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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
##### Keywords:
entire functions of exponential type