Cauchy-Gelfand problem for quasilinear conservation law

We obtain the precise asymptotic (t→∞) for solution f(x,t) of Cauchy-Gelfand problem for quasilinear conservation law ∂f∂t+φ(f)∂f∂x=0 with initial data of bounded variation f(x,0)=f0(x). The main theorem develops results of Liu (1981) [22], Kruzhkov, Petrosjan (1987) [20], Henkin, Shananin (2004) [10], Henkin (2012) [9]. Proofs are based on vanishing viscosity estimates and localized Maxwell type conservation laws. The main application consists in the reconstruction of parameters of initial data responsible for location of inviscid shock waves in the solution f(x,t).