A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.