A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability

TL;DR

This paper proves an observability inequality for a class of degenerate hyperbolic equations with mixed boundary conditions, enabling control via boundary and interior observations despite degeneracy.

Contribution

It introduces a novel approach combining weighted functional analysis and localized multipliers to establish observability for degenerate hyperbolic equations with mixed boundary conditions.

Findings

Established a mixed observability inequality for degenerate hyperbolic equations.

Demonstrated control of solutions through boundary and interior observations.

Developed a new analytical framework combining regularity, cutoff, and energy estimates.

Abstract

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $\Omega = \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the radial variable, whereas the periodic angular variable allows propagation with a strong tangential component, making a direct top boundary observation delicate. For $\alpha \in [1,2)$, we prove that the solution can be controlled by a boundary observation on the top boundary together with an interior observation on a narrow strip. The proof combines a weighted functional framework, improved regularity, a cutoff decomposition in the angular variable, a multiplier argument for the localized component, and an energy estimate for the remainder.