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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.

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