032785177230835},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5866588354110718},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5419275760650635},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5345133543014526},{"id":"https://openalex.org/keywords/phase-transition","display_name":"Phase transition","score":0.5332412719726562},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5300611853599548},{"id":"https://openalex.org/keywords/von-neumann-architecture","display_name":"Von Neumann architecture","score":0.492260605096817},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4889000952243805},{"id":"https://openalex.org/keywords/neumann-boundary-condition","display_name":"Neumann boundary condition","score":0.48683756589889526},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.4356752634048462},{"id":"https://openalex.org/keywords/phase-diagram","display_name":"Phase diagram","score":0.4195692241191864},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.4016990661621094},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3808513879776001},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.30442118644714355},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1617811620235443},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1096135675907135},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10647207498550415},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.08787474036216736},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08431842923164368}],"concepts":[{"id":"https://openalex.org/C89464430","wikidata":"https://www.wikidata.org/wiki/Q849516","display_name":"Metastability","level":2,"score":0.6888712048530579},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.678623616695404},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.6032785177230835},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5866588354110718},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5419275760650635},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5345133543014526},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.5332412719726562},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5300611853599548},{"id":"https://openalex.org/C80469333","wikidata":"https://www.wikidata.org/wiki/Q189088","display_name":"Von Neumann architecture","level":2,"score":0.492260605096817},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4889000952243805},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.48683756589889526},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.4356752634048462},{"id":"https://openalex.org/C85906118","wikidata":"https://www.wikidata.org/wiki/Q186693","display_name":"Phase diagram","level":3,"score":0.4195692241191864},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.4016990661621094},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3808513879776001},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.30442118644714355},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1617811620235443},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1096135675907135},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10647207498550415},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.08787474036216736},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08431842923164368},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/19m125827x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m125827x","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:spiral.imperial.ac.uk:10044/1/74264","is_oa":false,"landing_page_url":"http://hdl.handle.net/10044/1/74264","pdf_url":null,"source":{"id":"https://openalex.org/S4306401396","display_name":"Spiral (Imperial College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I47508984","host_organization_name":"Imperial College London","host_organization_lineage":["https://openalex.org/I47508984"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"261","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8299999833106995}],"awards":[{"id":"https://openalex.org/G1257954173","display_name":null,"funder_award_id":"EP/P031587/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4696308637","display_name":null,"funder_award_id":"11501460","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4787172370","display_name":"Nonlocal Partial Differential Equations: entropies, gradient flows, phase transitions and applications","funder_award_id":"EP/P031587/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6130878571","display_name":null,"funder_award_id":"153031/17P","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"},{"id":"https://openalex.org/G7894735467","display_name":null,"funder_award_id":"JBK1805001","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321592","display_name":"Research Grants Council, University Grants Committee","ror":"https://ror.org/00djwmt25"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"},{"id":"https://openalex.org/F4320335000","display_name":"Changjiang Scholar Program of Chinese Ministry of Education","ror":"https://ror.org/01mv9t934"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W1505542423","https://openalex.org/W1963640358","https://openalex.org/W1968835931","https://openalex.org/W1974037609","https://openalex.org/W1981384745","https://openalex.org/W1981659355","https://openalex.org/W1987347396","https://openalex.org/W1987588325","https://openalex.org/W1994880182","https://openalex.org/W1999083427","https://openalex.org/W2012405657","https://openalex.org/W2020870487","https://openalex.org/W2026979923","https://openalex.org/W2031088533","https://openalex.org/W2032168148","https://openalex.org/W2034145387","https://openalex.org/W2038366232","https://openalex.org/W2039770548","https://openalex.org/W2040380154","https://openalex.org/W2042892258","https://openalex.org/W2045783965","https://openalex.org/W2055569326","https://openalex.org/W2058747001","https://openalex.org/W2060469872","https://openalex.org/W2062302699","https://openalex.org/W2075018793","https://openalex.org/W2077289604","https://openalex.org/W2085327059","https://openalex.org/W2087757079","https://openalex.org/W2087873587","https://openalex.org/W2092696532","https://openalex.org/W2098870063","https://openalex.org/W2099613614","https://openalex.org/W2119414533","https://openalex.org/W2121679360","https://openalex.org/W2155385278","https://openalex.org/W2165485564","https://openalex.org/W2288245807","https://openalex.org/W2612414054","https://openalex.org/W2963031057","https://openalex.org/W2963599278","https://openalex.org/W4244836353"],"related_works":["https://openalex.org/W2004710695","https://openalex.org/W2115377877","https://openalex.org/W2795441417","https://openalex.org/W3102070013","https://openalex.org/W1982789862","https://openalex.org/W2034069575","https://openalex.org/W2190701989","https://openalex.org/W2066067786","https://openalex.org/W1967103343","https://openalex.org/W2508446691"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2],"the":[3,25],"Keller-Segel":[4],"model":[5],"in":[6],"one":[7],"dimension":[8],"with":[9],"Neumann":[10],"boundary":[11],"conditions":[12],"and":[13,38,57],"quadratic":[14],"cellular":[15],"diffusion":[16],"has":[17],"an":[18],"intricate":[19],"phase":[20],"transition":[21],"diagram":[22],"depending":[23],"on":[24],"chemosensitivity":[26],"strength.":[27],"Explicit":[28],"computations":[29],"allow":[30],"us":[31],"to":[32],"find":[33],"a":[34],"myriad":[35],"of":[36],"symmetric":[37],"asymmetric":[39],"stationary":[40],"states":[41],"whose":[42],"stability":[43],"properties":[44],"are":[45,62],"mostly":[46],"studied":[47],"via":[48,65],"free":[49,59],"energy":[50,60],"decreasing":[51],"numerical":[52,67],"schemes.":[53],"The":[54],"metastability":[55],"behavior":[56],"staircased":[58],"decay":[61],"also":[63],"illustrated":[64],"these":[66],"simulations.":[68]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}