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Global Fuchsian Cauchy problem. (English) Zbl 0967.35024
The Fuchsian Caychy problem for entire functions in \(\mathbb C_t \times \mathbb C^n _x\), \(x= (x_1,\ldots, x_n)\), is considered. Let \(P= P(t,x,D_t,D_x)\) be a linear partial differential operator of order \(m\) with entire coefficients. Here \(D_t= \partial/\partial t\), \(D_x= (D_1,\ldots,D_n)\). It is supposed that the operator has the form \[ \begin{gathered} P(t,x,D_t,D_x)= t^{m'}D_t ^m +a_{m-1} t^{m'-1}D_t ^{m-1}+\ldots+ a_{m-m'}D_t ^{m-m'} \\ +\sum_{j= 0} ^{m-1} \sum_{|\beta|= m-j} \sum_{k= \alpha(j)} t^k a_{jk\beta}(x)D_t ^j D_x ^{\beta} \\ + \sum_{j= 0} ^{m-1} \sum_{|\beta|= m-j-1} t^{\alpha(j)}a_{j\beta}(t,x) D_t ^j D_x ^{\beta}, \end{gathered} \] where \(0\leq m' \leq m\), \(\alpha(j)=\max [0,j-(m-m')+1]\), \(D_x ^{\beta}= D_x ^{\beta_1} \ldots D_x ^{\beta_n}\) and \(|\beta|= \beta_1+\ldots \beta_n\). Moreover, \(a_{m-j}\) are constants, \(a_{jk\beta}\) are polynomials in \(x\) of degree \(\leq m-j-1\), \(a_{j \beta}\) are entire functions in \({\mathbb C}_t \times {\mathbb C}^n _x\). Such an operator is called a Fuchsian one. We denote by \(C(\lambda)\) a characteristic polynomial of \(P\). The following theorem is proved.
Theorem. If \(C(\lambda) \neq 0\) for any integer \(\lambda \geq m-m'\), then for any entire function \(f(t,x)\) in \(\mathbb C_t \times \mathbb C^n _x\) and any entire function \(f_{\lambda}(x)\) \((0 \leq \lambda \leq m-m'-1)\) in \(\mathbb C^n _x\), the following Cauchy problem has a unique entire solution \(u(t,x)\): \[ Pu= 0, \quad D_t ^\lambda u(0,x)= f_\lambda(x). \] Note. If \(m= m'\), we have no conditions and the equation has a unique solution.
In conclusion the author investigates the action of a class of partial differential operators on entire functions of exponential type. The statement of the problem is a natural generalization of the analogous problems solved ealier by several authors for partial (not Fuchsian) differential equations.

35B60 Continuation and prolongation of solutions to PDEs
35A20 Analyticity in context of PDEs