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Entire solutions of non-quasi-monotone delayed reaction—diffusion equations with applications

Published online by Cambridge University Press:  03 October 2014

Shi-Liang Wu  and
Cheng-Hsiung Hsu
Shi-Liang Wu
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, People's Republic of China, (slwu@xidian.edu.cn)
Cheng-Hsiung Hsu
Affiliation:
Department of Mathematics, National Central University, Chungli 32001, Taiwan, (chhsu@math.ncu.edu.tw)
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Abstract

We are interested in finding the entire solutions of non-quasi-monotone delayed non-local reaction–diffusion equations. It is well known that the comparison principle is not applicable for such equations. To overcome this difficulty, we introduce two auxiliary quasi-monotone equations and establish some comparison arguments for the three systems. Some new types of entire solutions are then constructed using the comparison argument, the travelling wavefronts and a spatially independent solution of the auxiliary equations. We also extend our arguments to a delayed cellular neural network with non-monotonic output functions and a delayed non-local lattice differential equation with non-monotonic birth functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 1085 - 1112
Copyright
Copyright © Royal Society of Edinburgh 2014 

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References

1Ai, S.. Travelling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Diff. Eqns 232 (2007), 104–133.Google Scholar
2Al-Omari, J. F. M. and Gourley, S. A.. A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay. Eur. J. Appl. Math. 16(2005), 37–51.Google Scholar
3Chen, X. and Guo, J.-S.. Existence and uniqueness of entire solutions for a reaction-diffusion equation. J. Diff. Eqns 212 (2005), 62–84.Google Scholar
4Fang, J., Wei, J. J. and Zhao, X.-Q.. Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. Lond. A 466 (2010), 1919–1934.Google Scholar
5Faria, T., Huang, W. and Wu, J.. Travelling waves for delayed reaction-diffusion equations with global response. Proc. R. Soc. Lond. A 462 (2006), 229–261.Google Scholar
6Gourley, S. A. and Wu, J.. Delayed nonlocal diffusive systems in biological invasion and disease spread. In Noniinear dynamics and evoiution equations, Fields Institute Commununications, vol. 48, pp. 137–200 (Providence, RI: American Mathematical Society, 2006).Google Scholar
7Gourley, S. A., So, J. W.-H. and Wu, J.. Nonlocality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124 (2004), 5119–5153.Google Scholar
8Guo, J.-S. and Morita, Y.. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete Contin. Dynam. Syst. 12 (2005), 193–212.Google Scholar
9Guo, J.-S. and Wu, C.-H.. Entire solutions for a two-component competition system in a lattice. Tohoku Math. J. 62 (2010), 17–28.Google Scholar
10Hamel, F. and Nadirashvili, N.. Entire solutions of the KPP equation. Commun. Pure Appl. Math. 52 (1999), 1255–1276.Google Scholar
11Li, W. T., Ruan, S. and Wang, Z. C.. On the diffusive Nicholson's blowflies equation with nonlocal delay. J. Nonlin. Sci. 17 (2007), 505–525.Google Scholar
12Li, W. T., Wang, Z. C. and Wu, J.. Entire solutions in monostable reaction–diffusion equations with delayed nonlinearity. J. Diff. Eqns 245 (2008), 102–129.Google Scholar
13Ma, S.. Traveling waves for non-local delayed diffusion equations via auxiliary equation. J. Diff. Eqns 237 (2007), 259–277.Google Scholar
14Ma, S. and Wu, J.. Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation. J. Dynam. Diff. Eqns 19 (2007), 391–436.Google Scholar
15Martin, R. H. and Smith, H. L.. Abstract functional differential equations and reaction–diffusion system. Trans. Am. Math. Soc. 321 (1990), 1–44.Google Scholar
