TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 4, April 2009, Pages 2047–2084 S 0002-9947(08)04694-1 Article electronically published on October 23, 2008 ENTIRE SOLUTIONS IN BISTABLE REACTION-DIFFUSION EQUATIONS WITH NONLOCAL DELAYED NONLINEARITY ZHI-CHENG WANG, WAN-TONG LI, AND SHIGUI RUAN 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 ∈ R. Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the exis- tence 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 → -∞ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equa- tions with nonlocal delay arising from mathematical biology are considered. 1. Introduction and main results In this paper, we are concerned with entire solutions of the bistable reaction- diffusion equation with nonlocal delay of the form (1.1) ∂u ∂t = d∆u + g  u (x, t) ,  0 −τ  ∞ −∞ h (y, −s) S (u (x + y,t + s)) dyds  , where x ∈ R, t> 0, d> 0, ∆ is the Laplacian operator on R, τ> 0 is a given constant, and h ∈ L 1 (R × [0,τ ]) is a nonnegative kernel satisfying (H1)  τ 0  ∞ 0 h (y,s) dyds = 1 [normalization]; (H2) h (x, t)= h (−x, t) for (x, t) ∈ R × [0,τ ] [spatial symmetry]; (H3)  τ 0  ∞ 0 e λy h (y,s) dyds < ∞ for λ ≥ 0 [convergence]. Received by the editors May 10, 2007. 2000 Mathematics Subject Classification. Primary 35K57, 35R10; Secondary 35B40, 34K30, 58D25. Key words and phrases. Entire solution, traveling wave solution, reaction-diffusion equation, nonlocal delay, bistable nonlinearity. 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. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 2047