Parabolic PDE appears in transferring problems, where some mechanisms of dissipations exist, for instance, considering viscous fluids or considering problems of thermal conductivity.
The classical example of parabolic PDE is the equation of diffusion (or the equation of thermal conductivity).
In 1-dimensional case it looks like this
If, for instance, we give, similar to the case of hyperbolic equation, as an initial condition (at t=0) the function,
and in addition give the same boundary condition as in hyperbolic case, then the exact solution will be
It’s worth to compare the principally different behavior of the exact solution of hyperbolic equation and this solution of parabolic equation:
In wave equation, we had an oscillatory solution. In diffusion equation, we see exponential decay.
Some more examples:
The transport equation is a linear parabolic PDE.
The Burger’s equation is an example of non-linear parabolic PDE.
The unsteady Navier-Stokes equations are parabolic.
Many of the reduced forms of the Navier-Stokes equations are governed by parabolic PDEs as well.