Discontinuous solutions of nonlinear mixed problems for hyperbolic equations on a~plane

V. N. Gol'dberg

Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 159-180

Résumé

In $\overline\Pi_{\frac12}=\{0\leqslant x\leqslant1,\ 0\leqslant t\leqslant\frac12\}$ we consider the problem \begin{gather} u_{xx}-u_{tt}=A_1(x,t,u)u_x+A_2(x,t,u)u_t+A_3(x,t,u), \\ u(x,0)=\varphi_0(x),\quad u_t(x,0)=\varphi_1(x)\quad\text{for}\quad0\leqslant x\leqslant1, \\ a_i(u)u_x+b_i(u)u_t=f_i(t,u)\quad\text{for}\quad x=i\enskip(i=0,1), \end{gather} here $A_j, \varphi_i, a_i, b_i$ and $f_i$ are sufficiently smooth functions, $h_0=b_0-a_0$ has only isolated zeros on $R^1$, and $h_1=b_1+a_1$ does not have zeros on $R^1$. It is assumed that in $\Pi_{T^*}=\{0\leqslant x\leqslant 1,\ 0\leqslant t$, $0$, there exists a solution $\mathring u\in C_2(\Pi_{T^*})$ of problem (1)–(3), where $\sup|\mathring u|\infty$, $|h_0(\mathring u(0,t))|>0$ for $0\leqslant t $, and $\inf_{0\leqslant t$. It is shown that $\mathring u\in C(\overline\Pi_{T^*})$ and $\mathring v=\mathring u_x+\mathring u_t\in C(\overline\Pi_{T^*})$, and that if $\Gamma_0=f_0(T^*,u(0,T^*))-a_0(\mathring u(0,T^*))\mathring u(0,T^*)\ne0$, there are not even continuous generalized solutions of problem (1)–(3) in $\overline\Pi_T$ for any $T$, $T^*$. For $\Gamma_0\ne0$ the author introduces a definition and establishes existence and uniqueness theorems for the discontinuous solution of (1)–(3) in $\overline\Pi_{\frac12}$. Bibliography: 9 titles.