In this case, the time-dependent PDE is treated as a three-dimensional spatial PDE because an initial condition is missing. The three-dimensional PDE is convection dominated:

The solution is to add an initial condition:

Here is a convection-dominated PDE:

Solving the PDE with boundary conditions gives a warning:

Inspecting the solution reveals a kink in the solution:

To improve the situation, a finer mesh can be used:

Even though the message is still present, the solution looks much better: