The global Goursat problem and scattering for nonlinear wave equations

Abstract

The Goursat problem for nonlinear scalar equations on the Einstein Universe M ̃ , with finite-energy datum, has a unique global solution in the positive-energy, Sobolev-controllable case. Such equations include those of the form □ ϑ + H ′( ϑ ) = 0, where H denotes a hamiltonian that is a fourth-order polynomial, bounded below, in components of the multicomponent scalar section ϑ. In particular, the conformally invariant equation (□ + 1) ϑ + λϑ 3 = 0 ( λ ⩾ 0) is included. In the higher-dimensional analog R × S n to the Einstein Universe the same result holds under the stronger conditions on H required for Sobolev controllability. Irrespective of energy positivity, there is a unique local-in-time solution for arbitrary finite-energy Goursat datum, for all n ⩾ 3, establishing evolution from the given lightcone to any sufficiently close lightcone. These results show the existence of wave operators in the sense of scattering theory, and their continuity in the (Einstein) energy metric, for positive-energy equations of the indicated type. They also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in Minkowski space M 0 to the Goursat problem in M ̃ . In particular, any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction Lagrangian on multicomponent scalar sections, having finite Einstein energy at any one time, is asymptotic to solutions of the corresponding multicomponent free wave equation as the Minkowski time x 0 → ± ∞. Thus given a finite-Einstein-energy solution of the equation □ƒ + λƒ 3 = 0 on M 0 (λ ⩾ 0) there exist unique solutions ƒ ± of the free wave equation which approach ƒ in the Minkowski energy norm as x 0 → ± ∞, and every finite-Einstein-energy solution of the free wave equation is of the form ƒ + (or ƒ − ) for a unique solution ƒ of the nonlinear equation. This generalizes, in part in maximality sharp form, earlier results of Strauss for this equation.