Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity
Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity
复制标题
空间无限处具有不可积行为的 Keller-Segel 系统的解
DOI：
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发表时间：
2023
影响因子：
0.8
通讯作者：
M. Winkler
中科院分区：
文献类型：
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作者：
M. Winkler
Rndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {R}^n$$end{document} 中的柯西问题被考虑为凯勒-塞格尔系统 ut=Δu-∇·(u∇v),0=Δv+u,(⋆)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{文档}$$egin{对齐} 左{egin{数组}{l}u_t = Delta u -
abla cdot (u
abla v), \ 0 = Delta v + u, end{数组}
对了。 qquad qquad (star ) end{aligned}$$end{document}重点关注存在非负径向对称初始数据时的行为的详细描述 u0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u_0$$end{document} 在空间无限处具有不可积行为。结果表明，如果 u0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u_0$$end{document} 是连续且有界的，则(⋆documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$star $$end{document}) 承认本地时间经典解决方案，而如果u0(x)→+∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{文档}$$u_0(x)
ightarrow +infty $$end{文档} as |x|→∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{文档}$$|x|
ightarrow infty $$end{document}，则找不到这样的解决方案。此外，全局存在或全局不存在的三个充分标准的集合表明，对于有限时间爆炸的发生，显式奇异稳态的空间衰减特性起着关键作用。特别是，这强调了 (⋆documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$star $$end{document}) 中的爆炸不需要通过最初的高浓度来强制执行接近有限点，但可能完全是由于大尾部造成的。
The Cauchy problem in Rndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {R}^n$$end{document} is considered for the Keller–Segel system ut=Δu-∇·(u∇v),0=Δv+u,(⋆)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} left{ egin{array}{l}u_t = Delta u -
abla cdot (u
abla v), \ 0 = Delta v + u, end{array}
ight. qquad qquad (star ) end{aligned}$$end{document}with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data u0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u_0$$end{document} with non-integrable behavior at spatial infinity. It is shown that if u0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u_0$$end{document} is continuous and bounded, then (⋆documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$star $$end{document}) admits a local-in-time classical solution, whereas if u0(x)→+∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u_0(x)
ightarrow +infty $$end{document} as |x|→∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$|x|
ightarrow infty $$end{document}, then no such solution can be found. Furthermore, a collection of three sufficient criteria for either global existence or global nonexistence indicates that with respect to the occurrence of finite-time blow-up, spatial decay properties of an explicit singular steady state plays a critical role. In particular, this underlines that explosions in (⋆documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$star $$end{document}) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.