  Transport phenomena plays an important role in science and technology. In the
wide variety of applications both advection and diffusion may appear. Regarding
diffusion, for long times, different type of decay rates are possible for
different non-equilibrium systems. After summarizing the existing solutions of
the regular diffusion equation, we present not so well known solution derived
from three different trial functions, as a key point we present a family of
solutions for the case of infinite horizon. By this we tried to make a step
toward understanding the different long time decays for different diffusive
systems.
