A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ∂ t2u + (− Δ)mu = f in ℝn × (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 tα or In t as t → ∞. In the first case we study the limit \documentclass{article}\pagestyle{empty}\begin{document}$ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $\end{document} and give a uniqueness condition that characterizes u0 among the solutions of the polyharmonic equation ( − Δ)mu = f in ℝn. 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.