{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:35:14Z","timestamp":1760243714377,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2012,6,25]],"date-time":"2012-06-25T00:00:00Z","timestamp":1340582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The general quasi-linear autonomous fourth order diffusion equation ut = \u2212[G(u)uxxx  + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.<\/jats:p>","DOI":"10.3390\/e14071127","type":"journal-article","created":{"date-parts":[[2012,6,25]],"date-time":"2012-06-25T16:55:20Z","timestamp":1340643320000},"page":"1127-1139","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Fourth Order Diffusion Equations with Increasing Entropy"],"prefix":"10.3390","volume":"14","author":[{"given":"Naghmana","family":"Tehseen","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Philip","family":"Broadbridge","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1088\/0951-7715\/19\/3\/006","article-title":"An algorithmic construction of entropies in higher-order nonlinear PDEs","volume":"19","author":"Matthes","year":"2006","journal-title":"Nonlinearity"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1063\/1.1637353","article-title":"Theoretical and numerical results for spin coating of viscous liquids","volume":"16","author":"Schwartz","year":"2004","journal-title":"Phys. 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