A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ∂   t 2u + (− Δ) m u = f in ℝ 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 \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 u 0 among the solutions of the polyharmonic equation ( − Δ) m u = f in ℝ n . Furthermore we prove in the case 2 m ⩾ 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.