TY - JOUR
T1 - Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems
AU - Wu, Shi Liang
AU - Wang, Haiyan
N1 - Funding Information:
Acknowledgment The authors thank the anonymous referee for their valuable comments and suggestions that help the improvement of the manuscript. Shi-Liang Wu was supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 12JK0860) and the Fundamental Research Funds for the Central Universities (N0. K50511700002).
PY - 2013/6
Y1 - 2013/6
N2 - This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.
AB - This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.
KW - Cooperative system
KW - Entire solution
KW - Monostable nonlinearity
KW - Non-cooperative system
KW - Traveling wave solution
UR - http://www.scopus.com/inward/record.url?scp=84877799383&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84877799383&partnerID=8YFLogxK
U2 - 10.1007/s10884-013-9293-6
DO - 10.1007/s10884-013-9293-6
M3 - Article
AN - SCOPUS:84877799383
SN - 1040-7294
VL - 25
SP - 505
EP - 533
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
IS - 2
ER -