On the asymptotics of the solution of a~problem with a~small parameter

A. M. Il'in

Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 261-279

Résumé

The problem $\partial_tu+\partial_x\varphi(u)=\varepsilon\partial_x^2u$, $u(x,t_0)=\psi(x)$, is considered, where $\varphi,\psi\in C^\infty$, $\varphi''(u)>0$, $0\leqslant\varepsilon\ll1$. It is assumed that for $\varepsilon=0$ the problem has a generalized solution with one smooth line of discontinuity, so that this line, modeling a shock wave, appears within the strip $\Omega=\{t_0\leqslant t\leqslant T\}$. The asymptotics of a solution, uniform in $\Omega$ up to any degree in $\varepsilon$, is constructed and justified. Bibliography: 18 titles.