Compact embedding for p(x, t)-Sobolev spaces and existence theory to parabolic equations with p(x, t)-growth

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• 作者：André H. Erhardt
• 关键词：Existence theory ; Nonlinear parabolic problems ; Nonstandard growth ; Nonstandard parabolic Lebesgue and Sobolev spaces ; Compactness theorem
• 刊名：Revista Matemática Complutense
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：30
• 期：1
• 页码：35-61
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Algebra; Analysis; Applications of Mathematics; Geometry; Topology;
• 出版者：Springer Milan
• ISSN：1988-2807
• 卷排序：30

文摘

In this paper, we establish the compact embedding of p(x, t)-Sobolev spaces into p(x, t)-Lebesgue spaces. Moreover, we prove some existence results for nonlinear parabolic problems of the form $$\begin{aligned} \partial _tu-{\mathrm {div}\,}a(x,t,Du)=f-{\mathrm {div}\,}\left( |F|^{p(x,t)-2}F\right) \,\,\,\text {in}\,\Omega _T, \end{aligned}$$where the vector-field \(a(x,t,\cdot )\) satisfies certain p(x, t)-growth conditions.KeywordsExistence theoryNonlinear parabolic problemsNonstandard growthNonstandard parabolic Lebesgue and Sobolev spacesCompactness theoremMathematics Subject Classification35K8635A0146E3554C25References1.Acerbi, E., Mingione, G.: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)MathSciNetCrossRefMATHGoogle Scholar2.Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Rational Mech. Anal. 156(2), 121–140 (2001)MathSciNetCrossRefMATHGoogle Scholar3.Acerbi, E., Mingione, G.: Regularity results for a class of quasiconvex functionals with nonstandard growth. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(2), 311–339 (2001)MathSciNetMATHGoogle Scholar4.Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Rational Mech. Anal. 164(3), 213–259 (2002)MathSciNetCrossRefMATHGoogle Scholar5.Acerbi, E., Mingione, G., Seregin, G.A.: Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(1), 25–60 (2004)MathSciNetMATHGoogle Scholar6.Alkhutov, YuA, Zhikov, V.V.: Existence theorems for solutions of parabolic equations with a variable order of nonlinearity. (Russian) Tr. Mat. Inst. Steklova 270(2010), Differentsialnye Uravneniya i Dinamicheskie Sistemy, p. 21–32; translation in. Proc. Steklov Inst. Math. 270(1), 15–26 (2010)MathSciNetCrossRefGoogle Scholar7.Antontsev, S., Shmarev, S.: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53(2), 355–399 (2009)MathSciNetCrossRefMATHGoogle Scholar8.Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions. Atlantis Studies in Differential Equations. Atlantis Press, Amsterdam (2015)CrossRefMATHGoogle Scholar9.Antontsev, S., Shmarev, S.: Parabolic equations with anisotropic nonstandard growth conditions. Free Bound. Probl. 60(2), 33–44 (2007)MathSciNetCrossRefMATHGoogle Scholar10.Antontsev, S., Shmarev, S.: Vanishing solutions of anisotropic parabolic equations with variable nonlinearity. J. Math. Anal. Appl. 361(2), 371–391 (2010)MathSciNetCrossRefMATHGoogle Scholar11.Antontsev, S., Shmarev, S.: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)MathSciNetCrossRefMATHGoogle Scholar12.Antontsev, S., Zhikov, V.: Higher integrability for parabolic equations of \(p(z)\) Laplacian type. Adv. Differ. Equ. 10(9), 1053–1080 (2005)MathSciNetMATHGoogle Scholar13.Baroni, P.: New contributions to nonlinear Calderón–Zygmund theory. PhD Thesis, Scuola Normale Superiore (2013)14.Baroni, P., Bögelein, V.: Calderón–Zygmund estimates for parabolic \(p(x, t)\)-Laplacian systems. Rev. Mat. Iberoam. 30(4), 1355–1386 (2014)MathSciNetCrossRefMATHGoogle Scholar15.Bögelein, V., Duzaar, F.: Higher integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat. 55, 201–250 (2011)MathSciNetCrossRefMATHGoogle Scholar16.Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\cdot )}\) and \(W^{k, p(\cdot )}\). Math. Nachr. 268, 31–34 (2004)MathSciNetCrossRefMATHGoogle Scholar17.Diening, L., Nägele, P., Ru̇žička, M.: Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var. Elliptic Equ. 57(11), 1209–1231 (2012)MathSciNetCrossRefMATHGoogle Scholar18.Erhardt, A.: Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth. Adv. Nonlinear Anal. 3(1), 15–44 (2014)MathSciNetMATHGoogle Scholar19.Erhardt, A.: Existence and gradient estimates in parabolic obstacle problems with nonstandard growth. PhD Thesis, University Erlangen-Nürnberg (2013)20.Erhardt, A.: Existence of solutions to parabolic problems with nonstandard growth and irregular obstacle. Adv. Differ. Equ. 21(5–6), 463–504 (2016)MathSciNetMATHGoogle Scholar21.Erhardt, A.: Higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth. J. Math. Anal. Appl. 435(2), 1772–1803 (2016)MathSciNetCrossRefMATHGoogle Scholar22.Erhardt, A.: On the Calderón–Zygmund theory for nonlinear parabolic problems with nonstandard growth condition. J. Math. Res. 7, 10–36 (2015)Google Scholar23.Ettwein, F., Ru̇žička, M.: Existence of strong solutions for electrorheological fluids in two dimensions steady: Dirichlet problem. Geometric analysis and nonlinear partial differential equations. Springer, Berlin (2003)Google Scholar24.Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)25.Henriques, E., Urbano, J.M.: Intrinsic scaling for PDEs with an exponential nonlinearity. Indiana Univ. Math. J. 55, 1701–1722 (2006)MathSciNetCrossRefMATHGoogle Scholar26.Roubíček, T.: Nonlinear partial differential equations with applications. International Series of Numerical Mathematics, vol. 153, 2nd edn. Birkhäuser, Basel (2013)CrossRefMATHGoogle Scholar27.Ru̇žička, M.: Electrorheological fluids: modeling and mathematical theory. Springer, Heidelberg (2000)Google Scholar28.Ru̇žička, M.: Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 49(6), 565–609 (2004)MathSciNetCrossRefGoogle Scholar29.Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)Google Scholar30.Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publications, Providence (2001)CrossRefMATHGoogle Scholar31.Zhang, C., Zhou, S., Xue, X.: Global gradient estimates for the parabolic \(p(x, t)\)-Laplacian equation. Nonlinear Anal. 105, 86–101 (2014)MathSciNetCrossRefMATHGoogle Scholar32.Zhikov, V., Pastukhova, S.E.: On the property of higher integrability for parabolic systems of variable order of nonlinearity. Mat. Zametki 87(2):179–200 (2010). [Math. Notes 87(2010), no.1–2. p.169–188)]Copyright information© Universidad Complutense de Madrid 2016Authors and AffiliationsAndré H. Erhardt1Email author1.Institute of Applied Mathematics and StatisticsUniversity of HohenheimStuttgartGermany About this article CrossMark Publisher Name Springer Milan Print ISSN 1139-1138 Online ISSN 1988-2807 About this journal Reprints and Permissions Co-published with