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arXiv · 2607.24839

On the Wave of Klein-Gordon Type: Propagation Equation at a Boundary and Interior Observability

Siwar Ben Said
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Abstract

Understanding the transport of semiclassical measures along generalized bicharacteristics is a key ingredient in the proof of observability for the wave equation. Here, the coefficients are only assumed to be of class $C^1$, which is a limit case for the existence of bicharacteristics. We consider boundary conditions of the form $\partial_\nu u + i \partial_t u = 0$, as a first step towards more general Lopatinskii-type conditions. We derive the transport equation satisfied by the measure arising from the concentration of high-frequency waves that obstruct observability. Observability of solutions associated with positive time-frequencies is then obtained by contradiction under the geometrical control condition.

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Siwar Ben Said. 2026-07-24. On the Wave of Klein-Gordon Type: Propagation Equation at a Boundary and Interior Observability. https://arxiv.org/abs/2607.24839

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