Numerical solution of a fast diffusion equation
Numerical solution of a fast diffusion equation
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DOI：
10.1090/s0025-5718-99-01063-7
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发表时间：
1999-04
期刊：
影响因子：
--
通讯作者：
M. L. Roux;P. Maingé
中科院分区：
文献类型：
--
作者：
M. L. Roux;P. Maingé
在本文中，作者考虑了非线性反应扩散方程的第一个边值问题：u t - Δum = αu p 1 in Ω，R d (d > 1) 中的光滑有界域，具有零横向边界条件和正初始条件，m E ]0,1[（快速扩散问题），α > 0 且 p1 > m。获得初始数据的充分条件，使得解在有限时间内消失或变为无穷大。提出了该问题的时间离散化方案。数值格式保留了初始问题的基本属性；即存在灭绝或爆炸时间，对此已获得估计。并证明了该方法的收敛性。
In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: u t - Δu m = αu p 1 in Ω, a smooth bounded domain in R d (d > 1) with the zero lateral boundary condition and with a positive initial condition, m E ]0,1[ (fast diffusion problem), α > 0 and p1 > m. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.