On the existence of global solutions of a coupled nonlinear Klein-Gordon equations. (English) Zbl 0637.35055

Milla Miranda, M.; Medeiros, L. A.

Funkc. Ekvacioj, Ser. Int. 30, 147-161 (1987).

We study the existence and uniqueness of weak solutions of the mixed problem for the coupled system \[ (*)\quad \partial^ 2u/\partial t^ 2-\Delta u+u-| v|^{\rho +2}| u|^{\rho} u=f_ 1;\quad \partial^ 2v/\partial t^ 2-\Delta v+v-| u|^{\rho +2}| v|^{\rho} v=f_ 2 \] in \(\Omega\times]0,T[\), where \(\Omega\) is any open set of \({\mathbb{R}}^ n\) and \(\rho\) satisfies certain conditions.
We observe that the standard energy method to obtain solutions does not work for the system (*) because the nonlinear terms are negative. We determine the existence of solutions using the method of potential well introduced by D. H. Sattinger [Arch. Ration. Mech. Anal. 30, 148- 172 (1968; Zbl 0159.391)]. So the data belong to the stability set of (*) which implies that the norms of the data are small. The uniqueness is obtained with more restrictions on \(\rho\) and n.
This paper is a continuation of our work [Ann. Mat. Pura Appl., IV. Ser. 146, 173-183 (1987; Zbl 0633.35053)], where we study the existence and uniqueness of weak solutions of the mixed problem for the system \[ \partial^ 2u/\partial t^ 2-\Delta u+| v|^{\rho +2}| u|^{\rho} u=f_ 1;\quad \partial^ 2v/\partial t^ 2-\Delta v+| u|^{\rho +2}| v|^{\rho} u=f_ 2. \]

Reviewer: M.M.Miranda

MSC:

35L70	Second-order nonlinear hyperbolic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35L20	Initial-boundary value problems for second-order hyperbolic equations
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

Klein-Gordon equations; existence; uniqueness; weak solutions; mixed problem; coupled system; method of potential well; stability set