Resonance Phenomena for a Class of Partial Differential

Equations of Higher Order in Cylindrical Waveguides

P. Lesky jun., Stuttgart

Abstract.

We consider a domain Ω in IR^{n} of the form Ω = IR^{l} × Ω′ with bounded Ω′ ⊂ IR^{n−l}. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂_{t}^{2}u + [(−∂_{1}^{2} − ⋯ − ∂_{l}^{2})^{m} + (−∂_{l+1}^{2} − ⋯ − ∂_{n}^{2})^{m}] u = f e^{−iωt}. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^{α} (α rational, 0 < α < 1) or like ln t as t → ∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.

AMS subject classification: 35 B 40, 35 C 10, 35 E 99

Published in: Math. Meth. in the Appl. Sci. 11 (1989), 697-723