{"title":"一类导数型非线性广义Tricomi方程的爆破结果","authors":"Sandra Lucente, Alessandro Palmieri","doi":"10.1007/s00032-021-00326-x","DOIUrl":null,"url":null,"abstract":"<p>In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation <span>\\(\\mathcal {T}_\\ell u=|\\partial _t u|^p\\)</span>, where <span>\\(\\mathcal {T_\\ell }=\\partial _t^2-t^{2\\ell }\\Delta \\)</span>. Smooth solutions blow up in finite time for positive Cauchy data when the exponent <i>p</i> of the nonlinear term is below <span>\\(\\frac{\\mathcal {Q}}{\\mathcal {Q} -2}\\)</span>, where <span>\\(\\mathcal {Q}=(\\ell +1)n+1\\)</span> is the quasi-homogeneous dimension of the generalized Tricomi operator <span>\\(\\mathcal {T}_\\ell \\)</span>. Furthermore, we get also an upper bound estimate for the lifespan.</p>","PeriodicalId":49811,"journal":{"name":"Milan Journal of Mathematics","volume":"76 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":"{\"title\":\"A Blow-Up Result for a Generalized Tricomi Equation with Nonlinearity of Derivative Type\",\"authors\":\"Sandra Lucente, Alessandro Palmieri\",\"doi\":\"10.1007/s00032-021-00326-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation <span>\\\\(\\\\mathcal {T}_\\\\ell u=|\\\\partial _t u|^p\\\\)</span>, where <span>\\\\(\\\\mathcal {T_\\\\ell }=\\\\partial _t^2-t^{2\\\\ell }\\\\Delta \\\\)</span>. Smooth solutions blow up in finite time for positive Cauchy data when the exponent <i>p</i> of the nonlinear term is below <span>\\\\(\\\\frac{\\\\mathcal {Q}}{\\\\mathcal {Q} -2}\\\\)</span>, where <span>\\\\(\\\\mathcal {Q}=(\\\\ell +1)n+1\\\\)</span> is the quasi-homogeneous dimension of the generalized Tricomi operator <span>\\\\(\\\\mathcal {T}_\\\\ell \\\\)</span>. Furthermore, we get also an upper bound estimate for the lifespan.</p>\",\"PeriodicalId\":49811,\"journal\":{\"name\":\"Milan Journal of Mathematics\",\"volume\":\"76 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2021-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"15\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Milan Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00032-021-00326-x\",\"RegionNum\":3,\"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":"Milan Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00032-021-00326-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Blow-Up Result for a Generalized Tricomi Equation with Nonlinearity of Derivative Type
In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation \(\mathcal {T}_\ell u=|\partial _t u|^p\), where \(\mathcal {T_\ell }=\partial _t^2-t^{2\ell }\Delta \). Smooth solutions blow up in finite time for positive Cauchy data when the exponent p of the nonlinear term is below \(\frac{\mathcal {Q}}{\mathcal {Q} -2}\), where \(\mathcal {Q}=(\ell +1)n+1\) is the quasi-homogeneous dimension of the generalized Tricomi operator \(\mathcal {T}_\ell \). Furthermore, we get also an upper bound estimate for the lifespan.
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