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Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems

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Abstract

In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    ZhiCheng Wang & WanTong Li

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    ShiGui Ruan

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  1. ZhiCheng Wang
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  2. WanTong Li
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  3. ShiGui Ruan
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Correspondence to ShiGui Ruan.

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Wang, Z., Li, W. & Ruan, S. Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems. Sci. China Math. 59, 1869–1908 (2016). https://doi.org/10.1007/s11425-016-0015-x

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