Asymptotic analysis of extinction behaviour in fast nonlinear diffusion
Asymptotic analysis of extinction behaviour in fast nonlinear diffusion
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快速非线性扩散消光行为的渐近分析
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发表时间：
2010
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通讯作者：
J. King
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文献类型：
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作者：
J. King
摘要分析了快速扩散方程的许多初始边值问题（在不同的细节和完整性水平上），即
$$frac{部分u}{部分t} =
abla cdot (u^{-n})
abla u)$$n > 0，维度 N > 2，零狄利克雷边界数据，即 (i) 柯西问题（无边界），主要总结现有结果，(ii) 简单连通有界域的内部问题（很大程度上回顾了早期结果），(iii) 简单连通有界域外部的问题，以及 (iv) 半空间问题（其中我们包括 N =2）。 Yamabe 流中出现的临界（边界）情况 $${n = n_{s} equal 4/(N+2)}$$ 是特别关注的主题，部分原因是它提供了对亚临界情况 0 < n < ns 和超临界情况 ns < n < 1 的深入了解。结果基于形式渐近分析，并提出了一系列可能成为严格研究主题的猜想。不同类型的相似性解决方案的作用得到了强调。
AbstractA number of initial-boundary-value problems for the equation of fast diffusion are analysed (at varying levels of detail and completeness), i.e.,
$$frac{partial u}{partial t} =
abla cdot (u^{-n}
abla u)$$with n > 0, in dimension N > 2 and with zero-Dirichlet boundary data, namely (i) the Cauchy problem (no boundary), mainly summarising existing results, (ii) the interior problem for a simply connected bounded domain (in large part revisiting earlier results), (iii) the problem exterior to a simply connected bounded domain and (iv) the half-space problem (for which we include N =2). The critical (borderline) case $${n = n_{s} equiv 4/(N+2)}$$ , which arises in Yamabe flow, is the subject of particular focus, in part because it provides considerable insight into both the subcritical case, 0 < n < ns, and the supercritical one, ns < n < 1. The results are based on formal-asymptotic analysis and suggest a range of conjectures that could be the subject of rigorous studies. The role of distinct types of similarity solutions is highlighted.