Geodesic
Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws
Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 111-133 
A. Yu. Goritsky; L. V. Gargyants. Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws. Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 111-133. http://geodesic.mathdoc.fr/item/TSP_2019_32_32_a5/
@article{TSP_2019_32_32_a5,
     author = {A. Yu. Goritsky and L. V. Gargyants},
     title = {Nonuniqueness of unbounded solutions of the {Cauchy} problem for scalar conservation laws},
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     volume = {32},
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     url = {http://geodesic.mathdoc.fr/item/TSP_2019_32_32_a5/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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