Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

S. Fornaro; V. Vespri

摘要

We prove forward, backward and elliptic Harnack type inequalities for non-negative local weak solutions of singular parabolic differential equations of type $$u_t={rm div}{bf A}(x, t, u, Du)$$where A satisfies suitable structure conditions and a monotonicity assumption. The prototype is the parabolic p−Laplacian with 1 < p < 2. By using only the structure of the equation and the comparison principle, we generalize to a larger class of equations the estimates first proved by Bonforte et al. (Adv. Math. 224, 2151–2215, 2010) for the model equation. Mathematics Subject Classification (2000) 35K65 35B65 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (10) References1.Bonforte M., Vázquez J.L.: Positivity, local smoothing and Harnack inequalities for very fast diffusion equations. Adv. Math. 223, 529–578 (2010)MathSciNetMATHCrossRef2.Bonforte M., Iagar R.G., Vázquez J.L.: Local smoothing effects, positivity and Harnack inequalities for the fast p-Laplacian equation. Adv. Math. 224, 2151–2215 (2010)MathSciNetMATHCrossRef3.Chen, Y.Z., DiBenedetto, E.: On the Harnack inequality for nonnegative solutions of singular parabolic equations, Degenerate diffusions (Minneapolis, MN, 1991), 6169, IMA Vol. Math. Appl. 47 Springer, New York, 19934.DiBenedetto E.: Degenerate parabolic equations. Springer, New York (1993)MATHCrossRef5.DiBenedetto E., Gianazza U., Vespri V.: Local clustering of the non-zero set of functions in W 1,1(E). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 17, 223–225 (2006)MathSciNetCrossRef6.DiBenedetto E., Gianazza U., Vespri V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200, 181–209 (2008)MathSciNetMATHCrossRef7.DiBenedetto E., Gianazza U., Vespri V.: Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations. Manuscripta Math. 131, 231–245 (2010)MathSciNetMATHCrossRef8.DiBenedetto, E., Gianazza, U., Vespri, V.: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 Vol. IX, 385–422 (2010)9.DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, (2012)10.Kinnunen J., Lewis J.L.: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102(2), 253–271 (2000)MathSciNetMATHCrossRef About this Article Title Harnack estimates for non-negative weak solutions of a class of singular parabolic equations Journal Manuscripta Mathematica Volume 141, Issue 1-2 , pp 85-103 Cover Date2013-05 DOI 10.1007/s00229-012-0562-1 Print ISSN 0025-2611 Online ISSN 1432-1785 Publisher Springer-Verlag Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Algebraic Geometry Topological Groups, Lie Groups Geometry Number Theory Calculus of Variations and Optimal Control; Optimization Keywords 35K65 35B65 Industry Sectors Finance, Business & Banking IT & Software Telecommunications Authors S. Fornaro (1) V. Vespri (2) Author Affiliations 1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy 2. Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, viale Morgagni, 67/A, 50134, Firenze, Italy Continue reading... To view the rest of this content please follow the download PDF link above.

机译：我们证明了类型为$ u_t = {rm div} {bf A}（x，t，u，Du）$$的奇异抛物型微分方程的非负局部弱解的向前，向后和椭圆形的Harnack型不等式合适的结构条件和单调性假设。原型是具有1 <2的抛物线p-Laplacian。通过仅使用方程的结构和比较原理，我们将Bonforte等人首先证明的估计推广到更大的方程组。（Adv。Math。224，2151–2215，2010）中的模型方程式。数学主题分类（2000）35K65 35B65页％P关闭纯文本查找内部参考工具导出引用EndNote（.ENW）JabRef（.BIB）Mendeley（.BIB）论文（.RIS）Zotero（.RIS）BibTeX（.BIB）添加到文件中其他动作注册以获取期刊更新关于本期刊的转载和许可共享在Facebook上共享此内容在Twitter上共享此内容在LinkedIn上共享此内容相关内容补充材料（0）参考（10）参考1.Bonforte M.，VázquezJL：积极性，局部平滑和Harnack不等式，用于非常快的扩散方程。进阶数学。 223，529–578（2010）MathSciNetMATHCrossRef2.Bonforte M.，Iagar R.G.，VázquezJ.L .：快速p-Laplacian方程的局部平滑效应，正性和Harnack不等式。进阶数学。 224，2151–2215（2010）MathSciNetMATHCrossRef3.Chen，Y.Z.，DiBenedetto，E .:关于奇异抛物方程非负解的Harnack不等式，简并扩散（Minneapolis，MN，1991），6169，IMA Vol。数学。应用47 Springer，纽约，19934。DiBenedettoE .：退化的抛物线方程。纽约州斯普林格（1993）MATHCrossRef5.DiBenedetto E.，Gianazza U.，Vespri V .: W 1,1（E）中非零函数集的局部聚类。 Atti Accad。纳兹Lincei Cl。科学Fis。垫。自然。 end。 Lincei Mat。应用17，223–225（2006）MathSciNetCrossRef6.DiBenedetto E.，Gianazza U.，Vespri V .：准线性退化抛物型微分方程的Harnack估计。数学学报。 200，181–209（2008）MathSciNetMATHCrossRef7.DiBenedetto E.，Gianazza U.，Vespri V .：某些次临界奇异抛物型偏微分方程非负解的Harnack类型估计和Hölder连续性。手稿数学。 131，231–245（2010）MathSciNetMATHCrossRef8.DiBenedetto，E.，Gianazza，U.，Vespri，V .：某些奇异抛物型偏微分方程非负解的正向，反向和椭圆形Harnack不等式。 Scuola规范。喂比萨Cl。科学5卷IX，385–422（2010）9。DiBenedetto，E.，Gianazza，U.，Vespri，V .:退化和奇异抛物方程的Harnack不等式，数学史普林格专着，（2012）10。Kinnunen J.，Lewis JL ：p-Laplacian类型的抛物线系统的更高可积性。数学公爵。 J. 102（2），253–271（2000）MathSciNetMATHCrossRef关于本文标题一类奇异抛物方程的非负弱解的Harnack估计期刊《数学手册》第141卷，第1-2期，pp 85-103封面日期2013-05 DOI 10.1007 / s00229-012-0562-1打印ISSN 0025-2611在线ISSN 1432-1785出版商Springer-Verlag附加链接注册期刊更新编辑委员会关于本期刊论文投稿主题一般数学代数几何拓扑群，李群几何数论变分和最优控制演算；优化关键字35K65 35B65工业部门金融，商业和银行业IT与软件电信作者S. Fornaro（1）V. Vespri（2）作者关联1. Dipartimento di Matematica“ F.卡索拉蒂”，意大利大学帕维亚分校，通过Ferrata，1，27100，意大利帕维亚，2。Dipartimento di Matematica“ U. Dini”，University degli Studi di Firenze，viale Morgagni，67 / A，50134，Firenze，意大利继续阅读...要查看本内容的其余部分，请点击上面的下载PDF链接。

Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

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