How to compute an energy function of PDE ?

[xortdsc]

Hi,

I have a PDE of the form

f(x,y,z)'' = Δf(x,y,z) + f(x,y,z) * (1 - f(x,y,z)^2)

where f(x,y,z) is a 3 dimensional vector-field.
Now I want to compute an energy function for it such that for any state (f(x,y,z) and its first derivative f(x,y,z)') I can compute its corresponding energy (which should stay constant over the whole domain during evolution of the system, but may vary for parts of it).

How would I do this ? Can someone help me out here ?

Thanks and cheers

 

[Xiuh]

What does f(x,y,z)'' mean?

 

[xortdsc]

What does f(x,y,z)'' mean?

second derivative of f
The PDE can be interpreted as describing the "force" f(x,y,z)'' (acting on the "velocity" f(x,y,z)') being dependent on the "position" f(x,y,z) of itself (in the second term) and its direct neighborhood (the laplacian in the first term). This should be closely related to PDEs commonly encountered in physics, but I coudn't find a energy-function for this version anywhere.