Geodesic
A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities
Electronic journal of differential equations, Tome 2011 (2011)
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 
@article{EJDE_2011__2011__a5,
     author = {Deguchi,  Hideo},
     title = {A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities},
     journal = {Electronic journal of differential equations},
     year = {2011},
     volume = {2011},
     zbl = {1237.46027},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a5/}
}
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%A Deguchi,  Hideo
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%J Electronic journal of differential equations
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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/