Well-posedness results for hyperbolic operators with coefficients rapidly oscillating in time

2023/01/25 by Ferruccio Colombini, Colombini, Ferruccio, Daniele Del Santo +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.10854

openalex publication_date 2023/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the present paper, we consider second order strictly hyperbolic linear operators of the form Lu = ∂_t²u - \rm div(A(t,x)∇ u), for (t,x)∈[0,T]×ℝⁿ. We assume the coefficients of the matrix A(t,x) to be smooth in time on ]0,T]×ℝⁿ, but rapidly oscillating when t→ 0⁺; they match instead minimal regularity assumptions (either Lipschitz or log-Lipschitz regularity conditions) with respect to the space variable. Correspondingly, we prove well-posedness results for the Cauchy problem related to L, either with no loss of derivatives (in the Lipschitz case) or with a finite loss of derivatives, which is linearly increasing in time (in the log-Lipschitz case).

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