WIAS Preprint No. 59, (1993)

On the smoothness of the solution to a boundary value problem for a differential-difference equation.


  • Ivanova, I. P.
  • Kamenskij, G. A.




This paper deals with the first boundary value problem (BVP) for equations which are a differential with respect to one variable (t) and difference with respect to the other variable (s) in a bounded domain. The initial value problem for differential-difference equations of this type was studied in [1], [2]. The theory of the BVP under investigation is connected with the theory of the BVP for strongly elliptic differential-difference equations which are difference and differential with respect to the same variable (see [3]). Some questions of this work were considered earlier in the papers [4], [5], [8]. Section 1 considers the solvability of the BVP for differential-difference equations. In contrast to differential equations the smoothness of the generalized solutions can be broken in the domain Q and is preserved only in some subdomains Qr ⊂ Q where (∪rr = Q̅). In section 2 we construct such a set of Qr. Section 3 deals with the smoothness of generalized solutions in the subdomains Qr. Section 4 considers the conditions under which the smoothness is preserved when passing the boundaries between neighboring subdomains Qr.

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