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

Dong-Hui Yang, Jie Zhong

Abstract

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $Ω= \mathbb{T} \times (0,1)$ 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 $α\in [1,2)$, 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.

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

Abstract

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain 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 , 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.

Paper Structure

This paper contains 13 sections, 25 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Solution spaces
  4. Green's formula
  5. Neumann boundary condition
  6. The spectrum for the partial differential operator $\mathcal{A}$
  7. Spectrum
  8. Existence and uniqueness of weak solutions to equation \ref{['01.21.1']}
  9. Observability
  10. A motivating example
  11. Cut-off decomposition
  12. Preliminary identities
  13. Multiplier argument

Key Result

Theorem 1.3

Let $\varphi^0\in H_\Gamma^1(\Omega;w)$ and $\varphi^1\in L^2(\Omega)$. Let $\varphi$ be the solution of 01.21.1 with respect to $(\varphi^0, \varphi^1, f=0)$. If $T>\sqrt{2}/(2-\alpha)$, then there exists a constant $C=C(T,\alpha,\delta_0)>0$ such that

Theorems & Definitions (56)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 46 more