The evolution problem associated with the fractional first eigenvalue

TL;DR

This paper investigates the evolution problem linked to the first fractional eigenvalue, establishing well-posedness, decay rates, and properties of solutions within the framework of viscosity solutions.

Contribution

It proves existence, uniqueness, and decay properties of solutions for the fractional eigenvalue evolution problem, extending understanding of fractional operators.

Findings

Solutions decay exponentially fast as time approaches infinity.

The Dirichlet problem is well-posed with a comparison principle.

The first eigenvalue bounds the decay rate of solutions.

Abstract

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as $t\to \infty$ with a bound that is given by the first eigenvalue for this problem that we also study.