Semilinear geometric optics with boundary amplification

Abstract

We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude $O\left({\epsilon }^2\right)$ and wavelength $\epsilon$ give rise to reflected waves of amplitude $O\left(\epsilon \right)$, so the overall solution has amplitude $O\left(\epsilon \right)$. Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant.

An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash–Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form ${\partial }_{x^{\prime }}+\beta {\partial }_{{\theta }_0}∕\epsilon$, $x^{\prime }$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order (wavetrain) calculus of singular pseudodifferential operators constructed in a companion article (“Singular pseudodifferential calculus for wavetrains and pulses”, arXiv 1201.6202, 2012). These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash–Moser iteration.

The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^{\infty }$ on a fixed time interval independent of the (small) wavelength $\epsilon$. The approach using singular systems allows us to avoid constructing high-order expansions and making small divisor assumptions. Our analysis of the exact singular system applies with no change to the case of pulses, provided one substitutes the pulse calculus from the companion paper for the wavetrain calculus.