A Uniqueness Condition for the Polyharmonic Equation in Free Space^{1}

P. Lesky jun.

Abstract

Consider the polyharmonic wave equation ∂_{t}^{2}u + (−Δ)^{m}u = f in IR^{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 ln t as t → ∞. In the first case we study the limit u_{0}(x) := lim_{t→∞} u(x, t) and give a uniqueness condition that characterizes u_{0} among the solutions of the polyharmonic equation (−Δ)^{m}u = f in IR^{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 homogenous polyharmonic equation.

AMS subject classification: 35 J 30, 35 B 40, 35 E 15

Published in: Math. Meth. in the Appl. Sci. 12 (1990), 275-291

1Supported by the Deutsche Forschungsgemeinschaft (SFB 256, Bonn)