Abstract
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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Kinoshita, T., Taglialatela, G. Time regularity of the solutions to second order hyperbolic equations. Ark Mat 49, 109–127 (2011). https://doi.org/10.1007/s11512-009-0120-6
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DOI: https://doi.org/10.1007/s11512-009-0120-6