On the behaviour of blow-up interfaces for an inhomogeneous filtration equation
On the behaviour of blow-up interfaces for an inhomogeneous filtration equation
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DOI：
10.1093/imamat/57.1.53
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发表时间：
1996-08
影响因子：
1.2
通讯作者：
V. Galaktionov;J. King
中科院分区：
文献类型：
--
作者：
V. Galaktionov;J. King
研究了非齐次介质p(x)u 1 = (u m) xx在Q= R x R +条件下，p(x) = > 1为常数，p(x) = |x| -α（对于|x|≥1，α > 2）为有界正光滑对称函数的爆破界面渐近行为。假设初始数据是光滑的、有界的、紧支持的、对称的和单调的。众所周知,由于密度的快速衰减p (x) | x |→∞的支持解决方案增加无限制地在有限的时间内我们证明t→t -界面像O (t - t) - b),在指数b > 0(这取决于m和α)是由一个独特的第二种满足方程的自相似解x | | -αu t = (u) xx。相应的缩放后的轮廓也会收敛。对于指数密度p(x) = e -|x|，当|x|≥1时，建立了第二类自相似解的稳定性。本文给出了非自相似密度p(x) = e -|x|的爆破行为的形式化渐近分析。
We study the asymptotic behaviour of blow-up interfaces of the solutions to the one-dimensional nonlinear filtration equation in inhomogeneous media p(x)u 1 = (u m ) xx in Q= R x R + , where m > 1 is a constant and p(x) = |x| -α (for |x| ≥ 1, with α > 2) is a bounded, positive, smooth, and symmetric function. The initial data are assumed to be smooth, bounded, compactly supported, symmetric, and monotone. It is known that due to the fast decay of the density p(x) as |x| → ∞ the support of the solution increases unboundedly in a finite time T. We prove that as t → T - the interface behaves like O((T - t) -b ), where the exponent b > 0 (which depends on m and α only) is given by a unique self-similar solution of the second kind satisfying the equation |x| -α u t = (u m ) xx . The corresponding rescaled profiles also converge. We establish the stability of the self-similar solution of the second kind for the exponential density p(x) = e -|x| for |x| ≥ 1. We give a formal asymptotic analysis of the blow-up behaviour for the non-self-similar density p(x) = e -|x|2 Several exact self-similar solutions and their corresponding asymptotics are constructed.