Cubic NLW/NLKG on R

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  • Scaling is .
  • LWP for by energy estimates and Sobolev (solution is in ).
    • For one has ill-posedness (CtCoTa-p2), indeed it is not even possible to make sense of solutions in the distributional sense.
  • GWP for for defocussing NLKG (Bo1999)
    • For this is clear from energy conservation (for both NLKG and NLW).
    • Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
    • In the focussing case there is blowup from large data by the ODE method.
  • Remark: NLKG can be viewed as a symplectic flow with the symplectic form of . NLW is similar but with the homogeneous .
  • Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see De2001, LbSf-p.