Asymptotic analysis of extinction behaviour in fast nonlinear diffusion

作者：King J R^*

来源：Journal of Engineering Mathematics, 2010, 66(1-3): 65-86.

DOI：10.1007/s10665-009-9329-4

• Coagulation-fragmentation
• Extinction behaviour
• Fast nonlinear diffusion
• Self-similarity
• Spike dynamics
• Thin-film flows
• Yamabe flows

A number of initial-boundary-value problems for the equation of fast diffusion are analysed ( at varying levels of detail and completeness), i.e., partial derivative u/partial derivative t = del . (u(-n)del 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) = 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 < n(s), and the supercritical one, n(s) < 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.

• 出版日期2010-3