{"title":"非线性抛物线系统解的寿命期","authors":"Slim Tayachi","doi":"10.1007/s00030-024-00952-5","DOIUrl":null,"url":null,"abstract":"<p>In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system <span>\\(\\partial _t u=\\Delta u+|v|^{p-1} v,\\; \\partial _t v=\\Delta v+|u|^{q-1} u,\\)</span> <span>\\(q\\ge p\\ge 1,\\; q>1\\)</span> with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"98 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Life-span of solutions for a nonlinear parabolic system\",\"authors\":\"Slim Tayachi\",\"doi\":\"10.1007/s00030-024-00952-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system <span>\\\\(\\\\partial _t u=\\\\Delta u+|v|^{p-1} v,\\\\; \\\\partial _t v=\\\\Delta v+|u|^{q-1} u,\\\\)</span> <span>\\\\(q\\\\ge p\\\\ge 1,\\\\; q>1\\\\)</span> with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.</p>\",\"PeriodicalId\":501665,\"journal\":{\"name\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"volume\":\"98 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00030-024-00952-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-024-00952-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Life-span of solutions for a nonlinear parabolic system
In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system \(\partial _t u=\Delta u+|v|^{p-1} v,\; \partial _t v=\Delta v+|u|^{q-1} u,\)\(q\ge p\ge 1,\; q>1\) with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.