Speaker
Description
Many mathematical biology and population dynamics models involve nonlocal diffusion corresponding to long-range interactions in a system. These models are typically described by evolution problems with convolution-type integral operators and their qualitative and quantitative properties can be obtained by studying of the corresponding spectral problems.
We consider spectral problems
$
\quad \quad \quad \quad \quad \quad \quad -\frac{1}{\varepsilon^{d}}\int_\Omega J\Bigl({\frac{x-y}{\varepsilon}}\Bigr)
\kappa(x,y)
\rho_\varepsilon(y)dy+
a(x)\rho_\varepsilon(x)= \lambda_\varepsilon \rho_\varepsilon(x) \quad \quad \quad \quad \quad (1)
$
in a bounded domain in $\Omega\subset\mathbb{R}^d$, where $J(z)\geq 0$ is a continuous function on $\mathbb{R}^d$ decaying sufficiently fast as $|z|\to \infty$, $\kappa\in C^2(\overline{\Omega}\times\overline{\Omega})$, $\kappa >0$ and (the potential) $a\in C^2(\overline{\Omega})$; $\varepsilon>0$ is a scaling parameter. We study the asymptotic behavior of eigenvalues and eigenfunctions of (1) in the limit of small parameter $\varepsilon$.
We focus on the self-ajoint case when $J(z)=J(-z)$, $\kappa(x,y)=\kappa(y,x)$ and show that the principal eigenvalue of (1) exists for sufficiently small $\varepsilon$ and converges to the minimum $m(x^*)=\min\limits_{\overline{\Omega}} m(x)$, where $m(x)=a(x)-\kappa(x,x)$. More precise asymptotic description is obtained when $m$ satisfies some non-degeneracy conditions at $x^*$. Namely, if the minimum is strict and the point $x^*$ is an inner point of $\Omega$ then we suppose the positiveness of Hessian and via rescaling by $\varepsilon^{1/2}$ we derive a limit differential spectral problem of the form:
$
\quad \quad \quad \quad \quad \quad \quad \quad \quad -{\rm div} A\nabla\rho + \partial^2_{ij}m(x^*)z_iz_j\rho=\mu\rho \quad \text{in} \ \mathbb{R}^d.
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (2)
$
We prove that $\lambda_\varepsilon=m(x^*)+\mu_k\varepsilon +\bar o(\varepsilon)$, where $\mu_k$ are eigenvalues of (2). The case $x^*\in \partial\Omega$ is more sophisticated and we consider the situation when $\Omega$ is a polyhedron and $m(x)$ attains its strict minimum at $x^*$ on a face of $\partial \Omega$. Without loss of generality, we assume that $x^*=0$ and locally $\Omega$ is given by $x_1>0$ in a neighborhood of $0$. Then the non-degeneracy condition reads: $\partial_{x_1} m(0)>0$, \ $\partial^2_{x'_ix'_j} m(0)\, \xi_i'\xi'_j>0$ $\forall \xi'\in \mathbb{R}^{d-1}\setminus \{0\}$. Under these conditions, we establish the following asymptotic formula for the eigenvalues $\lambda_\varepsilon=m(0)+\Lambda_1\varepsilon^{2/3}+(\beta+\mu_k)\varepsilon +\bar o(\varepsilon)$, where $\Lambda_1$ is the principal eigenvalue of the 1D problem $-\theta\phi_0''(t) + \alpha t \phi_0(t)=\Lambda_1 \phi_0(t)$ on $\mathbb{R}_+$, $\phi_0(0)=0$, $\mu_k$ are eigenvalues of a harmonic oscillator in $\mathbb{R}^{d-1}$. In this case, eigenfunctions have the asymptotic form $\rho_\varepsilon(x)=\phi_0(\varepsilon^{-2/3}x_1)\, v(\varepsilon^{-1/2}x')+...$, that reveals emergence of two fine scales $\varepsilon^{2/3}$ and $\varepsilon^{1/2}$.
