{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:50Z","timestamp":1772253110217,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,17]],"date-time":"2018-03-17T00:00:00Z","timestamp":1521244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Using the nonclassical symmetry of nonlinear reaction\u2013diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen\u2013Cahn\u2013Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a \u201cmushy\u201d zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions.<\/jats:p>","DOI":"10.3390\/sym10030072","type":"journal-article","created":{"date-parts":[[2018,3,20]],"date-time":"2018-03-20T15:59:39Z","timestamp":1521561579000},"page":"72","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction\u2013Diffusion"],"prefix":"10.3390","volume":"10","author":[{"given":"Philip","family":"Broadbridge","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, Australia"}]},{"given":"Dimetre","family":"Triadis","sequence":"additional","affiliation":[{"name":"Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan"}]},{"given":"Dilruk","family":"Gallage","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, Australia"}]},{"given":"Pierluigi","family":"Cesana","sequence":"additional","affiliation":[{"name":"Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1016\/S1359-6454(01)00075-1","article-title":"Nanoscale phase field microelasticity theory of dislocations: Model and 3D simulations","volume":"49","author":"Wang","year":"2001","journal-title":"Acta Mater."},{"key":"ref_2","unstructured":"Green, H.S., and Hurst, C.A. 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