Cauchy problem for some non-Kovalewskian equations (Q2756112)

In this paper the autor considers the Cauchy problem for the partial differential equation with constant leading coefficients in \([-T,T] \times \mathbb{R}^n\) as follows: NEWLINE\[NEWLINE\Pi(D_t,D_x) u(t,x)+ f\biggl(t,x, \bigl\{ \partial^j_t \partial_x^\alpha u(t,x)\bigr\}_{j=0,1, \dots,M,|\alpha|\leq p(m-j)-q} \biggr)=0.NEWLINE\]NEWLINE Here \(\Pi(D_t,D_x)\) denotes \(D^m_t+ \sum^m_{j=1} \sum_{|\alpha|=pj}a_{j \alpha}D^j_t D^\alpha_x\), and it is assumed that NEWLINE\[NEWLINE\Pi(\tau, \xi)=\prod^r_{j=1} \prod^{s_j}_{i=1} (\tau-\lambda^i_j(\xi)),NEWLINE\]NEWLINE where \(\lambda^i_j\) are real-valued, \(\lambda^i_j (\xi)\neq \lambda_k^h (\xi)\) if \(i\neq h\) and \(\xi\neq 0\), \(\lambda^i_j (\xi)=\lambda^i_k (\xi)\) for some \(\xi \neq 0\) and \(s_r\geq s_{r-1} \geq\cdots \geq s_1\). Then, when \(q=pr\), the above Cauchy problem is solved locally in time in usual Sobolev space \(H^\infty\) for given initial data in \(H^\infty\), and, when \(q<pr\), the Cauchy problem can be solved locally in time in Gevrey class of order \(\sigma(1 \leq\sigma <1/(p-q/r))\) for initial data in Gevrey class of order \(\sigma\).