Well-posedness of the Cauchy problem for some weakly hyperbolic operators in Gevrey classes
The problem studied in this paper is concerned with the determination of a function space in which the Cauchy problem for a given weakly hyperbolic operator is well-posed. The operator is supposed to be of the form \[ L=L_ 0(t,x,D_ t,D_ x)+L_ 1(t,x,D_ t,D_ x),\quad where \] \[ L_ 0(t,x,D_ t,D_ x)=D^ m_ t +\sum^{m}_{k=1}\sigma (x)^{k\mu} (\sum_{| \alpha | =k}a_{k\alpha}(t,x)D_ x^{\alpha})D_ t^{m-k},\quad and \] \[ L_ 1(t,x,D_ t,D_ x)=\sum^{m}_{k=1}\sum^{k}_{j=1}\sigma (x)^{\mu_ k-j} (\sum_{| \alpha | =k-j}b_{jk\alpha}(t,x)D_ x^{\alpha})D_ t^{m-k} \] while \(t\in (0,T]\), \(x\in {\mathbb{R}}^ n\) and \(\mu\) and \(\mu_ k\) are non-negative integers. It is proven that under certain conditions involving also the lower order terms the Cauchy problem \[ Lu(t,x)=f(t,x)\quad in\quad (0,T]\times {\mathbb{R}}^ n;\quad D^ i_ t u(t,x)=u^ i(x)\quad on\quad {\mathbb{R}}^ n,\quad t=0;\quad i=0,...,m-1 \] is well-posed in some sense.
scientific article; zbMATH DE number 3935609