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 β in the hyperbolic region. As a consequence of this
degeneracy there is an amplification phenomenon: outgoing waves of amplitude
O(\eps^2) and wavelength \eps give rise to reflected waves of amplitude
O(\eps), so the overall solution has amplitude O(\eps). 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'}+\beta\frac{\partial_{\theta_0}}{\eps}, x′ being the
tangential variables with respect to the boundary. Tame estimates for the
linearization of that problem are proved using a first-order calculus of
singular pseudodifferential operators constructed in the companion article
\cite{CGW2}. 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∞ on a fixed time interval
independent of the (small) wavelength \eps. The approach using singular
systems allows us to avoid constructing high order expansions and making small
divisor assumptions
