Abstract
In this paper, we study an initial-boundary value problem for a semilinear hyperbolic system with the initial date having a possibly continuous oscillatory spectrum in the half-space $R^{1+2}_+=\{x=(t,y_{1},y_{2}):t>0,y_{2}>0\}. $ The goal of this paper is to rigorously justify the asymptotic analysis for the reflection of wave trains with such a continuous oscillatory spectrum.
Highlights
The study of highly oscillatory waves in hyperbolic problems is an important topic
There is a rich literature devoted to the formal analysis and rigorous justification for the propagation and interaction of highly oscillatory waves
For a rigorous mathematical theory, this requires to introduce new spaces of solutions, which are Wiener algebras associated to spaces of vector-valued measures with bounded total variation
Summary
The study of highly oscillatory waves in hyperbolic problems is an important topic. There is a rich literature devoted to the formal analysis and rigorous justification for the propagation and interaction of highly oscillatory waves (see [8, 10, 16, 17] and references cited there). Let λ ∈ BV (R2, B2) and T be a λ− integrable function on R2 with values in B1 := LC(B2, B3), the space of continuous linear operators from B2 to B3, the product Tλ defined as. The Fourier transforms of functions in Wiener algebras are measures of bounded variation, and the supports of such measure can be either countable or continuous. 1) The space As0 denotes the set of functions defined on R+2 × RX1+2 with values in CN whose Fourier transforms with respect to the X variable are in BV (Rξ1+2, Hs(R+2 )N ). 2) The values in CspNacwe hAosste denotes Fourier the set of functions defined transforms with respect to on the.
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