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On the solvability of a spatial problem of Darboux type for the wave equation. (English) Zbl 0842.35052
The solvability of the Darboux problem for the wave equation (1) $$\square u = F$$ with the boundary condition (2) $$u|_{S_1} = f_1$$, (3) $${\partial u \over \partial n} |_{S_0} = 0$$ is proved, where $$S_0$$, $$S_1$$ are some given surfaces.
The main result is: For any $$f_1 \in W^1_2 (S_1)$$, $$F \in L_2 (D_+)$$ there exists a unique strong solution $$u$$ of the problem (1)–(3) in the class $$W^1_2$$. The estimate $$|u |_{W^1_2 (D_+)}\leq C (|f_1 |_{W^1_2 (S_1)} + |F |_{L_2 (D_+)})$$ is also proved.

##### MSC:
 35L20 Initial-boundary value problems for second-order hyperbolic equations
##### Keywords:
Darboux problem; unique strong solution
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##### References:
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