We have shown that the solution to the diffusion equation in an unbounded
domain with a prescribed initial condition is, at any given time, the
convolution of the initial spatial distribution 
with a Gaussian
kernel. The variance of the Gaussian kernel, moreover, is proportional to
the time t. Now, convolution is, in general, a smoothing operation.
Convolution with a narrow kernel roughly preserves the functional shape.
Consider, for example, the extreme case: for very small variance, we have
shown the Gaussian kernel to approximate an impulse, and convolution with an
impulse does not change the functional shape at all. On the
other hand, convolution with a broad kernel spreads out and smooths the
functional shape. Thus, we see, that as time evolves, the spatial
distribution of the solution to the diffusion equation is a broadened and
smoothed version of the initial distribution. This process is called blurring, in general, and in this case is called Gaussian blurring.
Let us take a different approach. If we imagine examining the solution u
at some time T, 
, and
let time run backwards, towards 0, we see that the solution sharpens.
That is, 
has sharper, more highly defined
features that 
for t<T. Viewed another way, we
have:
Consider, for the moment, the case where the conduction coefficient is negative: c<0. Thus, we have:
Let us replace t by -t; then we get:
This suggests that when c<0:
Thus, the solution sharpens as time evolves in this case. This is called backward diffusion; for the usual case where c>0, considered up to this point, we have forward diffusion. Forward diffusion is blurring, and backward diffusion is sharpening.