Geodesic
First Darboux Problem for Nonlinear Hyperbolic Equations of Second Order
Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 693-712

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We study the first Darboux problem for hyperbolic equations of second order with power nonlinearity. We consider the question of the existence and nonexistence of global solutions to this problem depending on the sign of the parameter before the nonlinear term and the degree of its nonlinearity. We also discuss the question of local solvability of the problem.
Keywords: first Darboux problem, nonlinear hyperbolic equation of second order, integral equation of Volterra type, Green–Hadamard function, Leray–Schauder theorem. 
@article{MZM_2008_84_5_a5,
     author = {O. M. Dzhokhadze and S. S. Kharibegashvili},
     title = {First {Darboux} {Problem} for {Nonlinear} {Hyperbolic} {Equations} of {Second} {Order}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {693--712},
     publisher = {mathdoc},
     volume = {84},
     number = {5},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_84_5_a5/}
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O. M. Dzhokhadze; S. S. Kharibegashvili. First Darboux Problem for Nonlinear Hyperbolic Equations of Second Order. Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 693-712. http://geodesic.mathdoc.fr/item/MZM_2008_84_5_a5/