Abstract: The problem of defining δ-shock wave type solutions of hyperbolic systems of conservation laws in connection with the constructing singular superpositions products of distributions is studied. We illustrate this problem by constructing δ-shock wave type solutions for two systems. One of them,ut+ (f(u)-v)x=0, vt+ g(u)x=0,is a generalization of the well-known Keyfitz--Kranzer system, where f(u) and g(u) are polynomials of degree n and n+1, respectively, n is an even integer. The other one is the systemut+f(u)x=0, vt+(vg(u))x=0,where f(u), g(u) are smooth functions. As far as we know, exact -shock wave type solutions for the first system have never been constructed.
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