Geodesic
Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a~First-Order Quasilinear Equation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 120-133

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Generalized entropy solutions for a first-order quasilinear partial differential equation are studied. It is shown that the Cauchy problem for this equation is ill-posed in the class of locally bounded functions. The examples of nonexistence and nonuniqueness of solutions are constructed. Moreover, a uniqueness theorem, which holds for solutions integrable with respect to the spatial variable, is proved. 
@article{TM_2002_236_a12,
     author = {A. Yu. Goritskii and E. Yu. Panov},
     title = {Locally {Bounded} {Generalized} {Entropy} {Solutions} to the {Cauchy} {Problem} for {a~First-Order} {Quasilinear} {Equation}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {120--133},
     publisher = {mathdoc},
     volume = {236},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_236_a12/}
}
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A. Yu. Goritskii; E. Yu. Panov. Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a~First-Order Quasilinear Equation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 120-133. http://geodesic.mathdoc.fr/item/TM_2002_236_a12/