{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T14:20:40Z","timestamp":1775485240095,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We study the stability of a unique weak solution to certain parabolic systems with nonstandard growth condition, which are additionally dependent on a cross-diffusion term. More precisely, we show that two unique weak solutions of the considered system with different initial values are controlled by their initial values.<\/jats:p>","DOI":"10.3390\/axioms10010014","type":"journal-article","created":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T03:31:30Z","timestamp":1611286290000},"page":"14","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Stability of Weak Solutions to Parabolic Problems with Nonstandard Growth and Cross\u2013Diffusion"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4389-8554","authenticated-orcid":false,"given":"Andr\u00e9 H.","family":"Erhardt","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion","volume":"2020","author":"Arumugam","year":"2020","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1515\/anona-2016-0055","article-title":"On a class of fully nonlinear parabolic equations","volume":"8","author":"Antontsev","year":"2019","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s13163-013-0126-2","article-title":"Mathematical analysis of fluids in motion: From well-posedness to model reduction","volume":"26","author":"Feireisl","year":"2013","journal-title":"Rev. Mat. Complut."},{"key":"ref_4","first-page":"253","article-title":"On a climate model with a dynamic nonlinear diffusive boundary condition","volume":"1","author":"Tello","year":"2008","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"key":"ref_5","first-page":"x+509","article-title":"Lebesgue and Sobolev spaces with variable exponents","volume":"Volume 2017","author":"Diening","year":"2011","journal-title":"Lecture Notes in Mathematics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/s10492-004-6432-8","article-title":"Modeling, mathematical and numerical analysis of electrorheological fluids","volume":"49","year":"2004","journal-title":"Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/S1631-073X(02)02337-3","article-title":"Regularity results for electrorheological fluids: The stationary case","volume":"334","author":"Acerbi","year":"2002","journal-title":"C. R. Math. Acad. Sci. Paris"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s00205-002-0208-7","article-title":"Regularity results for stationary electro-rheological fluids","volume":"164","author":"Acerbi","year":"2002","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s13163-011-0088-1","article-title":"Global regularity and stability of solutions to obstacle problems with nonstandard growth","volume":"26","author":"Eleuteri","year":"2013","journal-title":"Rev. Mat. Complut."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1016\/j.camwa.2008.01.017","article-title":"New diffusion models in image processing","volume":"56","author":"Aboulaich","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1137\/050624522","article-title":"Variable exponent, linear growth functionals in image restoration","volume":"66","author":"Chen","year":"2006","journal-title":"SIAM J. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.aml.2012.03.032","article-title":"Critical variable exponent functionals in image restoration","volume":"26","author":"Harjulehto","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.na.2004.09.026","article-title":"A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions","volume":"60","author":"Antontsev","year":"2005","journal-title":"Nonlinear Anal."},{"key":"ref_14","first-page":"463","article-title":"Existence of solutions to parabolic problems with nonstandard growth and irregular obstacles","volume":"21","author":"Erhardt","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","first-page":"xviii+409","article-title":"Evolution PDEs with nonstandard growth conditions","volume":"Volume 4","author":"Antontsev","year":"2015","journal-title":"Atlantis Studies in Differential Equations"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.na.2017.11.002","article-title":"On the continuity of solutions of the nonhomogeneous evolution p(x,t)-Laplace equation","volume":"167","author":"Shmarev","year":"2018","journal-title":"Nonlinear Anal."},{"key":"ref_17","unstructured":"Arora, R., and Shmarev, S. (2020). Double phase parabolic problem with variable growth. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/j.matpur.2018.04.003","article-title":"Regularity for parabolic equations with time dependent growth","volume":"120","author":"Ok","year":"2018","journal-title":"J. Math. Pures Appl."},{"key":"ref_19","first-page":"32","article-title":"Gradient bounds for solutions to irregular parabolic equations with (p,q)-growth","volume":"59","year":"2020","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1017\/prm.2018.63","article-title":"On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p,q)-growth","volume":"150","author":"Giannetti","year":"2020","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"103090","DOI":"10.1016\/j.nonrwa.2020.103090","article-title":"Existence of weak solutions to the Keller\u2013Segel chemotaxis system with additional cross-diffusion","volume":"54","author":"Arumugam","year":"2020","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s41808-020-00078-6","article-title":"Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion","volume":"6","author":"Arumugam","year":"2020","journal-title":"J. Elliptic Parabol. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s13163-016-0211-4","article-title":"Compact embedding for p(x,t)-Sobolev spaces and existence theory to parabolic equations with p(x,t)-growth","volume":"30","author":"Erhardt","year":"2017","journal-title":"Rev. Mat. Complut."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Erhardt, A.H. (2017). The stability of parabolic problems with nonstandard p(x,t)-growth. Mathematics, 5.","DOI":"10.3390\/math5040050"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:50Z","timestamp":1760159630000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,22]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["axioms10010014"],"URL":"https:\/\/doi.org\/10.3390\/axioms10010014","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,22]]}}}