Time regularity of the solutions to second order hyperbolic equations

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≤s₁≤2−1/s₀. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s₁≤s₀.