Geodesic
A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities
Electronic Journal of Differential Equations, Tome 2011 (2011).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: It is well-known that distributional solutions to the Cauchy problem for $u_t + (b(t,x)u)_{x} = 0$ with $b(t,x) = 2H(x-t)$, where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied.
Classification : 46F30, 35L03, 35A21
Keywords: first-order hyperbolic equation, discontinuous coefficient, generalized solutions 
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     author = {Deguchi, Hideo},
     title = {A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities},
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Deguchi, Hideo. A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities. Electronic Journal of Differential Equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a5/