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

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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. 
@article{SM_1972_17_4_a9,
     author = {A. S. Kalashnikov},
     title = {A~problem without initial conditions for linear degenerate hyperbolic equations of second order with infinite domain of dependence},
     journal = {Sbornik. Mathematics},
     pages = {603--616},
     publisher = {mathdoc},
     volume = {17},
     number = {4},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_17_4_a9/}
}
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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/