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We show global existence backwards from scattering data at infinity forsemilinear wave equations satisfying the null condition or the weak nullcondition. Semilinear terms satisfying the weak null condition appear in manyequations in physics. The scattering data is given in terms of the radiationfield, although in the case of the weak null condition there is an additionallogarithmic term in the asymptotic behaviour that has to be taken into account.Our results are sharp in the sense that the solution has the same spatial decayas the radiation field does along null infinity, which is assumed to decay at arate that is consistent with the forward problem. The proof uses a higher orderasymptotic expansion together with a new fractional Morawetz estimate withstrong weights at infinity.

Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition