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Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

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Abstract

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.

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Acknowledgments

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).

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  1. Department of Mathematics, Xidian University, Xi’an, Shaanxi, 710071, People’s Republic of China

    Shi-Liang Wu

  2. School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ, 85069, USA

    Haiyan Wang

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  1. Shi-Liang Wu
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  2. Haiyan Wang
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Correspondence to Shi-Liang Wu.

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Wu, SL., Wang, H. Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems. J Dyn Diff Equat 25, 505–533 (2013). https://doi.org/10.1007/s10884-013-9293-6

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  • DOI: https://doi.org/10.1007/s10884-013-9293-6

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