I'd like to find some biharmonic and triharmonic time dependent problem.

For example,

u_t = u_{xxxx}

and

u_t = u_{xxxxxx} in one dimensional case.

Would you recommend some references?

Here's a good place to start:

http://www.amazon.com/exec/obidos/AS...287138-4896024

Maybe this is of use????

http://scholar.google.com/scholar?hl...&q=triharmonic+biharmonic

Your first example is an ill-posed problem ( you require u_t = -u_{xxxx} for well-posedness).

Any paper on hyperdiffusion should help. You could also look at papers on elasticity where the biharmonic equation turns up quite naturally.