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Hyperbolic systems with discontinuous coefficients: Generalized solutions and a transmission problem in acoustics. (English) Zbl 0705.35146
The paper is concerned with hyperbolic systems of the form $(S)\quad (\partial_ t+\Lambda (x,t)\partial_ x)V+F(x,t)V+G(x,t),\quad (x,t)\in R^ 2,$ under initial condition $$V(x,0)=A(x)$$, $$x\in R$$. Basic problems like existence, uniqueness and regularity of generalized solutions are dealt with for the system (S), in which F and G are n- vectors and the matrices involved in (S) are n by n. The attention is concentrated on the case of coefficients with discontinuities. Sobolev spaces and Colombeau’s algebra $$G(R^ 2)$$ are used.
A sample result is the following: Assume $$\Lambda,F,G\in G(R^ 2)$$, with $$\Lambda$$ globally bounded; moreover, $$\partial_ x\Lambda$$ and F are supposed to be of locally of logarithmic growth. Then for every initial function $$A\in G(R)$$, the system (S) has a unique solution $$V\in G(R^ 2)$$. The second part of the paper is dedicated to a special case, encountered in the acoustics.
Reviewer: C.Corduneanu

##### MSC:
 35R05 PDEs with low regular coefficients and/or low regular data 35D05 Existence of generalized solutions of PDE (MSC2000) 35L45 Initial value problems for first-order hyperbolic systems
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##### References:
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