The Cauchy problem for semilinear hyperbolic systems with discontinuous data.(English)Zbl 0631.35056

The author proves short time existence of the solution to the strictly hyperbolic Cauchy problem $Lu\equiv u_ t+\sum^{n}_{i=1}A_ i(t,x)u_ i=F(t,x,u),\quad u_{t=0}=g,$ where g is piecewise smooth with jumps only over a hypersurface $$\Gamma \subset R^ n$$. The solution is obtained by means of the successive approximation scheme $$Lu_{k+1}=F(t,x,u_ k)$$, $$u_{k+1}(t=0)=g$$ in the space, roughly speaking, $$L_{\infty}\cap$$ conormal distributions (piecewise Sobolev functions cannot be used here because of the lack of stability (solvability of linear system), which is discussed more explicitly).
Reviewer: A.Doktor

MSC:

 35L60 First-order nonlinear hyperbolic equations 35L67 Shocks and singularities for hyperbolic equations 35R05 PDEs with low regular coefficients and/or low regular data 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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References:

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