Smoothness of the solutions.

  Parabolic equations such as the heat equation usually have smooth solutions. Even if we prescribe an initial condition which is singular, the heat equation will immediately smooth out this singularity. As an example, assume that we prescribe an initial condition with a jump in temperature, in which the wall and the first half of the bar are at 100 degrees and the second half at zero degrees. This initial condition is shown as a temperature profile, or T,x-plane, in figure 3. The jump is singular: at it, the second and higher derivatives are infinite. But at a time $t=\epsilon$ that is even slightly greater than zero, diffusion of heat has caused the jump to diffuse out into a steep but smooth slope, and all derivitives do now exist.

  

Figure 3: Smoothing of a singularity.