Exponential decay for fractional Schrödinger parabolic problems

We discuss exponential decay in $L^p(R^N)$, $1\le p\le \mathrm{\infty }$, of solutions of a fractional Schrödinger parabolic equation with a locally uniformly integrable potential. The exponential type of the semigroup of solutions is considered and its dependence in $1\le p\le \mathrm{\infty }$ is addressed. We characterise a large class of potentials for which solutions decay exponentially.