Celebratio Mathematica

F. Cagnac and Y. Choquet-Bruhat: “Solution globale d’une équation non linéaire sur une variété hyperbolique,” C. R. Acad. Sci. Paris Sér. I Math. 296 : 20 (1983), pp. 845–849. A paper by the same title was subsequently published in J. Math. Pures Appl. 63:4 377–390 (1984). MR 712755 Zbl 0529.58032 article

The authors prove the existence of a global, unique solution for the equation \[ \square_g u + au + bu^3=0 \] on a globally hyperbolic 4-dimensional manifold \( (V,g) \) with given Cauchy data on a space-like section \( S \).The case when \( a \) and \( b \) are given positive functions on \( V \) is considered. First, the definitions and the hypothesis used in the work are given. The space \( (V,g) \) is organized as a global hyperbolic manifold and some regularity hypotheses are introduced. Secondly, a local theorem regarding the existence of a solution for the linear equation \[ \square_g u + au = f \] is given. The existence of the solution is proved using the energy majoring inequalities. Then it is shown that the equation \[ \square_g u + au + bu^3 = 0 \] admits a unique solution with the Cauchy data and the hypotheses imposed. Finally, the existence of a global solution for the equation \[ \square_g u + au + bu^3 = 0 \] is proved. The existence theorem is proved using an a priori condition of majoring and also the lemmas and theorems proven previously in the work.

F. Cagnac and Y. Choquet-Bruhat: “Solution globale d’une équation non linéaire sur une variété hyperbolique,” J. Math. Pures Appl. (9) 63 : 4 (1984), pp. 377–390. MR 789558 Zbl 0554.58049 article

F. Cagnac, Y. Choquet-Bruhat, and N. Noutchegueme: “Solutions of the Einstein equations with data at past infinity,” pp. 35–54 in Seventh Italian conference on general relativity and gravitational physics (Rapallo, 1986). Edited by U. Bruzzo, R. Cianci, and E. Massa. World Sci. Publ. (Singapore), 1987. MR 892547 incollection

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