{"title":"高维排斥性趋化消耗系统的全局有界性和爆破性","authors":"Jaewook Ahn , Kyungkeun Kang , Dongkwang Kim","doi":"10.1016/j.jde.2025.113503","DOIUrl":null,"url":null,"abstract":"<div><div>This paper investigates the repulsive chemotaxis-consumption model<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>)</mo><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>,</mo><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo></math></span></span></span> in an <em>n</em>-dimensional ball, <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, where the diffusion coefficient <em>D</em> is an appropriate extension of the function <span><math><mn>0</mn><mo>≤</mo><mi>ξ</mi><mo>↦</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mi>m</mi><mo>></mo><mn>0</mn></math></span>. Under the boundary conditions<span><span><span><math><mi>ν</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>+</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext> and </mtext><mspace></mspace><mi>v</mi><mo>=</mo><mi>M</mi><mo>></mo><mn>0</mn><mo>,</mo></math></span></span></span> we demonstrate that for <span><math><mi>m</mi><mo>></mo><mn>1</mn></math></span>, or <span><math><mi>m</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo><</mo><mi>M</mi><mo><</mo><mn>2</mn><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>, the system admits globally bounded classical solutions for any choice of sufficiently smooth radial initial data. This result is further extended to the case <span><math><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><mn>1</mn></math></span> when <em>M</em> is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for <span><math><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, the system exhibits blow-up behavior for sufficiently large <em>M</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113503"},"PeriodicalIF":2.3000,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions\",\"authors\":\"Jaewook Ahn , Kyungkeun Kang , Dongkwang Kim\",\"doi\":\"10.1016/j.jde.2025.113503\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper investigates the repulsive chemotaxis-consumption model<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>)</mo><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>,</mo><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo></math></span></span></span> in an <em>n</em>-dimensional ball, <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, where the diffusion coefficient <em>D</em> is an appropriate extension of the function <span><math><mn>0</mn><mo>≤</mo><mi>ξ</mi><mo>↦</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mi>m</mi><mo>></mo><mn>0</mn></math></span>. Under the boundary conditions<span><span><span><math><mi>ν</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>+</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext> and </mtext><mspace></mspace><mi>v</mi><mo>=</mo><mi>M</mi><mo>></mo><mn>0</mn><mo>,</mo></math></span></span></span> we demonstrate that for <span><math><mi>m</mi><mo>></mo><mn>1</mn></math></span>, or <span><math><mi>m</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo><</mo><mi>M</mi><mo><</mo><mn>2</mn><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>, the system admits globally bounded classical solutions for any choice of sufficiently smooth radial initial data. This result is further extended to the case <span><math><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><mn>1</mn></math></span> when <em>M</em> is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for <span><math><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, the system exhibits blow-up behavior for sufficiently large <em>M</em>.</div></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":\"443 \",\"pages\":\"Article 113503\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-06-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039625005303\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625005303","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions
This paper investigates the repulsive chemotaxis-consumption model in an n-dimensional ball, , where the diffusion coefficient D is an appropriate extension of the function for . Under the boundary conditions we demonstrate that for , or and , the system admits globally bounded classical solutions for any choice of sufficiently smooth radial initial data. This result is further extended to the case when M is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for , the system exhibits blow-up behavior for sufficiently large M.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics