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Article Dans Une Revue Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Année : 2023

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

Résumé

In the present paper, we consider second order strictly hyperbolic linear operators of the form $Lu\,=\,\partial_t^2u\,-\,{\rm div}\big(A(t,x)\nabla u\big)$, for $(t,x)\in[0,T]\times\mathbb{R}^n$. We assume the coefficients of the matrix $A(t,x)$ to be smooth in time on $\,]0,T]\times\mathbb{R}^n$, but rapidly oscillating when $t\to 0^+$; 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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Dates et versions

hal-03960829 , version 1 (15-03-2024)

Identifiants

Citer

Ferruccio Colombini, Daniele Del Santo, Francesco Fanelli. Well-posedness results for hyperbolic operators with coefficients rapidly oscillating in time. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2023, pp.18. ⟨10.2422/2036-2145.202301_017⟩. ⟨hal-03960829⟩
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