{"title":"Global [formula omitted]-estimates and dissipative [formula omitted]-estimates of solutions for retarded reaction–diffusion equations","authors":"Ruijing Wang, Chunqiu Li","doi":"10.1016/j.aml.2024.109423","DOIUrl":null,"url":null,"abstract":"This paper is concerned with the retarded reaction–diffusion equation <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mi>∂</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> in a bounded domain. We allow both the nonlinear terms <mml:math altimg=\"si2.svg\" display=\"inline\"><mml:mi>f</mml:mi></mml:math> and <mml:math altimg=\"si3.svg\" display=\"inline\"><mml:mi>G</mml:mi></mml:math> to be supercritical, in which case the solutions may blow up in finite time, making it difficult to obtain global estimates. Here we employ some appropriate structure conditions to deal with this problem. In particular, we establish detailed global <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msup></mml:math>-estimates and dissipative <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>-estimates for the solutions and further enhance the regularity results.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"31 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.aml.2024.109423","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is concerned with the retarded reaction–diffusion equation ∂tu−Δu=f(u)+G(t,ut)+h(x) in a bounded domain. We allow both the nonlinear terms f and G to be supercritical, in which case the solutions may blow up in finite time, making it difficult to obtain global estimates. Here we employ some appropriate structure conditions to deal with this problem. In particular, we establish detailed global L∞-estimates and dissipative H2-estimates for the solutions and further enhance the regularity results.
期刊介绍：
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.
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