English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the present paper, we consider second order strictly hyperbolic linearoperators of the form $Lu\,=\,\partial_t^2u\,-\,{\rm div}\big(A(t,x)\nablau\big)$, for $(t,x)\in[0,T]\times\mathbb{R}^n$. We assume the coefficients ofthe matrix $A(t,x)$ to be smooth in time on $\,]0,T]\times\mathbb{R}^n$, butrapidly oscillating when $t\to 0^+$; they match instead minimal regularityassumptions (either Lipschitz or log-Lipschitz regularity conditions) withrespect to the space variable.  Correspondingly, we prove well-posedness results for the Cauchy problemrelated to $L$, either with no loss of derivatives (in the Lipschitz case) orwith a finite loss of derivatives, which is linearly increasing in time (in thelog-Lipschitz case).

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