{"title":"涉及边界上规定信号浓度的趋化-消耗系统的小质量解的长时间渐近线","authors":"Soo-Oh Yang , Jaewook Ahn","doi":"10.1016/j.nonrwa.2024.104129","DOIUrl":null,"url":null,"abstract":"<div><p>This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity <span><math><mrow><mi>χ</mi><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></math></span> under no-flux/Dirichlet boundary conditions. For general <span><math><mi>χ</mi></math></span> which may allow singularities at <span><math><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span>, the global existence and boundedness of radial large data solutions are established in dimensions <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. In particular, when <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math></span>, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial <span><math><mrow><mo>(</mo><mi>d</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></math></span> and radial <span><math><mrow><mo>(</mo><mi>d</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></math></span> solutions at suitably small mass levels.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary\",\"authors\":\"Soo-Oh Yang , Jaewook Ahn\",\"doi\":\"10.1016/j.nonrwa.2024.104129\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity <span><math><mrow><mi>χ</mi><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></math></span> under no-flux/Dirichlet boundary conditions. For general <span><math><mi>χ</mi></math></span> which may allow singularities at <span><math><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span>, the global existence and boundedness of radial large data solutions are established in dimensions <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. In particular, when <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math></span>, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial <span><math><mrow><mo>(</mo><mi>d</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></math></span> and radial <span><math><mrow><mo>(</mo><mi>d</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></math></span> solutions at suitably small mass levels.</p></div>\",\"PeriodicalId\":49745,\"journal\":{\"name\":\"Nonlinear Analysis-Real World Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Real World Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1468121824000695\",\"RegionNum\":3,\"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":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824000695","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary
This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity under no-flux/Dirichlet boundary conditions. For general which may allow singularities at , the global existence and boundedness of radial large data solutions are established in dimensions . In particular, when , we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with , a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial and radial solutions at suitably small mass levels.
期刊介绍:
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.