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