Geodesic
The Cauchy problem with modified initial data for the generalized Euler–Poisson–Darboux equation
Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 141-157 
F. T. Baranovskii. The Cauchy problem with modified initial data for the generalized Euler–Poisson–Darboux equation. Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 141-157. http://geodesic.mathdoc.fr/item/SM_1984_48_1_a6/
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     author = {F. T. Baranovskii},
     title = {The {Cauchy} problem with modified initial data for the generalized {Euler{\textendash}Poisson{\textendash}Darboux} equation},
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     url = {http://geodesic.mathdoc.fr/item/SM_1984_48_1_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the equation $$ \varphi(y-\tau(x))\frac{\partial^2u}{\partial x\partial y}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}+c(x,y)u=f(x,y), $$ where $\varphi(t)$ is an increasing function with $\varphi(0)=0$, consider the Cauchy problem in different formulations determined by specifying the initial data in various forms on the curve $y=\tau(x)$. It is proved that the problems considered are uniquely solvable. Bibliography: 12 titles.

[1] Bitsadze A. V., “K teorii odnogo klassa uravnenii smeshannogo tipa”, Nekotorye problemy matematiki i mekhaniki, Nauka, L., 1970, 112–119

[2] Bitsadze A. V., Nakhushev A. M., “K teorii vyrozhdayuschikhsya giperbolicheskikh uravnenii v mnogomernykh oblastyakh”, DAN SSSR, 204:6 (1972), 1289–1291 | Zbl

[3] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981 | MR

[4] Weinstein A., “The generalized radiation problem and the Euler–Poisson–Darboux equation”, Summa brasil. math., 3:7 (1955), 126–147 | MR

[5] Blum E. K., “The solutions of the Euler–Poisson–Darboux equation for negative values of the parameter”, Duke Math., 21:2 (1954), 257–270 | DOI | MR

[6] Tersenov S. A., “O singulyarnoi zadache Koshi”, DAN SSSR, 196:5 (1971), 1032–1035 | MR | Zbl

[7] Gavrilova N. V., Yurchuk N. I., “Zadacha Koshi dlya differentsialno-operatornykh uravnenii tipa Eilera–Puassona–Darbu”, Diff. uravneniya, 17:5 (1981), 789–795 | MR | Zbl

[8] Baranovskii F. T., “O zadache Koshi dlya giperbolicheskogo uravneniya vtorogo poryadka s osobennostyu v koeffitsiente”, UMN, 17:2 (1962), 167–174 | MR | Zbl

[9] Baranovskii F. T., “O zadache Koshi dlya giperbolicheskogo uravneniya vtorogo poryadka s osobennostyu v koeffitsiente (sluchai $1\alpha2$)”, Diff. uravneniya, 16:8 (1970), 1459–1466 | MR

[10] Baranovskii F. T., “Zadacha Koshi giperbolichnogo rivnyannya drugogo poryadku, scho silno virodzhuetsya”, DAN USSR, seriya A, 1971, no. 1, 11–16 | Zbl

[11] Protter M. H., “The Gauchy problem for a hyperbolic second order equation with data the parabolic line”, Canad. J. Math., 6:4 (1954), 542–553 | MR | Zbl

[12] Kurant R., Fridrikhs K., Levi G., “O raznostnykh uravneniyakh matematicheskoi fiziki”, UMN, 1941, no. 8, 125–160