论文标题
通过非局部扩散方程在一个空间维度的自由边界问题中近似随机扩散方程
Approximation of random diffusion equation by nonlocal diffusion equation in free boundary problems of one space dimension
论文作者
论文摘要
我们展示了如何通过非局部扩散的相应的自由边界问题近似于一个空间维度的Stefan类型自由边界问题。近似问题是[4,8]中考虑的自由边界的非局部扩散问题的一个稍微修改的版本。证明依赖于引入几个辅助自由边界问题以及针对这些问题精致的上和下解决方案的构建。与往常一样，通过在$j_ε（x）= \ frac1εj（\ fracxε）的非本地扩散项中选择内核函数来实现近似值。我们还通过$ε$的某些正功率对近似的错误项进行了估计。
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_ε(x)=\frac 1εJ(\frac xε)$ for small $ε>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $ε$.
