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<p>It is well-known that the propagation phenomena of nonlocal dispersal equations have been extensively studied, and the known results on the interface dynamics of this equation are under the <b>compactly supported initial value</b>. Moreover, there was no explicit formula regarding the interface due to the peculiarity of nonlocal dispersal operators. A natural question is whether it is possible to provide a precise characterization of the interface with respect to small parameter for the <i>general initial values</i> (including exponentially bounded and unbounded). This paper is concerned with the interface dynamics of the non-local dispersal equation with scaling parameter. For the exponentially bounded initial value, by choosing the hyperbolic scaling, we show that at a very small time, the interface is confined within a generated layer whose thickness is at most <i>O</i>(√<i>ɛ</i>|ln <i>ɛ</i>|), and subsequently, the interface propagates at a <b>linear speed</b> determined by the decay rate of initial value. For a class of exponentially unbounded initial value, by introducing the nonlinear scaling based on the decay of initial value, we deduce the corresponding Hamilton-Jacobi equation and describe precisely the propagation of the interface, which provides a <b>superlinear speed</b> of the interface. The investigation of the interface dynamics under different scaling refletcs multiplex propagation modes in spatial dynamics and provides a new perspective on the wave propagation in nonlocal dispersal equations.</p>

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Interface dynamics in nonlocal dispersal Fisher-KPP equations

  • 期刊论文

  • 发表日期：

    2025年6月18日

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  • Wen Tao,
  • Wantong Li,
  • Jianwen Sun,
  • Wenbing Xu

摘要

It is well-known that the propagation phenomena of nonlocal dispersal equations have been extensively studied, and the known results on the interface dynamics of this equation are under the compactly supported initial value. Moreover, there was no explicit formula regarding the interface due to the peculiarity of nonlocal dispersal operators. A natural question is whether it is possible to provide a precise characterization of the interface with respect to small parameter for the general initial values (including exponentially bounded and unbounded). This paper is concerned with the interface dynamics of the non-local dispersal equation with scaling parameter. For the exponentially bounded initial value, by choosing the hyperbolic scaling, we show that at a very small time, the interface is confined within a generated layer whose thickness is at most O(√ɛ|ln ɛ|), and subsequently, the interface propagates at a linear speed determined by the decay rate of initial value. For a class of exponentially unbounded initial value, by introducing the nonlinear scaling based on the decay of initial value, we deduce the corresponding Hamilton-Jacobi equation and describe precisely the propagation of the interface, which provides a superlinear speed of the interface. The investigation of the interface dynamics under different scaling refletcs multiplex propagation modes in spatial dynamics and provides a new perspective on the wave propagation in nonlocal dispersal equations.