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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity


Authors: Zhi-Cheng Wang, Wan-Tong Li and Shigui Ruan
Journal: Trans. Amer. Math. Soc. 361 (2009), 2047-2084
MSC (2000): Primary 35K57, 35R10; Secondary 35B40, 34K30, 58D25
Published electronically: October 23, 2008
MathSciNet review: 2465829
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Abstract: This paper is concerned with entire solutions for bistable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain. Here the entire solutions are defined in the whole space and for all time $ t\in \mathbb{R}$. Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the $ x$-axis and annihilating at a finite time. Furthermore, we show that such an entire solution is unique up to space-time translations and is Liapunov stable. A key idea is to characterize the asymptotic behavior of the solutions as $ t\to -\infty$ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are considered.


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Additional Information

Zhi-Cheng Wang
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Email: wangzhch@lzu.edu.cn

Wan-Tong Li
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Email: wtli@lzu.edu.cn

Shigui Ruan
Affiliation: Department of Mathematics, University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250
Email: ruan@math.miami.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04694-1
PII: S 0002-9947(08)04694-1
Keywords: Entire solution, traveling wave solution, reaction-diffusion equation, nonlocal delay, bistable nonlinearity
Received by editor(s): May 10, 2007
Published electronically: October 23, 2008
Additional Notes: The research of the first author was partially supported by NSF of Gansu Province of China (0710RJZA020).
The second author is the corresponding author and was partially supported by NSFC (10571078) and NSF of Gansu Province of China (3ZS061-A25-001).
The research of the third author was partially supported by NSF grants DMS-0412047 and DMS-0715772.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.