16Mei, M. and So, J. W.-H. Stability of strong traveling waves for a nonlocal time-delayed reaction–diffusion equation. Proc. R. Soc. Edinb. A 138 (2008), 551–568.Google Scholar
17Morita, Y. and Tachibana, K.. An entire solution to the Lotka-Volterra competition–diffusion equations. SIAM J. Math. Analysis 40 (2009), 2217–2240.Google Scholar
18Ou, C. and Wu, J.. Persistence of wavefronts in delayed non-local reaction–diffusion equations. J. Diff. Eqns 235 (2007), 216–239.Google Scholar
19Ruan, S.. Spatial–temporal dynamics in nonlocal epidemiological models. In Mathematics for life science and medicine (ed. Iwasa, Y., Sato, K. and Takeuchi, Y.), pp. 99–122 (Springer, 2007).Google Scholar
20So, J. W.-H., Wu, J. and Zou, X.. A reaction–diffusion model for a single species with age structure, I, travelling wavefronts on unbounded domains. Proc. R. Soc. Lond. A 457 (2001), 1841–1853.Google Scholar
21Wang, H.. On the existence of traveling waves for delayed reaction–diffusion equations. J. Diff. Eqns 247 (2009), 887–905.Google Scholar
22Wang, Z. C. and Li, W. T.. Dynamics of a nonlocal delayed reaction–diffusion equation without quasi-monotonicity. Proc. R. Soc. Edinb. A 140 (2010), 1081–1109.Google Scholar
23Wang, M. X. and Lv, G. Y.. Entire solutions of a diffusive and competitive Lotka–Volterra type system with nonlocal delay. Nonlinearity 23 (2010), 1609–1630.Google Scholar
24Wang, Z. C., Li, W. T. and Ruan, S.. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays. J. Diff. Eqns 222 (2006), 185–232.Google Scholar
25Wang, Z. C., Li, W. T. and Ruan, S.. Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Diff. Eqns 238 (2007), 153–200.Google Scholar
26Wang, Z. C., Li, W. T. and Ruan, S.. Travelling fronts in monostable equations with nonlocal delayed effects. J. Dynam. Diff. Eqns 20 (2008), 563–607.Google Scholar
27Wang, Z. C., Li, W. T. and Ruan, S.. Entire solutions in bistable reaction–diffusion equations with nonlocal delayed nonlinearity. Trans. Am. Math. Soc. 361 (2009), 2047–2084.Google Scholar
28Wang, Z. C., Li, W. T. and Ruan, S.. Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Analysis 40 (2009), 2392–2420.Google Scholar
29Weng, P. X., Huang, H. and Wu, J.. Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68 (2003), 409–439.Google Scholar
30Wu, S. L.. Entire solutions in a bistable reaction–diffusion system modeling man–environment-man epidemics. Nonlin. Analysis 13 (2012), 1991–2005.Google Scholar
31Wu, S. L. and Hsu, C.-H.. Entire solutions of nonlinear cellular neural networks with distributed time delays. Nonlinearity 25 (2012), 2785–2801.Google Scholar
32Wu, S. L. and Wang, H. Y.. Front-like entire solutions for monostable reaction-diffusion systems. J. Dynam. Diff. Eqns 25 (2013), 505–533.Google Scholar
33Wu, S. L. and Weng, P. X.. Entire solutions for a multi-type SIS nonlocal epidemic model in ℝ or ℤ. J. Math. Analysis Applic. 394 (2012), 603–615.Google Scholar
34Wu, S. L., Li, W. T. and Liu, S. Y.. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete Contin. Dynam. Syst. B 17 (2012), 347–366.Google Scholar
35Wu, S. L., Sun, Y. J. and Liu, S. Y.. Traveling fronts and entire solutions in partially degenerate reaction–diffusion systems with monostable nonlinearity. Discrete Contin. Dynam. Syst. 33 (2013), 921–946.Google Scholar
36Yu, Z. X., Yuan, R., Hsu, C.-H. and Jiang, Q.. Traveling waves for nonlinear cellular neural networks with distributed delays. J. Diff. Eqns 251 (2011), 630–650.Google Scholar
37Yu, Z. X., Yuan, R., Hsu, C.-H. and Peng, M. S.. Traveling waves for delayed cellular neural networks with non-monotonic output functions. (Submitted.)Google Scholar
38Zhang, J. and Peng, Y.. Traveling waves of diffusive Nicholson's blowflies equation with strong generic kernel and nonlocal effect. Nonlin. Analysis TMA 68 (2008), 1263–1270.Google Scholar