A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities
A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities
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DOI：
10.4310/cms.2006.v4.n2.a1
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发表时间：
2004-09
影响因子：
1
通讯作者：
J. Dolbeault;I. Gentil;A. Jungel
中科院分区：
文献类型：
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作者：
J. Dolbeault;I. Gentil;A. Jungel
研究一维空间中具有周期边界条件的非线性四阶抛物型方程。这个方程出现在固定非平衡界面的涨落和量子半导体器件的建模中。证明了全局非负弱解的存在性。给出了非负弱解唯一性的一个判据。最后，利用一个新的高阶导数的最优对数Sobolev不等式证明了该解在“熵范数”中以指数速度收敛于其均值。
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the “entropy norm” using a new optimal logarithmic Sobolev inequality for higher derivatives.