On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 2: Time‐dependent theory
K. Morgenröther, Peter Christian Werner · Mathematical Methods in the Applied Sciences · 1989
Abstract In part 1 Math. meth. in the Appl. Sci, 10, 125–144 (1988). we studied the principle of limiting absorption for local perturbations Ω of the n ‐dimensional domain Ω 0 = ℝ n −1 × (0, π). In this second part we extend our investigations to the time‐dependent theory and show that absence of admissible standing waves implies the validity of the principle of limiting amplitude for every frequency ω≥0 if n ≠ 3 and for ω ≠ 2, 3,… if n = 3, respectively. In particular, the principle of limiting amplitude holds for every ω≥0 in the case n ≠ 3 and for every ω ≠ 2, 3,… in the case n = 3 if Ω≠Ω 0 and ν · x ′ ⩽0 on ∂Ω, where x ′ = ( x 1 ,…, x n −1 , 0) and ν is the normal unit vector on ∂Ω pointing into the complement of Ω This result stands in remarkable contrast to the fact that both principles are violated in the case of the unperturbed domain Ω 0 at the frequencies ω = 1, 2,… if n ⩽3. The question of the asymptotic behaviour of the solution as t →∞ for n = 3 and ω = 2, 3,… will be discussed in two subsequent papers.