The evolution problem associated with the fractional first eigenvalue

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 $t\to \mathrm{\infty }$ with a bound that is given by the first eigenvalue for this problem that we also study.