Spectral Analysis of a Multistratified Acoustic Strip Part II: Asymptotic Behavior of Solutions for a Simple Stratification

Abstract

We consider the acoustic propagator $A = - \nabla \cdot c^2 \nabla $ in $\Omega = \{ {(x,z) \in {{\mathbb{R}^2 } / {0 < z < H}}} \}$. The velocity c, which describes the stratification of the strip $\Omega $, depends only on the variable z: it is assumed to be a function in $L^\infty ((0,H))$ bounded from below by $c_m > 0$. Let A be the self-adjoint operator associated with the Neumann or Dirichlet condition at $z = 0$ and $z = H$; let $\mu $ be a real number in the spectrum of A; and let u be the solutions of the equation $(A - \mu I)u = f$ locally in the domain of A, which are determined by the limiting absorption principle in [E. Croc and Y. Dermenjian, SIAM J. Math. Anal., 26 (1995), pp. 880–924] and made explicit with trace operators. Thanks to accurate Hölder properties for the trace operators, we control the asymptotic behavior of u with so-called “zero-trace” conditions for f.