Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity
复制标题
DOI：
10.1090/s0002-9947-08-04694-1
复制
发表时间：
2008-10
影响因子：
1.3
通讯作者：
Zhi-Cheng Wang;Wan-Tong Li;S. Ruan
中科院分区：
文献类型：
--
作者：
Zhi-Cheng Wang;Wan-Tong Li;S. Ruan
研究了一维空间域中具有非局部时滞的反应扩散方程的整体解。这里的整体解定义在整个空间和所有时间t ∈ R。假设该方程存在一个波速不为零的递增行波解，利用比较论证，证明了整体解的存在性，其表现为两个来自x轴两端的行波解在有限时刻零化.此外，我们表明，这样的整个解决方案是唯一的时空平移，是李雅普诺夫稳定。一个关键的思想是描述的渐近行为的了解t → -∞的适当的下解和上解。为了说明我们的主要结果，考虑了两个数学生物学中的具有非局部时滞的反应扩散方程模型。
This paper is concerned with entire solutions for bistable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain. Here the entire solutions are defined in the whole space and for all time t ∈ R. Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x-axis and annihilating at a finite time. Furthermore, we show that such an entire solution is unique up to space-time translations and is Liapunov stable. A key idea is to characterize the asymptotic behavior of the solutions as t → -∞ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are considered.