{"title":"一类具有信号依赖运动和广义logistic源的趋化系统全局可解性条件的改进","authors":"Changfeng Liu , Jianping Gao","doi":"10.1016/j.aml.2025.109470","DOIUrl":null,"url":null,"abstract":"<div><div>This paper deals with a chemotaxis system with signal-dependent motility <span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>χ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>λ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>l</mi></mrow></msup><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><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>u</mi><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>∂</mi></mrow><mrow><mi>ν</mi></mrow></msub><mi>u</mi><mo>=</mo><msub><mrow><mi>∂</mi></mrow><mrow><mi>ν</mi></mrow></msub><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> under homogeneous Neumann boundary conditions in a bounded 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><mo>(</mo><mi>n</mi><mo>></mo><mn>2</mn><mo>)</mo></mrow></math></span>. If <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span> and <span><math><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math></span> are constants, we prove that this problem possesses a global classical solution that is uniformly bounded under the conditions that <span><math><mrow><mi>l</mi><mo>></mo><mo>min</mo><mfenced><mrow><mn>3</mn><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>.</mo></mrow></math></span> This result partially improved the work of Lv and Wang (Proc Roy Soc Edinburgh Sect A. 2021, 151 (2): 821-841), in which, the global boundedness of solution is established for <span><math><mrow><mi>l</mi><mo>></mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Our results show that there is a consistent decay rate that effectively rules out the occurrence of blow-up phenomena across all spatial dimensions in the system.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109470"},"PeriodicalIF":2.9000,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Improvement of conditions for global solvability in a chemotaxis system with signal-dependent motility and generalized logistic source\",\"authors\":\"Changfeng Liu , Jianping Gao\",\"doi\":\"10.1016/j.aml.2025.109470\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper deals with a chemotaxis system with signal-dependent motility <span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>χ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>λ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>l</mi></mrow></msup><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><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>u</mi><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>∂</mi></mrow><mrow><mi>ν</mi></mrow></msub><mi>u</mi><mo>=</mo><msub><mrow><mi>∂</mi></mrow><mrow><mi>ν</mi></mrow></msub><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> under homogeneous Neumann boundary conditions in a bounded 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><mo>(</mo><mi>n</mi><mo>></mo><mn>2</mn><mo>)</mo></mrow></math></span>. If <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span> and <span><math><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math></span> are constants, we prove that this problem possesses a global classical solution that is uniformly bounded under the conditions that <span><math><mrow><mi>l</mi><mo>></mo><mo>min</mo><mfenced><mrow><mn>3</mn><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>.</mo></mrow></math></span> This result partially improved the work of Lv and Wang (Proc Roy Soc Edinburgh Sect A. 2021, 151 (2): 821-841), in which, the global boundedness of solution is established for <span><math><mrow><mi>l</mi><mo>></mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Our results show that there is a consistent decay rate that effectively rules out the occurrence of blow-up phenomena across all spatial dimensions in the system.</div></div>\",\"PeriodicalId\":55497,\"journal\":{\"name\":\"Applied Mathematics Letters\",\"volume\":\"163 \",\"pages\":\"Article 109470\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2025-01-23\",\"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/S0893965925000175\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965925000175","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Improvement of conditions for global solvability in a chemotaxis system with signal-dependent motility and generalized logistic source
This paper deals with a chemotaxis system with signal-dependent motility under homogeneous Neumann boundary conditions in a bounded domain . If and are constants, we prove that this problem possesses a global classical solution that is uniformly bounded under the conditions that This result partially improved the work of Lv and Wang (Proc Roy Soc Edinburgh Sect A. 2021, 151 (2): 821-841), in which, the global boundedness of solution is established for . Our results show that there is a consistent decay rate that effectively rules out the occurrence of blow-up phenomena across all spatial dimensions in the system.
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.