arXiv subjects

Siwar Ben Said

Publications and source records attributed to Siwar Ben Said.

2 recordsLinked to original sources

arXiv2026

Lower bound of the energy norm for type II blow-up solutions to the energy-critical wave equation in non-euclidean geometries

This article is concerned with the study of the focusing energy-critical wave equation on the three-dimensional Riemannian manifold $\mathcal{M}=\mathbb{R}^3$ equipped with a smooth time-independent Riemannian metric $g$, which coincides with the Euclidean metric outside a compact set. Our focus is on Type II blow-up solutions, namely finite-time solutions that remain bounded in the energy space, with the goal of revealing their fundamental properties. We establish in particular lower bounds for the energy norm as the solution approaches its maximal time of existence in terms of the bounds of the ground state of the Euclidean problem.

math.AP↗

arXiv2026

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

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.

math.AP↗