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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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