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Pré-Publication, Document De Travail Année : 2020

Propagation phenomena with nonlocal diffusion in presence of an obstacle

Julien Brasseur
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Jérôme Coville

Résumé

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. Lastly, we study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors.
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Dates et versions

hal-02555983 , version 1 (27-04-2020)
hal-02555983 , version 2 (14-05-2020)

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Julien Brasseur, Jérôme Coville. Propagation phenomena with nonlocal diffusion in presence of an obstacle. 2020. ⟨hal-02555983v1⟩
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