{"title":"Superlinear transmission in an indirect signal production chemotaxis system","authors":"","doi":"10.1016/j.aml.2024.109235","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, the indirect signal production system with nonlinear transmission is considered <span><span><span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a bounded smooth domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>) associated with homogeneous Neumann boundary conditions, where <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> satisfies <span><math><mrow><mn>0</mn><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≤</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span>. It is known from <span><span>[1]</span></span> that the system possesses a global bounded solution if <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span> when <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>. In the case <span><math><mrow><mi>n</mi><mo>≤</mo><mn>3</mn></mrow></math></span> and if we consider superlinear transmission, no regularity of <span><math><mi>w</mi></math></span> or <span><math><mi>v</mi></math></span> can be derived directly. In this work, we show that if <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mo>min</mo><mrow><mo>{</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>}</mo></mrow></mrow></math></span>, the solution is global and bounded via an approach based on the maximal Sobolev regularity.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965924002556","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
In this paper, the indirect signal production system with nonlinear transmission is considered in a bounded smooth domain () associated with homogeneous Neumann boundary conditions, where satisfies with . It is known from [1] that the system possesses a global bounded solution if when . In the case and if we consider superlinear transmission, no regularity of or can be derived directly. In this work, we show that if , the solution is global and bounded via an approach based on the maximal Sobolev regularity.
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.