  We consider solutions to linear parabolic equations with initial data
decaying at spatial infinity. For a class of advection-diffusion equations with
a spatially dependent velocity field, we study the behavior of solutions as
time tends to infinity. We characterize velocity fields, so that positive
solutions decay or lift-off at spatial infinity as time tends to infinity. This
addresses the question of stability of the zero solution for decaying
perturbations.

Analysis of PDEs