Two by two strongly hyperbolic systems and Gevrey classes (Q2756117)

The authors consider a first order \(2\times 2\)-system of the form NEWLINE\[NEWLINE L=D_{t}+ \sum_{j=1 }^{ n } A _{j} (t) D _{j} +B(t,x), \quad D _{j} = - \sqrt{-1} (\partial /\partial x _{j}),\quad D_{t}= - \sqrt{-1} (\partial /\partial t), NEWLINE\]NEWLINE where the \(A _{j} \) are \( 2\times 2\)-matrices depending only on \(t \in [0,T]\). They assume that the matrix \(A(t, \xi)= \sum_{j=1 }^{ n } A _{j} (t) \xi _{j} \) is uniformly diagonalizable for every \(t\) and \( \xi \) and show under some additional assumptions that the Cauchy problem is stronlgy hyperbolic in a certain number of cases: NEWLINENEWLINENEWLINEa) assuming that the \(A(t)\) are real analytic and that \(B(t,x) \in {\mathcal C} ^{1}([0,T]; {\mathcal C}^{\infty})\), they obtain \({\mathcal C}^{\infty} \)-well-posedness, NEWLINENEWLINENEWLINEb) assuming that \( A _{j} \in {\mathcal C} ^{k}([0,T])\) for some \(k\) and \(B \in {\mathcal C} ^{1}([0,T]; \gamma ^{ s(k) })\), (here \(s(1)=1\), \(s(2)=3/2\), \(s(k)= (\sqrt{2k+1}+1)\), \(k \geq 3\) and \( \gamma ^{s } \) is the Gevrey class of order \(s\)) they obtain well-posedness of the Cauchy problem in \( \gamma ^{ s(k) }\), NEWLINENEWLINENEWLINEc) if \(B(t,x)=B(t) \in {\mathcal C} ^{1}([0,T])\), then the Cauchy problem is well-posed in \( \gamma ^{s } \) for \( s < 1+k/2\). NEWLINENEWLINENEWLINEAn example shows that the assumption that the coefficients \(A _{j} \) be analytic cannot be removed completely if one wants to obtain the stated results.