Geodesic
Cauchy--Gelfand Problem and the Inverse Problem for a First-Order Quasilinear Equation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 2, pp. 61-74

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Gelfand's problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.
Keywords: quasilinear equation, Cauchy problem, asymptotics, vanishing viscosity method, Maxwell's rule. 
@article{FAA_2016_50_2_a3,
     author = {G. M. Henkin and A. A. Shananin},
     title = {Cauchy--Gelfand {Problem} and the {Inverse} {Problem} for a {First-Order} {Quasilinear} {Equation}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {61--74},
     publisher = {mathdoc},
     volume = {50},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2016_50_2_a3/}
}
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G. M. Henkin; A. A. Shananin. Cauchy--Gelfand Problem and the Inverse Problem for a First-Order Quasilinear Equation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 2, pp. 61-74. http://geodesic.mathdoc.fr/item/FAA_2016_50_2_a3/