Poincaré inequalities for linearizations of very fast diffusion equations
Poincaré inequalities for linearizations of very fast diffusion equations
复制标题
极快扩散方程线性化的庞加莱不等式
DOI：
10.1088/0951-7715/15/3/303
复制
发表时间：
2002
期刊：
影响因子：
1.7
通讯作者：
G. Toscani
中科院分区：
文献类型：
--
作者：
J. Carrillo;C. Lederman;P. Markowich;G. Toscani
本文研究了具有和不具有位势限制的线性化甚快扩散方程的大时间渐近性。这些方程不满足，在一般情况下，对数Sobolev不等式，但是，正如我们所示，通过使用`Bakry-Emery反向方法'，在有限的情况下，他们有一个积极的频谱间隙在本征值为零。我们提出了估计这一频谱差距，并得出结论的时间衰减的解决方案，我们证明是指数的问题与限制和代数的纯扩散的情况下。这些结果适用于任意代数大的扩散速度，如果解决方案具有质量守恒属性。
In this paper we investigate the large-time asymptotic of linearized very fast diffusion equations with and without potential confinements. These equations do not satisfy, in general, logarithmic Sobolev inequalities, but, as we show by using the `Bakry-Emery reverse approach', in the confined case they have a positive spectral gap at the eigenvalue zero. We present estimates for this spectral gap and draw conclusions on the time decay of the solution, which we show to be exponential for the problem with confinement and algebraic for the pure diffusive case. These results hold for arbitrary algebraically large diffusion speeds, if the solutions have the mass-conservation property.