The Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

We consider the problem posed by Gelfand on the asymptotics in time of the solution of the Cauchy problem for a first-order quasilinear equation with Riemann-type initial conditions. Based on the method of vanishing viscosity with uniform estimates, exact asymptotics in the Cauchy-Gelfand problem are obtained without a priori assumption of monotonicity of the initial data. The initial data parameters responsible for the localization of shock waves are described.