Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle

Abstract

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form R N \ K, where K ⊂ 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 → −∞ 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. JULIEN BRASSEUR AND JÉRÔME COVILLE 7. On the impact of the geometry 40 7.1. A Liouville type result 40 7.2. The stationary solution encodes the geometry 41 8. Large time behaviour of super-level sets 44 8.1. An upper bound: the super-level sets move at speed at most c 44 8.2. A lower bound: the super-level sets move asymptotically at speed c 46 8.3. Conclusion: the super-level sets move exactly at speed c 51 9. The entire solution u(t, x) is a generalised transition wave 55 Acknowledgments 57 Appendix A. The J-covering property 58 References 61

Pyramidal traveling waves for a Belousov-Zhabotinskii system with obstacles

“…Moreover, Guo and Monobe [20], Han and Chang [26] extended the results of [5] to V -shaped traveling waves in R 2 and pyramidal traveling waves in R 3 , respectively. Furthermore, some results on the bistable equations set with heterogeneous media, for the Liouville type results [7], with time-period heterogeneities [33], for non-local diffusion [8,45] and on lattice differential equations [28].…”