Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides
Peter H. Lesky, Peter Christian Werner · Mathematical Methods in the Applied Sciences · 1989
Abstract We consider a domain Ω in ℝn of the form Ω = ℝl × Ω′ with bounded Ω′ ⊂ ℝn−l. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂ u + [(− ∂ −… − ∂ )m + (− ∂ −… − ∂ )m]u = fe−iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.