Regularly hyperbolic systems and Gevrey classes (Q1070117)

This paper deals with the first order Cauchy problem \[ (1)\quad \partial U/\partial t=\sum A_ h(t,x) \partial U/\partial x_ h+B(t,x),\quad U(0,x)=g(x), \] \(0\leq t\leq T\), \(x\in {\mathbb{R}}^ n\), where \(A_ h\) (1\(\leq h\leq n)\) and \(B\) are \(N\times N\) real matrices, while U and g are real \(N\)-vectors. System (1) is assumed to be regularly hyperbolic. Suppose that the coefficients \(A_ h(t,x)\) are Hölder continuous of order \(\alpha\) in t \((0<\alpha <1)\) and belong to the Gevrey class of order s in x and that \(B(t,x)\) is locally bounded and belongs to the Gevrey class of order s in x. Then the author proves that the Cauchy problem is well posed in the Gevrey class of order s provided that \(1\leq s<1/(1-\alpha)\). The method of energy inequalities is used.