{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:40:21Z","timestamp":1760208021956,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,20]],"date-time":"2017-01-20T00:00:00Z","timestamp":1484870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"7th European Community Framework Programme","award":["BVP symmetry 912563"],"award-info":[{"award-number":["BVP symmetry 912563"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A simplified Keller\u2013Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller\u2013Segel type system, together with the correctly-specified boundary and\/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller\u2013Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller\u2013Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.<\/jats:p>","DOI":"10.3390\/sym9010013","type":"journal-article","created":{"date-parts":[[2017,1,20]],"date-time":"2017-01-20T10:10:12Z","timestamp":1484907012000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A (1 + 2)-Dimensional Simplified Keller\u2013Segel Model: Lie Symmetry and Exact Solutions. II"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1733-5240","authenticated-orcid":false,"given":"Roman","family":"Cherniha","sequence":"first","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maksym","family":"Didovych","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,20]]},"reference":[{"key":"ref_1","first-page":"328","article-title":"Uber integration durch bestimmte integrale von einer Klasse lineare partiellen differentialgleichungen","volume":"8","author":"Lie","year":"1881","journal-title":"Arch. 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