Uniqueness for inhomogeneous solution from fundamental solution

Question

I am looking at the differential operator $$ L := \Delta^2-\frac{\partial^2}{\partial t^2} \ \ . $$ I have found a fundamental solution $F_L$ and know that the inhomogeneous solution $L u = f $ can be described by $$ u := F_L * f \ \ .$$ However I don't know how to find conditions that make this problem unique? Is there a general strategy for this, or is it specifically tied to the operator?