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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
Keywords:
well-posedness
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