Errors in approximate solutions of Cauchy's problem for a first-order quasilinear equation

The proximity is investigated of the solution of Cauchy's problem for the equation $u_t^\varepsilon+(\varphi(u^\varepsilon))_x=\varepsilon u_{xx}^\varepsilon$ ($\varphi''(u^\varepsilon)>0$) to the solution of Cauchy's problem for the equation $u_t+(\varphi(u))_x=0$, when the solution of the latter problem has a finite number of lines of discontinuity in the strip $0\leqslant t\leqslant T$. It is proved that, everywhere outside a fixed neighborhood of the lines of discontinuity, we have $|u^\varepsilon-u|\leqslant C\varepsilon$, where the constant $C$ is independent of $\varepsilon$. Similar inequalities are derived for the first derivatives of $u^\varepsilon-u$.