论文标题
来自定向聚合物的非局部模型的长期行为
Long-time behavior for a nonlocal model from directed polymers
论文作者
论文摘要
我们考虑了针对定向聚合物研究中产生的非局部反应扩散方程的溶液的长时间行为。该模型的特征是用内核$ r $和$ l^2 $内部产品进行卷积。在一个空间维度中，我们扩展了作者[Arxiv：2002.02799]的先前结果，其中仅考虑了$ r =δ$的情况；特别是，我们表明，根据$ 2/3 $的功率定律，解决方案与直接聚合物的KPZ缩放率一致。在特殊情况下，当$ r =δ$时，我们在重新定化的坐标中找到解决方案的确切曲线。我们还考虑了更高维度的行为。当尺寸为三个或更大时，我们表明长期行为与热方程式相同，因为解决方案会收敛到标准高斯。相反，当尺寸为两个时，我们构建了一个非高斯自相似解。
We consider the long time behavior of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers. The model is characterized by convolution with a kernel $R$ and an $L^2$ inner product. In one spatial dimension, we extend a previous result of the authors [arXiv:2002.02799], where only the case $R =δ$ was considered; in particular, we show that solutions spread according to a $2/3$ power law consistent with the KPZ scaling conjectured for direct polymers. In the special case when $R = δ$, we find the exact profile of the solution in the rescaled coordinates. We also consider the behavior in higher dimensions. When the dimension is three or larger, we show that the long-time behavior is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.
