Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 76, pp. 1-25.

Title: A linear first-order hyperbolic equation
       with discontinuous coefficient: distributional shadows
       and propagation of singularities
       
Author: Hideo Deguchi (Univ. of Toyama, Japan)

Abstract:
 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.

Submitted January 19, 2011. Published June 16, 2011.
Math Subject Classifications: 46F30, 35L03, 35A21.
Key Words: First-order hyperbolic equation;
           discontinuous coefficient; generalized solutions.