an:03933641
Zbl 0583.35074
Jannelli, Enrico
Regularly hyperbolic systems and Gevrey classes
EN
Ann. Mat. Pura Appl., IV. Ser. 140, 133-145 (1985).
00150857
1985
j
35L45 35F10 35R25 35L40 35L85
Cauchy problem; regularly hyperbolic; Gevrey class; energy inequalities
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 H??lder 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.
P.Jeanquartier