Some remarks on the \(C^{\infty}\)-Goursat problem
Let \(L=\sum_{i+j+| \alpha | \leq m}a_{ij\alpha}D^ i_ tD^ j_ xD^{\alpha}_ y\), \(t\geq 0\), \(x\in {\mathbb{R}}\), \(y\in {\mathbb{R}}^ n\) be a partial differential operator with constant coefficients. Under the assumption \[ (A)\quad a_{ij\alpha}=0\text{ for } i+j+| \alpha | =m,\quad i>m-s\text{ and } \sum_{j+| \alpha | =s}a_{m- s,j\alpha} \xi^ j\eta^{\alpha}\neq 0, \] the following \(C^{\infty}\)- Goursat problem for \(t\geq 0\) is considered \[ (P)\quad Lu=0, \] \[ D^ i_ tu(0,x,y)=\phi_ i(x,y)\in \epsilon_{(x,y)},\quad 0\leq i\leq m-s- 1, \] \[ D^ j_ xu(t,0,y)=\psi_ j(t,y),\quad 0\leq j\leq s-1,\quad t\geq 0, \] with the compatibility assumption \(D^ j_ x\phi_ i(0,y)=D^ i_ t\psi_ j(0,y)\), \(0\leq i\leq m-s-1\), \(0\leq j\leq s-1.\) The main result says: if the Goursat problem (P) is \(\epsilon\)-well posed then the roots \(\alpha_ i\) of the equation \(\sum^{m-s}_{i=0}a_{i m-i 0} \tau^ i=0\) are all real.
scientific article; zbMATH DE number 4148678