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Entire solutions for bistable lattice differential equations with obstacles
About this Title
A. Hoffman, Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, Massachusetts 02492, H. J. Hupkes, Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands and E. S. Van Vleck, Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 250, Number 1188
ISBNs: 978-1-4704-2201-1 (print); 978-1-4704-4200-2 (online)
DOI: https://doi.org/10.1090/memo/1188
Published electronically: August 26, 2017
Keywords: Travelling waves,
multi-dimensional lattice differential equations,
obstacles,
sub and super-solutions
MSC: Primary 34K31
Table of Contents
Chapters
- 1. Introduction
- 2. Main Results
- 3. Preliminaries
- 4. Spreading Speed
- 5. Large Disturbances
- 6. The Entire Solution
- 7. Various Limits
- 8. Proof of Theorem
- 9. Discussion
- Acknowledgments
Abstract
We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by “holes”) are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in Berestycki, Hamel, and Matuno (2009) for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.- D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
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