本文探讨了一维和二维空间中非线性扩散方程ae(x)ut=ΔA(u)的长期行为及界面特征，A函数为渗流型，ae可能在某些u值处导数消失。当ae不远离零时，与常数ae相比，解决方案的行为有显著差异。文章扩展了Rosenau和Kamin的结果，详细描述了一维下解决方案的长期渐近行为，并在某些情况下解析了二维界面特性。特别地，当ae单调递增时，一维中接口i(t)的极限行为与积分 axe(x)dx趋近于1密切相关。
摘要：
We study the large-time behaviour and the behaviour of the interfaces of the nonlinear diffusion equation ae(x)u t = DeltaA(u) in one and two space dimensions. The function A is of porous media type, smooth but with a vanishing derivative at some values of u, and ae ? 0 is supposed continuous and bounded from above. If ae is not bounded away from zero, the large-time behaviour of solutions and their interfaces can be essentially different from the case when ae is constant. We extend results by Rosenau and Kamin [13] and derive the large-time asymptotic behaviour of solutions, as well as a precise characterisation of the behaviour of the interfaces of solutions in one space dimension and in some cases in two space dimensions. In one space dimension and when ae is monotonic the result states that the interface i(t) = supfx 2 R : u(x; t) ? 0g tends to infinity in finite time if and only if R 1 0 xae(x) dx ! 1. 1 Introduction In this article we study some properties of solutions of ...
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