Geodesic
Discontinuous solutions of nonlinear mixed problems for hyperbolic equations on a plane
Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 159-180 
V. N. Gol'dberg. Discontinuous solutions of nonlinear mixed problems for hyperbolic equations on a plane. Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 159-180. http://geodesic.mathdoc.fr/item/SM_1972_16_2_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] V. E. Abolinya, A. D. Myshkis, “Smeshannaya zadacha dlya pochti lineinoi giperbolicheskoi sistemy na ploskosti”, Matem. sb., 50(92) (1960), 423–442 | MR | Zbl

[2] V. N. Goldberg, Yu. I. Neimark, “Korrektnost postanovki nelineinoi smeshannoi zadachi dlya volnovogo uravneniya na ploskosti”, Matem. sb., 67(109) (1965), 16–54 | MR | Zbl

[3] V. N. Goldberg, “Vozniknovenie razryvov pri prodolzhenii reshenii nelineinykh smeshannykh zadach dlya uravneniya struny”, DAN SSSR, 176:6 (1967), 1229–1232 | MR

[4] V. N. Goldberg, “Vozniknovenie razryvov pri prodolzhenii reshenii nelineinykh smeshannykh zadach dlya giperbolicheskikh uravnenii na ploskosti”, DAN SSSR, 182:6 (1968), 1257–1260 | MR

[5] E. Bekkenbakh, R. Bellman, Neravenstva, Nauka, Moskva, 1965

[6] Dzh. Sansone, Obyknovennye differentsialnye uravneniya, t. II, IL, Moskva, 1954

[7] A. A. Vitt, “K teorii skripichnoi struny”, Zh. teor. fiziki, VI:9 (1936), 1459–1479

[8] A. N. Tikhonov, “O sistemakh differentsialnykh uravnenii, soderzhaschikh parametry”, Matem. sb., 27(69) (1950), 147–156 | Zbl

[9] A. N. Tikhonov, “Sistemy differentsialnykh uravnenii, soderzhaschie malye parametry pri proizvodnykh”, Matem. sb., 31(73):3 (1952), 575–586 | MR | Zbl