2024/10/01 by Ru-Yu Lai, Hanming Zhou, Lai, Ru-Yu +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2410.00369
openalex publication_date 2024/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this work, we investigate inverse problems of recovering the time-dependent coefficient in the nonlinear transport equation in both cases: two-dimensional Riemannian manifolds and Euclidean space ℝn, n≥ 2. Specifically, it is shown that its initial boundary value problem is well-posed for small initial and incoming data. Moreover, the time-dependent coefficient appearing in the nonlinear term can be uniquely determined from boundary measurements as well as initial and final data. To achieve this, the central techniques we utilize include the linearization technique and the construction of special geometrical optics solutions for the linear transport equation. This allows us to reduce the inverse coefficient problem to the inversion of certain weighted light ray transforms. Based on the developed methodology, the inverse source problem for the nonlinear transport equation in the scattering-free media is also studied.