The null condition and global existence to nonlinear wave equations.

*(English)*Zbl 0599.35105Nonlinear systems of partial differential equations in applied mathematics, Proc. SIAM-AMS Summer Semin., Santa Fe/N.M. 1984, Lect. Appl. Math. 23, Pt. 1, 293-326 (1986).

[For the entire collection see Zbl 0579.00008.]

The global existence of solutions to the nonlinear wave equation $\square u=F(u,{u}^{\text{'}},{u}^{\text{'}\text{'}})$ in the three dimensional space case is discussed. Since John’s example on blow-up of solution to nonlinear wave equation in three dimensional space case shows that generally the global existence is not valid, the author introduces a ”null condition” on the nonlinear function F, which can ensure the existence of global solution of Cauchy problem with sufficiently small data. Meanwhile, some decay estimates of the solution are also obtained.

Reviewer: S.Chen

##### MSC:

35L70 | Nonlinear second-order hyperbolic equations |

35L15 | Second order hyperbolic equations, initial value problems |

35L05 | Wave equation (hyperbolic PDE) |

35B40 | Asymptotic behavior of solutions of PDE |