Cauchy problem on a characteristic conoid for semilinear hyperbolic equations

The aim of this work is to find a semi-global solution to the Cauchy problem (P) on a characteristic conoid C-0, that is which is defined riot only neighbourhood of the sumitt O, but all around the conoid parts C-0 which shows the Cauchy data. In this respect, we will use the method of Kirchhoff's formulae constructed by Y. Choquet-Bruhat [4] and the results obtained from the linear case by F. Cagnac [2]. Here, it is shown that all solution of (P), Jive times derivable, verify as such, as its derivatives Lcp to the third order, a system of integral equations. Next this system is solved by the method of successive approximations. In this work, we do not shown that (P) has a solution. However, in a particular semilinear case (where f do not depend on the first partial derivatives), we can show that the solution of the integral system is a generalized solution of the Cauchy problem. (C) Academie des Sciences/Elsevier, Paris.