论文标题
在存在障碍物的情况下，非本地扩散的传播现象
Propagation phenomena with nonlocal diffusion in presence of an obstacle
论文作者
论文摘要
我们考虑$ \ mathbb {r}^n \ setMinus k $的连接外部域上的非局部半线性抛物线方程，其中$ k \ subset \ subset \ mathbb {r}^n $是一个紧凑的“障碍”。我们研究的模型是由生物学应用的动机，并考虑到可能是各向异性的远距离传播事件，具体取决于给定人群如何看待环境。为了以有意义的方式提出这一点，我们引入了一个具有数学和生物学兴趣的新理论框架。本文的主要目的是构建一个像平面行动波一样的整个解决方案，就像$ t \ to-to-to \ infty $，并研究该解决方案如何根据障碍物的形状传播。我们表明，解决方案是否在较大的时间限制中恢复平面前端的形状等同于是否满足某些Liouville类型的属性。我们研究了这种liouville型特性的有效性，并扩展了Hamel，Valdinoci和作者的一些先前结果。最后，我们证明整个解决方案是广义过渡方面。
We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a compact "obstacle". The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as $t\to-\infty$ and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.
