A Nonlinear Fourth-order Parabolic Equation with Nonhomogeneous Boundary Conditions
A Nonlinear Fourth-order Parabolic Equation with Nonhomogeneous Boundary Conditions
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DOI：
10.1137/s0036141004444615
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发表时间：
2006-02
期刊：
影响因子：
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通讯作者：
M. Gualdani;A. Jüngel;G. Toscani
中科院分区：
文献类型：
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作者：
M. Gualdani;A. Jüngel;G. Toscani
研究了一维非线性四阶抛物型方程的非齐次Dirichlet-Neumann边值问题.例如，这个方程出现在量子半导体模型中。证明了该平稳问题严格正古典解的存在唯一性。进一步证明了该瞬态问题整体非负弱解的存在性。证明是基于变量的指数变换和新的“熵”估计。利用熵-熵产生方法证明了在稳态对数为凹的条件下，当时间趋于无穷大时，暂态解在L^1范数下指数快速收敛到稳态.数值例子表明，这个条件似乎是纯粹的技术。
A nonlinear fourth‐order parabolic equation with nonhomogeneous Dirichlet–Neumann boundary conditions in one space dimension is analyzed. This equation appears, for instance, in quantum semiconductor modeling. The existence and uniqueness of strictly positive classical solutions to the stationary problem are shown. Furthermore, the existence of global nonnegative weak solutions to the transient problem is proved. The proof is based on an exponential transformation of variables and new “entropy” estimates. Moreover, it is proved by the entropy–entropy production method that the transient solution converges exponentially fast to its steady state in the $L^1$ norm as time goes to infinity, under the condition that the logarithm of the steady state is concave. Numerical examples show that this condition seems to be purely technical.