A uniqueness condition for the polyharmonic equation in free space

✍ Scribed by P. Lesky Jr

• Publisher: John Wiley and Sons
• Year: 1990
• Tongue: English
• Weight: 595 KB
• Volume: 12
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670120402

✦ Synopsis

Abstract

Consider the polyharmonic wave equation ∂u + (− 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.