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Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel

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Abstract

This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of f′(0) = 0, we can no longer use the common method which mainly depends on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, so we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671180 and 11371179) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2016-ct12). The authors thank the referees for their valuable comments.

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Authors and Affiliations

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

    Li Zhang, WanTong Li & ZhiCheng Wang

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  1. Li Zhang
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  2. WanTong Li
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  3. ZhiCheng Wang
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Correspondence to WanTong Li.

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Zhang, L., Li, W. & Wang, Z. Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel. Sci. China Math. 60, 1791–1804 (2017). https://doi.org/10.1007/s11425-016-9003-7

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  • DOI: https://doi.org/10.1007/s11425-016-9003-7

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