Stationary profiles of degenerate problems when a parameter is large
Stationary profiles of degenerate problems when a parameter is large
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参数较大时退化问题的平稳分布
DOI：
10.57262/die/1356061124
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发表时间：
2000
影响因子：
1.4
通讯作者：
J. S. D. Lis
中科院分区：
文献类型：
--
作者：
J. García;J. S. D. Lis
本文讨论了一类非线性扩散问题正解的结构，|ru| pi 2 ru）= π f（u），在π中，u = 0 on @ π，p > 1，π/2 R N是一个有界的光滑区域，被精确地研究为π！+1，对于一类Logistic型非线性函数f（u）.通过逻辑，可以理解f（u）/u pi 1在u > 0时减小，f（u）>> mu pi 1，m > 0，因为u！0+，而f具有k阶的正零点u = u 0。结果表明，正解u向u 0均匀化为u！+1，并在@100附近形成边界层，其宽度被精确测量。另一方面，在参数区域k <pi 1中示出了对于large的“死核”{u = u 0}的出现，距离dist（{u = u 0}，@ pi）到@ pi的距离也被精确地估计为pi！+1。因此，（12）、（22）中的较早结果基本上是尖锐的。此外，适当的低阶扰动在无穷远的问题进行了研究。1.导论.本文研究了一类非线性扩散问题正解的内部和边界性质：1/2 i <$pu = 1/2 f（u）x 2 1/2
The structure of positive solutions to nonlinear diffusion problems of the form idiv (|ru| pi2 ru) = ¸f(u), in , u = 0 on @, p > 1, ½ R N a bounded, smooth domain, is precisely studied as ¸ ! +1, for a class of logistic- type nonlinearities f(u). By logistic it is understood that f(u)/u pi1 is decreasing in u > 0, f(u) » mu pi1 , m > 0, as u ! 0+, while f has a positive zero u = u0 of order k. It is shown that the positive solution uhomogenizes towards u0 as ¸ ! +1, and develops a boundary layer near @ whose width is exactly measured. On the other hand, the arising of "dead cores" {u¸ = u0} forlarge is shown in the parameters regime k < pi1, the distance dist({u¸ = u0},@) to @ being also exactly estimated as ¸ ! +1. Thus, earlier results in (12), (22) are substantially sharpened. In addition, suitable lower-order perturbations at infinity of the problem are studied. 1. Introduction. The present work is devoted to performing a detailed analysis of the inner and boundary behaviour of positive solutions to the following class of nonlinear diffusion problems: ½ i¢pu = ¸f(u) x 2