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

DOI:
https://doi.org/10.1090/S0002-9947-08-04694-1

Published electronically:
October 23, 2008

MathSciNet review:
2465829

Full-text PDF

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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 . 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 -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 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:
https://doi.org/10.1090/S0002-9947-08-04694-1

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.