Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws

This article is aimed at studying the Cauchy problem for a first-order quasi-linear equation with a flow function of power type and unbounded initial data of power or exponential type. It is known that the Cauchy problem in the class of locally bounded functions may have several solutions. We describe all entropy solutions of this problem, which can be represented in a special form. It is shown that after the first discontinuity line (shock wave), these solutions eventually exhibit the same behavior, and their nonuniqueness actually amounts to the choice of the first shock wave.