**REFERENCES**

[1] S.A. Aldashev, Correctness of multidimensional Darboux problems for the

wave equation, Ukr. Math. J., 45 (1993), 1456-1464.

[2] S.A. Aldashev, Special Darboux-Protter problems for a class of multidimensional

hyperbolic equations, Ukr. Math. J., 55 (2003), no. 1, 126-135.

[3] S.A. Aldashev, A criterion for the existence of eigenfunctions of the DarbouxProtter

spectral problem for degenerating multidimensional hyperbolic equations,

Differ. Equations, 41 (2005), no. 6, 833-839.

[4] A.K. Aziz, M. Schneider, Frankl-Morawetz problems in R3 , SIAM J. Math. Anal., 10 (1979), 913-921.

[5] Ar.B. Bazarbekov, Ak.B. Bazarbekov, Goursat and Darboux problems for the

three-dimensional wave equation, Differ. Equations, 38 (2002), 695-701.

[6] A.V. Bitsadze, Some Classes of Partial Differential Equations, Gordon and

Breach Science Publishers, New York, 1988.

[7] Jong Bae Choi, On Darboux-Protter problems for the hyperbolic equations

with Bessel operators, Far East J. Appl. Math., 5 (2001), no. 1, 75-85.

[8] Jong Bae Choi, Jong Yeoul Park, On the conjugate Darboux-Protter problems

for the two dimensional wave equations in the special case, J. Korean Math.

Soc., 39 (2002), no. 5, 681-692.

[9] D.E. Edmunds, N.I. Popivanov, A nonlocal regularization of some overdetermined

boundary value problems I, SIAM J. Math. Anal., 29 (1998), no. 1, 85-105.

[10] P. Garabedian, Partial differential equations with more than two variables in

the complex domain, J. Math. Mech., 9 (1960), 241-271.

[11] M.K. Grammatikopoulos, T. Hristov, N. Popivanov, Singular solutions to Protter’s

problem for the 3-D wave equation involving lower order terms, Electron.

J. Diff. Eqns., 2003 (2003), no. 03, 31p.; url: http://ejde.math.swt.edu

[12] M.K. Grammatikopoulos, N. Popivanov, T. Popov, New singular solutions of

Protter’s problem for the 3-D wave equation, Abstract and Applied Analysis, 2004 (2004), no. 4, 315-335.

[13] L. Hormander, The Analysis of Linear Partial Differential Operators III,

Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.

[14] G. Karatoprakliev, Uniqueness of solutions of certain boundary-value problems

for equations of mixed type and hyperbolic equations in space, Differ. Equations, 18 (1982), 49-53.

[15] S. Kharibegashvili, On the solvability of a spatial problem of Darboux type for

the wave equation, Georgian Math. J., 2 (1995), 385-394.

[16] Khe Kan Cher, On nontrivial solutions of some homogeneous boundary value

problems for the multidimensional hyperbolic Euler-Poisson-Darboux equation

in an unbounded domain, Differ. Equations, 34 (1998), 139-142.

[17] Khe Kan Cher, On the conjugate Darboux-Protter problem for the twodimensional

wave equation in the special case. Nonclassical equations in mathematical

physics (Russian) (Novosibirsk, 1998), 17-25, Izdat. Ross. Akad. Nauk

Sib. Otd. Inst. Mat., Novosibirsk, 1998.

[18] D. Lupo and K.R. Payne, Critical exponents for semilinear equations of mixed

elliptic-hyperbolic and degenerate types, Comm. Pure Appl. Math., 56 (2003), 403-424.

[19] D. Lupo and K.R. Payne, Conservation laws for equaitions of mixed elliptichyperbolic

type, Duke Math. J., 127 (2005), 251-290.

[20] D. Lupo, K.R. Payne and N. Popivanov, Nonexistence of nontrivial solutions

for supercritical equations of mixed elliptic-hyperbolic type, In: Progress in

Non-Linear Differential Equations and Their Applications, Series published

by Birkhauser, To Appear.

[21] C.S. Morawetz, Mixed equations and transonic flow, Journal of Hyperbolic

Differential Equations, 1 (2004), no. 1, 1-26.

[22] A. Nakhushev, A multidimensional analogue of the problem of Darboux for

hyperbolic equations, Sov. Math. Dokl., 11 (1970), 1162-1165.

[23] A. Nakhushev, Criteria for continuity of the gradient of the solution to the

Darboux problem for the Gellerstedt equation, Differ. Equations, 28 (1992), 1445-1457.

[24] N. Popivanov and M. Schneider, The Darboux problem in R3

for a class of

degenerated hyperbolic equations, Comptes Rend. de l’Acad. Bulg. Sci., 41 (1988), No11, 7-9.

[25] N.I. Popivanov and M. Schneider, The Darboux problems in R3 for a class of

degenerated hyperbolic equations, J. Math. Anal. Appl., 175 (1993), 537-579.

[26] N. Popivanov, M. Schneider, On M.H. Protter problems for the wave equation in R3 , J. Math. Anal. Appl., 194 (1995), 50-77.

[27] N. Popivanov, T. Popov, Exact behavior of the singularities for the 3-D Protter’s

problem for the wave equation, In: Inclusion Methods for Nonlinear

Problems with Applications in Engineering, Economics and Physics (Ed. J.

Herzberger), “Computing”, Supplement 16 (2002), 213-236.

[28] N. Popivanov, T. Popov, Estimates for the singular solutions of the 3-D Protter’s

problem, Annuaire de l’Universite de Sofia, 96 (2003), 117-139.

[29] N. Popivanov, T. Popov, Singular solutions of Protter’s problem for the 3+1-D

wave equation, Integral Transforms and Special Functiona, 15 (2004), no. 1,

73-91.

[30] M. Protter, A boundary value problem for the wave equation and mean value

problems, Annals of Math. Studies, 33 (1954), 247-257.

[31] M. Protter, New boundary value problem for the wave equation and equations

of mixed type, J. Rat. Mech. Anal., 3 (1954), 435-446.

[32] Tong Kwang-Chang, On a boundary value problem for the wave equation,

Science Record, New Series, 1 (1957), no. 1, 1-3.