Geodesic
A problem without initial conditions for linear degenerate hyperbolic equations of second order with infinite domain of dependence
Sbornik. Mathematics, Tome 17 (1972) no. 4, pp. 603-616 
A. S. Kalashnikov. A problem without initial conditions for linear degenerate hyperbolic equations of second order with infinite domain of dependence. Sbornik. Mathematics, Tome 17 (1972) no. 4, pp. 603-616. http://geodesic.mathdoc.fr/item/SM_1972_17_4_a9/
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     title = {A~problem without initial conditions for linear degenerate hyperbolic equations of second order with infinite domain of dependence},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The equation $$ \psi^2(t,x)u_{tt}+\varphi(t,x)u_t-M\biggl(t,x,\frac{\partial}{\partial x}\biggr)u=f(t,x) $$ is considered on the strip $H=(0,T]\times\mathbf R_x^n$. Here $M$ is a linear elliptic operator of the second order, and $\psi$ and $\varphi$ are nonnegative on $H$ and have a zero at least of the first order on a hyperplane $t=0$. Hence for $t=0$ we cannot give the initial values. Precise restrictions on the growth of the desired function for $|x|\to\infty$ are found guaranteeing the existence and uniqueness of a generalized solution of the problem without initial conditions. Bibliography: 11 titles.

[1] M. M. Smirnov, Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, Moskva, 1966 | MR

[2] S. A. Tersenov, “O singulyarnoi zadache Koshi”, DAN SSSR, 1971, no. 5, 1032–1035 | MR | Zbl

[3] N. P. Kulikov, “Zadacha Koshi dlya uravneniya giperbolicheskogo tipa s dannymi na linii parabolichnosti”, Volzhsk. matem. sb., 1968, no. 6, 95–103 | MR

[4] V. N. Nikolaenko, I. X. Khairullin, “Ob odnoi smeshannoi zadache dlya vyrozhdayuschegosya uravneniya giperbolicheskogo tipa”, Uchenye zapiski Kazansk. un-ta, 129:3 (1969), 131–134 | MR

[5] V. A. Eleev, “O nekotorykh kraevykh zadachakh dlya odnogo vyrozhdayuschegosya giperbolicheskogo uravneniya vtorogo roda”, Diff. uravneniya, 7:1 (1971), 25–33 | MR | Zbl

[6] A. N. Tikhonov, “Teoremy edinstvennosti dlya uravneniya teploprovodnosti”, Matem. sb., 42:2 (1935), 199–216

[7] A. S. Kalashnikov, “O lineinykh vyrozhdayuschikhsya giperbolicheskikh uravneniyakh vtorogo poryadka s beskonechnoi oblastyu zavisimosti”, Uspekhi matem. nauk, XXVI:4(160) (1971), 231–232 | MR

[8] A. S. Kalashnikov, “Zadacha bez nachalnykh uslovii v klassakh rastuschikh funktsii dlya nekotorykh lineinykh vyrozhdayuschikhsya parabolicheskikh sistem vtorogo poryadka. I, II”, Vestnik MGU, seriya matem., mekhan., 1971, no. 2, 42–48 | Zbl

[9] O. A. Oleinik, E. V. Radkevich, “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Itogi nauki. Matematicheskii analiz. 1969, 8, VINITI, Moskva, 1971, 7–252 | MR | Zbl

[10] V. I. Smirnov, Kurs vysshei matematiki, t. 5, Fizmatgiz, Moskva, 1959 | MR

[11] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, LGU, Leningrad, 1950