I have been working through the solution, and as suggested by Ackbeet at the beginning, the PDE drops to a ODE in the complex plane.... and with the initial condition... the general form drops to:

where 'A' is the arbitary constant.

Following from the discussion of the limit theorems, would it be possible to say that the value of:

can be found from using the limit of ? from our expression, will this leave ... will this imply 0 in the time domain (u) then?