# Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

## TL;DR

This paper investigates the long-term behavior of solutions to logistic equations involving the fractional Laplacian, demonstrating convergence to obstacle problem solutions and analyzing eigenvalue properties under perturbations.

## Contribution

It provides new asymptotic results for fractional Laplacian logistic equations and establishes the cut-off property for eigenvalues with perturbed potentials.

## Key findings

- Solutions converge strongly in energy space
- Uniform convergence to obstacle problem solutions
- Eigenvalues exhibit cut-off property under perturbations

## Abstract

We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.