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A free boundary problem with nonlocal diffusion and unbounded initial range

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Abstract

We consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is the interval \((-\infty ,h(t))\) with h(t) representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posedness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. Moreover, the rate of accelerated spreading and accurate estimates on longtime behaviors of solution are derived.

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, People’s Republic of China

    Lei Li

  2. School of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, People’s Republic of China

    Xueping Li

  3. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, People’s Republic of China

    Mingxin Wang

  4. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Mingxin Wang

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  1. Lei Li
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Correspondence to Mingxin Wang.

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This work was supported by NSFC Grants 12171120, 11901541.

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Li, L., Li, X. & Wang, M. A free boundary problem with nonlocal diffusion and unbounded initial range. Z. Angew. Math. Phys. 73, 192 (2022). https://doi.org/10.1007/s00033-022-01835-4

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  • DOI: https://doi.org/10.1007/s00033-022-01835-4

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