Inverse problems for first-order hyperbolic equations with time-dependent coefficients

TL;DR

This work addresses inverse problems for first-order hyperbolic equations with coefficients depending on space and time, focusing on global Lipschitz stability. The authors introduce a novel Carleman weight built from the length of integral curves of $A(\cdot,0)$, enabling stability results without strong structure assumptions on the time-dependent part. They prove Lipschitz stability for (i) an inverse source problem, (ii) a zeroth-order inverse coefficient problem, and (iii) a simultaneous reconstruction of time-independent principal parts, by combining the Carleman estimate with energy methods. The results extend inverse-problem theory to time-varying principal parts and provide tools for applications in conservation laws with evolving coefficients, with explicit boundary data frameworks and observability-time conditions $T_0<T$.

Abstract

We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.

Figures (2)

• Figure 1: $c_x$ is the maximal integral curve of $X$ through $x$.
• Figure 2: Pictures of $X$ (left) and $Y$ (right).

Theorems & Definitions (26)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Example 2.7
• Lemma 2.8
• proof
• Remark 2.9