高维排斥性趋化消耗系统的全局有界性和爆破性

IF 2.3 2区 数学 Q1 MATHEMATICS
Jaewook Ahn , Kyungkeun Kang , Dongkwang Kim
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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>&gt;</mo><mn>0</mn><mo>,</mo></math></span></span></span> we demonstrate that for <span><math><mi>m</mi><mo>&gt;</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>&lt;</mo><mi>M</mi><mo>&lt;</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>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span> when <em>M</em> is chosen to be sufficiently small, depending on the initial conditions. 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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>&gt;</mo><mn>0</mn><mo>,</mo></math></span></span></span> we demonstrate that for <span><math><mi>m</mi><mo>&gt;</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>&lt;</mo><mi>M</mi><mo>&lt;</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>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span> when <em>M</em> is chosen to be sufficiently small, depending on the initial conditions. 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引用次数: 0

摘要

本文研究了n≥3的n维球中,∂tu=∇⋅(D(u)∇u)+∇⋅(u∇v),0=Δv - uv,其中扩散系数D是函数0≤ξ≠(1+ξ)m−1对m>;0的适当扩展的排斥性趋化消耗模型。在边界条件ν⋅(D(u)∇u+u∇v)=0和v=M>;0下,我们证明了对于M>; 1或m=1和0<; M> 2/(n−2),系统对于任何选择的充分光滑的径向初始数据都允许全局有界经典解。根据初始条件的不同,将此结果进一步推广到选择M足够小的情况0<;m<1。结果表明,对于0<;m<2n,当M足够大时,系统表现出爆炸行为。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions
This paper investigates the repulsive chemotaxis-consumption modeltu=(D(u)u)+(uv),0=Δvuv, in an n-dimensional ball, n3, where the diffusion coefficient D is an appropriate extension of the function 0ξ(1+ξ)m1 for m>0. Under the boundary conditionsν(D(u)u+uv)=0 and v=M>0, we demonstrate that for m>1, or m=1 and 0<M<2/(n2), the system admits globally bounded classical solutions for any choice of sufficiently smooth radial initial data. This result is further extended to the case 0<m<1 when M is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for 0<m<2n, the system exhibits blow-up behavior for sufficiently large M.
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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
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
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