Asymptotic limit of the principal eigenvalue of asymmetric nonlocal diffusion operators and propagation dynamics
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| Format: | Preprint |
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2025
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| _version_ | 1866911111340621824 |
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| author | Du, Yihong Fang, Xiangdong Ni, Wenjie |
| author_facet | Du, Yihong Fang, Xiangdong Ni, Wenjie |
| contents | For fixed $c\in\mathbb R$, $l>0$ and a general non-symmetric kernel function $J(x)$ satisfying a standard assumption, we consider the nonlocal diffusion operator \begin{align*} \bf{L}^{J, c}_{(-l,l)}[ϕ](x):=\int_{-l}^lJ(x-y)ϕ(y)\,dy+cϕ'(x), \end{align*} and prove that its principal eigenvalue $λ_p(\bf{L}^{J, c}_{(-l,l)})$ has the following asymptotic limit: \begin{equation*}\label{l-to-infty-c} \lim\limits_{l\to \infty}λ_p(\bf {L}^{J, c}_{(-l,l)})=\inf\limits_{ν\in\mathbb{R}}\big[\int_{\mathbb{R}}J(x)e^{-νx}\,dx+cν\big]. \end{equation*} We then demonstrate how this result can be applied to determine the propagation dynamics of the associated Cauchy problem \begin{equation*}
\label{cau}
\left\{
\begin{array}{ll}
\displaystyle u_t = d \big[\int_{\mathbb{R}} J(x-y) u(t,y) \, dy - u(t,x)\big] + f(u), & t > 0, \; x \in \mathbb{R},
u(0, x) = u_0(x), & x \in \mathbb{R},
\end{array}
\right. \end{equation*} with a KPP nonlinear term $f(u)$. This provides a new approach to understand the propagation dynamics of KPP type models, very different from those based on traveling wave solutions or on the dynamical systems method of Weinberger (1982). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22062 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic limit of the principal eigenvalue of asymmetric nonlocal diffusion operators and propagation dynamics Du, Yihong Fang, Xiangdong Ni, Wenjie Analysis of PDEs 35K57, 35R20 For fixed $c\in\mathbb R$, $l>0$ and a general non-symmetric kernel function $J(x)$ satisfying a standard assumption, we consider the nonlocal diffusion operator \begin{align*} \bf{L}^{J, c}_{(-l,l)}[ϕ](x):=\int_{-l}^lJ(x-y)ϕ(y)\,dy+cϕ'(x), \end{align*} and prove that its principal eigenvalue $λ_p(\bf{L}^{J, c}_{(-l,l)})$ has the following asymptotic limit: \begin{equation*}\label{l-to-infty-c} \lim\limits_{l\to \infty}λ_p(\bf {L}^{J, c}_{(-l,l)})=\inf\limits_{ν\in\mathbb{R}}\big[\int_{\mathbb{R}}J(x)e^{-νx}\,dx+cν\big]. \end{equation*} We then demonstrate how this result can be applied to determine the propagation dynamics of the associated Cauchy problem \begin{equation*} \label{cau} \left\{ \begin{array}{ll} \displaystyle u_t = d \big[\int_{\mathbb{R}} J(x-y) u(t,y) \, dy - u(t,x)\big] + f(u), & t > 0, \; x \in \mathbb{R}, u(0, x) = u_0(x), & x \in \mathbb{R}, \end{array} \right. \end{equation*} with a KPP nonlinear term $f(u)$. This provides a new approach to understand the propagation dynamics of KPP type models, very different from those based on traveling wave solutions or on the dynamical systems method of Weinberger (1982). |
| title | Asymptotic limit of the principal eigenvalue of asymmetric nonlocal diffusion operators and propagation dynamics |
| topic | Analysis of PDEs 35K57, 35R20 |
| url | https://arxiv.org/abs/2503.22062 |