We consider the well-posedness of the Cauchy problem in Gevrey spaces for $N imes N$ first order weakly hyperbolic systems. The question is to know wether the general results of M.D.Bron stein cite{Br} and K.Kajitani cite{Ka2} can be improved when the coefficients depend only on time and are smooth, as it has been done for the scalar wave equation in cite{CJS}. The anwser is no for general systems, and yes when the system is uniformly diagonalizable: in this case we show that the Cauchy problem is well posed in all Gevrey classes $G^s$ when the coefficients are $C^infty$. Moreover, for $2 imes 2$ systems and some other special cases, we prove that the Cauchy problem is well posed in $G^s$ for $s < 1+k$ when the coefficients are $C^k$, which is sharp following the counterexamples of S.Tarama cite{Tar}. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, $S_eps$, the coefficients of which are polynomials of $eps$ and the coefficients of $A$ and $A^*$.
The Cauchy problem for weakly hyperbolic systems
COLOMBINI, FERRUCCIO
In corso di stampa
Abstract
We consider the well-posedness of the Cauchy problem in Gevrey spaces for $N imes N$ first order weakly hyperbolic systems. The question is to know wether the general results of M.D.Bron stein cite{Br} and K.Kajitani cite{Ka2} can be improved when the coefficients depend only on time and are smooth, as it has been done for the scalar wave equation in cite{CJS}. The anwser is no for general systems, and yes when the system is uniformly diagonalizable: in this case we show that the Cauchy problem is well posed in all Gevrey classes $G^s$ when the coefficients are $C^infty$. Moreover, for $2 imes 2$ systems and some other special cases, we prove that the Cauchy problem is well posed in $G^s$ for $s < 1+k$ when the coefficients are $C^k$, which is sharp following the counterexamples of S.Tarama cite{Tar}. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, $S_eps$, the coefficients of which are polynomials of $eps$ and the coefficients of $A$ and $A^*$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
