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Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle
Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2021-04-01 , DOI: 10.1007/s10884-021-09988-y
Julien Brasseur , Jérôme Coville

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 \(t\rightarrow -\infty \) 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.



中文翻译:

存在障碍物的非局部扩散传播现象

我们考虑形式为\({\ mathbb {R}} ^ N \ setminus K \)的连接外部域上的非局部半线性抛物方程,其中\(K \ subset {\ mathbb {R}} ^ N \ )是一个紧凑的“障碍”。我们研究的模型是受生物学应用启发的,并考虑到了长期分散事件,这些事件可能是各向异性的,具体取决于给定人群对环境的感知。为了以有意义的方式阐述这一点,我们引入了一个既具有数学意义又具有生物学意义的新理论框架。本文的主要目标是构造一个行为像平面行波为\(t \ rightarrow-\ infty \)的完整解决方案并研究此解决方案如何根据障碍物的形状传播。我们证明了该解决方案是否在较大的时限内恢复了平面前沿的形状,这等同于是否满足一定的Liouville型性质。我们研究了这种Liouville型性质的有效性,并扩展了Hamel,Valdinoci和作者的一些先前结果。最后,我们证明了整个解决方案都是广义的过渡前沿。

更新日期:2021-04-02
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