{"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. 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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.
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