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On a characteristic problem for the wave equation. (English) Zbl 1018.35001
This paper is devoted to the characteristic problem for the wave equation $\begin{cases} \square u=u_{tt}- u_{x_1x_1}- u_{x_2x_2}=F,\\ u|_{S_i}= f_i,\end{cases} \tag{1}$ where $$f_1$$, $$f_2$$ are given real functions on $$S_1$$ and $$S_2$$ respectively. Here $$S_i$$, $$i=1,2$$ are the characteristic surfaces for (1). Under the assumption $$f_1-f_2|_{S_1\cap S_2}=0$$, the author studies the correctness of (1) in the Sobolev spaces $$W^1_2$$ and $$W^2_2$$.

##### MSC:
 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35L15 Initial value problems for second-order hyperbolic equations 35L05 Wave equation
well-posedness