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Global solutions of quasilinear wave equations. (English) Zbl 1144.35035

The global existence for a class of quasilinear wave equations related to Einstein’s equations in harmonic coordinates for sufficiently small data is established. The considered equations do not satisty the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity. The decay estimates for the wave equation on a curved background, estimates for the radial characteristics and eikonal equation, the sharp decay estimates for the some class of nonlinear problem, for higher order derivatives; decay estimate for one and for more vector fields, energy estimates for the nonlinear problem are given.

MSC:

35L05 Wave equation
35L30 Initial value problems for higher-order hyperbolic equations
35L75 Higher-order nonlinear hyperbolic equations
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