Conditions of Osgood type in uniqueness problem for generalized entropy solution to Cauchy problem for quasilinear first order equations (Q2719460)

The paper deals with the problem of uniqueness of generalized entropy solutions to the Cauchy problem in nonlocal statements for the conservation law of the form NEWLINE\[NEWLINE \begin{gathered} u_t+\text{div}_x\phi(u) = 0,\quad x=(x_1,\dots,x_n)\in \mathbb{R}^n,\quad t\in \mathbb{R}^1,\\ u\equiv u(t,x),\quad \phi(u)\equiv(\phi_1(u),\dots,\phi_n(u))\in \mathbb{R}^n,\quad \phi_i(u)\in C(\mathbb{R}^1) \end{gathered} NEWLINE\]NEWLINE in the layer \((0,T]\times \mathbb{R}^n \) with the initial condition \(u(0,x) = u_0(x)\in L_\infty(\mathbb{R}^n)\). It is assumed that the main uniqueness condition characterizing ``total smoothness'' of flow functions has the form of the integral condition of Osgood type known in the theory of ODE.