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

The evolution problem associated with the fractional first eigenvalue