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Time regularity of the solutions to second order hyperbolic equations

Tamotu Kinoshita Institute of Mathematics, University of Tsukuba Giovanni Taglialatela Dipartimento di Scienze Economiche e Metodi Matematici, University of Bari

Analysis of PDEs mathscidoc:1701.03018

Arkiv for Matematik, 49, (1), 109-127, 2009.5
We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class $\gamma^{s_{0}}$ and the Cauchy data belong to $\gamma^{s_{1}}$ , then the Cauchy problem has a solution in $\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R}))$ for some$T$^{*}>0, provided 1≤$s$_{1}≤2−1/$s$_{0}. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤$s$_{1}≤$s$_{0}.
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@inproceedings{tamotu2009time,
  title={Time regularity of the solutions to second order hyperbolic equations},
  author={Tamotu Kinoshita, and Giovanni Taglialatela},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203629360519994},
  booktitle={Arkiv for Matematik},
  volume={49},
  number={1},
  pages={109-127},
  year={2009},
}
Tamotu Kinoshita, and Giovanni Taglialatela. Time regularity of the solutions to second order hyperbolic equations. 2009. Vol. 49. In Arkiv for Matematik. pp.109-127. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203629360519994.
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