{"title":"一类双非线性抛物型方程正性的展开式","authors":"E. Henriques","doi":"10.14232/ejqtde.2022.1.15","DOIUrl":null,"url":null,"abstract":"<jats:p>We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>div</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mtext> </mml:mtext> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext>and</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> considering separately the two possible cases <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:math>. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>p</mml:mi> </mml:math>-Laplacian equation.</jats:p>","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Expansion of positivity to a class of doubly nonlinear parabolic equations\",\"authors\":\"E. Henriques\",\"doi\":\"10.14232/ejqtde.2022.1.15\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <mml:msub> <mml:mi mathvariant=\\\"normal\\\">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>div</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width=\\\"2em\\\" /> <mml:mtext> </mml:mtext> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext>and</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> considering separately the two possible cases <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> and <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:math>. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>p</mml:mi> </mml:math>-Laplacian equation.</jats:p>\",\"PeriodicalId\":50537,\"journal\":{\"name\":\"Electronic Journal of Qualitative Theory of Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Qualitative Theory of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.14232/ejqtde.2022.1.15\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Qualitative Theory of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2022.1.15","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
摘要
我们建立了一类双非线性抛物型方程的非负、局部、弱解的正性展开式？t（uq）−div （|D u |p−2 D u）=0，p>1和q>0，分别考虑两种可能的情况q+1−p>0和q+1−p0。证明依赖于DiBenedetto、Gianazza和Vespri对退化和奇异抛物型p-Laplacian方程提出的程序。
Expansion of positivity to a class of doubly nonlinear parabolic equations
We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations ∂t(uq)−div(|Du|p−2Du)=0,p>1andq>0 considering separately the two possible cases q+1−p>0 and q+1−p<0. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic p-Laplacian equation.
期刊介绍：
The Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE) is a completely open access journal dedicated to bringing you high quality papers on the qualitative theory of differential equations. Papers appearing in EJQTDE are available in PDF format that can be previewed, or downloaded to your computer. The EJQTDE is covered by the Mathematical Reviews, Zentralblatt and Scopus. It is also selected for coverage in Thomson Reuters products and custom information services, which means that its content is indexed in Science Citation Index, Current Contents and Journal Citation Reports. Our journal has an impact factor of 1.827, and the International Standard Serial Number HU ISSN 1417-3875.
All topics related to the qualitative theory (stability, periodicity, boundedness, etc.) of differential equations (ODE''s, PDE''s, integral equations, functional differential equations, etc.) and their applications will be considered for publication. Research articles are refereed under the same standards as those used by any journal covered by the Mathematical Reviews or the Zentralblatt (blind peer review). Long papers and proceedings of conferences are accepted as monographs at the discretion of the editors.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
