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Global stability and convergence rate of traveling waves for a nonlocal model in periodic media

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  • In this paper, we study the stability and convergence rate of traveling wavefronts for a nonlocal population model in a periodic habitat \[ \left\{ \begin{array}{ll} \displaystyle\frac{\partial u(t,x)}{\partial t}=D(x)\frac{\partial ^2u(t,x)}{% \partial x^2}-d(x,u(t,x))+\int_R\Gamma (\tau ,x,y)b(y,u(t-\tau ,y))dy, & \\ u(\theta ,x)=\varphi (\theta ,x),\theta \in [-\tau ,0],& \end{array} \right. \] where $D(x), d(x,\cdot ), b(x,\cdot ), \Gamma (\tau ,x,y)$ are L-periodic functions with respect to space $x$ (and $y$) for some positive real constant $L $. Using the analysis of the principal eigenvalue of a non-local linear operator, we show that all noncritical wavefronts are globally exponentially stable, as long as the initial perturbation is uniformly bounded in a weighted space. This result can be generalized to n-dimensional case and three applications of our main results are also presented.
    Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D25.


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