Irregularities of nonlinear hyperbolic equations

Abstract. We consider the well-posedness of semilinear hyperbolic Cauchy problems for Gevrey functions. To obtain ageneral result, we define the notion of irregularities, and we give acriterion for the well-posedness. 1Introduction We assume $m\geq 1 $ and $n\geq 2 $. $1\mathrm{f}1 $ $<s<\infty $ , $R $ $>0 $ and ci $\subset \mathrm{R}^{n} $ is open, then we define $\mathcal{E}_{R}^{s}(\omega)=\{f(x)\in C^{\infty}(\omega) $ ; for $\exists C>0 $ and $\forall\alpha\in \mathrm{Z}_{+}^{n}$ we have $|\partial_{x}^{\alpha}f(x)|\leq CR^{|\alpha|}\alpha!^{s}\} $. lf 8=\infty then $\mathcal{E}_{R}^{s}(\omega)=\{f(x)\in C^{\infty}(\omega) $ ; for $\forall\alpha\in \mathrm{Z}_{+}^{n} $ and $\exists C_{\alpha}>0$ we have $|\partial_{x}^{\alpha}f(x)|\leq C_{\alpha}\} $, although it does not depend on $R $. We define $\mathcal{E}^{s}(\omega)=1\dot{\mathfrak{B}}^{R>0}\mathcal{E}_{R}^{s}(\omega) $ for $1<s\leq\infty $. The usual set $\overline{\mathcal{E}}^{s}(\omega) $ of Gevrey functions on $\omega $ is defined by $\overline{\mathcal{E}}^{s}(\omega)=.k^{\mathrm{m}_{\omega \mathrm{C}\mathrm{C}\omega}} ’ \mathcal{E}^{s}(\omega’) $ , but for the sake of convenience we consider $\mathcal{E}^{s}(\omega) $ instead of $\overline{\mathcal{E}}^{s}(\omega) $. Our aim is to determine when asemilinear hyperbolic Cauchy prob-lem is well-posed for such functions. M. D. Bronstein [1] and K. Ka-jitani [5] gave a sufficient condition for this problem (See Theorem 1 below). However, their result is not satisfactory for some important cases, especially in case of weakly hyperbolic equations. To recover this defect, we give amore refined criterion (See Theorem 2below)