Discontinuous solutions of nonlinear mixed problems for hyperbolic equations on a plane
V. N. Gol'dberg
Abstract:
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 $\Pi_{T^*}=\{0\leqslant x\leqslant 1,\ 0\leqslant t<T^*\}$,
$0<T^*<\frac12$, 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
$0\leqslant t <T^*$, and $\inf_{0\leqslant t<T^*}|h_0(\mathring u(0,t))|=0$. 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
$T$,
$T^*<T\leqslant\frac12$. 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.
UDC:
517.9
MSC: Primary
35L20,
35B99; Secondary
35D05 Received: 09.04.1970
Fulltext:
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