  Motivated by [7], we study the advection-hyperdiffusion equation in the whole
space in two and three dimensions with the goal of understanding the decay in
time of the $H^{-1}$- and $L^2$-norm of the solutions. We view the advection
term as a perturbation of the hyperdiffusion equation and employ the
Fourier-splitting method first introduced by Schonbek in [8] for scalar
parabolic equations and later generalized to a broader class of equations
including Navier-Stokes equations and magneto-hydrodynamic systems. This
approach consists of decomposing the Fourier space along a sphere with radius
decreasing in time. Combining the Fourier-splitting method with classical PDE
techniques applied to the hyperdiffusion equation we find a lower bound for the
$H^{-1}$-norm by interpolation.
