Transactions of the American Mathematical Society

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

Author(s): Zhi-Cheng Wang; Wan-Tong Li; Shigui Ruan
Journal: Trans. Amer. Math. Soc. 361 (2009), 2047-2084.
MSC (2000): Primary 35K57, 35R10; Secondary 35B40, 34K30, 58D25
Posted: October 23, 2008
Retrieve article in: PDF

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

-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.

1.: S. Ai, Travelling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations 232 (2007), 104-133. MR 2281191 (2007h:34052)
2.: P. B. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys. 53 (2002), 103-122. MR 1889183 (2002m:35097)
3.: J. F. M. Al-Omari and S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, Euro. J. Appl. Math. 16 (2005), 37-51. MR 2148679 (2006c:35146)
4.: J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity 17 (2004), 313-346. MR 2023445 (2004m:35142)
5.: N.F. Britton, Spatial structures and periodic travelling waves in an integro-deferential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990), 1663-1688. MR 1080515 (91m:92020)
6.: J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004), 2433-2439. MR 2052422 (2005b:35125)
7.: X. Chen, Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), 125-160. MR 1424765 (98f:35069)
8.: X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212 (2005), 62-84. MR 2130547 (2006b:35179)
9.: X. Chen, J. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 1207-1237. MR 2290131 (2007k:35256)
10.: D. Daners and P. K. McLeod, Abstract evolution equations, periodic problems and applications, Pitman Res. Notes Math. Ser. 279, Longman Scientific - Technical, Harlow, 1992.
11.: O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Analysis TMA 2 (1978), 721-737. MR 512163 (80c:45015)
12.: P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling wave solutions, Arch. Rational Mech. Anal. 65 (1977), 335-361. MR 0442480 (56:862)
13.: P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1980/81), 281-314. MR 607901 (83b:35085)
14.: T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. 462A (2006), 229-261. MR 2189262 (2006i:35182)
15.: T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations 228 (2006), 357-376. MR 2254435 (2007f:35159)
16.: Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math. 8 (2004), 15-32. MR 2057634 (2005a:35159)
17.: S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. Roy. Soc. Lond. 459A (2003), 1563-1579. MR 1994271 (2004f:92024)
18.: S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model, SIAM J. Math. Anal. 35 (2003), 806-822. MR 2048407 (2005i:35142)
19.: S. A. Gourley, J. H. W. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci. 124 (2004), 5119-5153. MR 2129130 (2006b:35182)
20.: J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Cont. Dyn. Systems 12 (2005), 193-212. MR 2122162 (2006f:35145)
21.: F. Hamel and N. Nadirashvili, Entire solutions of the KPP Equation, Comm. Pure Appl. Math. LII (1999), 1255-1276. MR 1699968 (2000e:35088)
22.: F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$ , Arch. Rational Mech. Anal. 157 (2001), 91-163. MR 1830037 (2002d:35090)
23.: W. T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity 19 (2006), 1253-1273. MR 2229998 (2007a:92055)
24.: W. T. Li and Z. C. Wang, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. angew. Math. Phys. 58 (2007), 571-591. MR 2341686
25.: D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci. 13 (2003), 289-310. MR 1982017 (2004d:92024)
26.: S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation, J. Dynam. Differential Equations 19 (2007), 391-436. MR 2333414
27.: R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1-44. MR 967316 (90m:35194)
28.: Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations 18 (2006), 841-861. MR 2263404 (2007h:35192)
29.: C. Ou and J. Wu, Persistence of wavefronts in delayed non-local reaction diffusion equations, J. Differential Equations 235 (2007), 216-239. MR 2309573 (2008c:35151)
30.: A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
31.: P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), 89-110. MR 997611 (90f:58025)
32.: P. Poláčik, Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle, J. Reine Angew. Math. 400 (1989), 32-56. MR 1013724 (90i:58099)
33.: M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, 1983. MR 0219861 (36:2935)
34.: S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in ``Mathematics for Life Science and Medicine'', Y. Iwasa, K. Sato and Y. Takeuchi, eds., Springer-Verlag, New York, 2007, pp. 99-122. MR 2309365
35.: S. Ruan and D. Xiao, Stability of steady states and existence of traveling waves in a vector disease model, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 991-1011. MR 2099575 (2005i:35152)
36.: K. W. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc. 302 (1987), 587-615. MR 891637 (88g:35190)
37.: H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995. MR 1319817 (96c:34002)
38.: H. L. Smith and X. Q. Zhao, Global asymptotic stability of travelling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal. 31 (2000), 514-534. MR 1740724 (2001c:35239)
39.: A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling wave solutions of parabolic systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994. MR 1297766 (96c:35092)
40.: Z. C. Wang and W. T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay, Nonlinear Analysis RWA 8(2007), 699-712. MR 2289581 (2007i:35129)
41.: Z. C. Wang, W. T. Li and S. Ruan, Travelling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185-232. MR 2200751 (2006k:35156)
42.: Z. C. Wang, W. T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations 238 (2007), 153-200. MR 2334595
43.: Z. C. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynamics and Differential Equations 20 (2008), 563-607.
44.: D. V. Widder, The Laplace transform, Princeton University Press, Princeton, NJ, 1941. MR 0005923 (3:232d)
45.: J. Wu, Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996. MR 1415838 (98a:35135)
46.: J. Wu and X. Zou, Travelling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), 651-687. (Erratum, J. Dynam. Differential Equations, DOI: 10.1007/s10884-007-9090-1.) MR 1845097 (2003a:35114)
47.: H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci. 39 (2003), 117-164. MR 1935462 (2003h:35142)
48.: X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comp. Appl. Math. 146 (2002), 309-321. MR 1925963 (2004c:35233)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K57, 35R10, 35B40, 34K30, 58D25

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: 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
Posted: 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.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS