Semilinear NLW

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To analyze these equations in we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that is smooth, or that is a p^th-power type non-linearity with .

The scaling regularity is . Notable powers of include the -critical power , the -critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1), and the -critical power .

Dimension d

Strauss exponent (NLKG)

-critical exponent

Strauss exponent (NLW)

H^{1/2}-critical exponent

H^1-critical exponent

























The following necessary conditions for LWP are known. Firstly, for focussing NLW/NLKG one has blowup in finite time for large data, as can be seen by the ODE method. One can scale this and obtain ill-posedness for any focussing NLW/NLKG in the supercritical regime s < s_c; this has been extended to the defocusing case in [CtCoTa-p2]. By using Lorentz scaling instead of isotropic scaling one can also obtain ill-posedness whenever s is below the conformal regularity

in the focusing case; the defocusing case is still open. In the -critical power or below, this condition is stronger than the scaling requirement.

  • When and 1 < p < p_{H^{1/2}} with the focusing sign, blowup is known to occur when a certain Lyapunov functional is negative, and the rate of blowup is self-similar MeZaa2003; earlier results are in AntMe2001, CafFri1986, Al1995, KiLit1993, KiLit1993b.

To make sense of the non-linearity in the sense of distributions we need s \geq 0 (indeed we have illposedness below this regularity by a high-to-low cascade, see [CtCoTa-p2]). In the one-dimensional case one also needs the condition to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.

Finally, in three dimensions one has ill-posedness when and Lb1993.

  • In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.
  • For d>4 sufficiency is only known assuming the condition


and excluding the double endpoint when (*) holds with equality and s=s_{conf} Ta1999. The main tool is two-scale Strichartz estimates.

    • By using standard Strichartz estimates this was proven with (*) replaced by
; (**)

see KeTa1998 for the double endpoint when (**) holds with equality and s=s_{conf}, and LbSo1995 for all other cases. A slightly weaker result also appears in Kp1994.

GWP and scattering for NLW is known for data with small norm when is at or above the -critical power (and this has been extended to Besov spaces; see [Pl-p4]. This can be used to obtain self-similar solutions, see [MiaZg-p2]). One also has GWP in in the defocussing case when p is at or below the -critical power. (At the critical power this result is due to Gl1992; see also SaSw1994. For radial data this was shown in Sw1988). For more scattering results, see below.

For the defocussing NLKG, GWP in , , is known in the following cases:

  • references:KnPoVe-p2 KnPoVe-p2
  • [MiaZgFg-p]
  • , and

[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition and the condition (**).

  • [Fo-p]; this is for the NLW instead of NLKG.
  • [Fo-p]; this is for the NLW instead of NLKG.

GWP and blowup has also been studied for the NLW with a conformal factor


the significance of this factor is that it behaves well under conformal compactification. See Aa2002, BcKkZz2002, Gue2003 for some recent results.

Scattering theory for semilinear NLW

[Thanks to Kenji Nakanishi for many helpful additions to this section - Ed.]

The Strauss exponent

plays a key role in the GWP and scattering theory. We have ; ; note that is always between the and critical powers, and is always between the and critical powers.

Another key power is

which lies between the critical power and .

Caveats: the cases may be somewhat different from what is stated here (partly because some of the powers here are not well-defined). Also, in many of the NLW results one needs some additional decay at spatial infinity (e.g. finiteness of the conformal energy), except in the special -critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the norm (even with conservation of energy).

Scattering for small data for arbitrary NLW:

  • Known for Sr1981.
  • For one has blow-up Si1984.
  • When this is extended to , but scattering fails for [Hi-p3]
  • When this is extended to , but scattering fails for [Hi-p3]
  • An alternate argument based on conformal compactification but giving slightly different results are in BcKkZz1999

Scattering for large data for defocussing NLW:

  • Known for BaSa1998, BaGd1997 (GWP was established earlier in GiVl1987).
  • Known for , BaeSgZz1990
  • When this is extended to [Hi-p3]
  • When this is extended to [Hi-p3]
  • For one expects scattering when , but this is not known.

Scattering for small smooth compactly supported data for arbitrary NLW:

  • GWP and scattering when GeLbSo1997
    • For this is in Jo1979
  • Blow-up for arbitrary nonzero data when Si1984 (see also Rm1987, JiZz2003
    • For this is in Gs1981b
    • For this is in Jo1979
  • At the critical power there is blowup for non-negative non-trivial data [YoZgq-p2]
    • For and arbitrary nonzero data this is in Scf1985
    • For large data and arbitrary this is in Lev1990

Scattering for small data for arbitrary NLKG:

  • Decay estimates are known when MsSrWa1980, Br1984, Sr1981, Pe1985.
  • Known when Na1999c, Na1999d, [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.

Scattering for large data for defocussing NLKG:

  • In this case one has an a priori bound and one does not need decay at spatial infinity.
  • Scattering is known for Na1999c, Na1999d, [Na-p5]
    • For and not -critical this is in Br1985 GiVl1985b
    • The -critical case is an interesting open problem.

Scattering for small smooth compactly supported data for arbitrary NLKG:

  • GWP and scattering for when LbSo1996
    • When this can be obtained by energy estimates and decay estimates.
    • In principle this extends to higher dimensions but there is a difficulty with lack of smoothness in the nonlinearity.
  • Blowup in the non-Hamiltonian case when KeTa1999. The endpoint remains open but one probably also has blow-up here.
    • Failure of scattering for was shown in Gs1973.

An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For [#nlw-5_on_R^3 NLW-5 on ] one has global regularity for convex obstacles SmhSo1995, and for smooth non-linearities there is the [#gwp_qnlw general quasilinear theory]. If one adds a suitable damping term near the obstacle then one can recover some global existence results Nk2001.

On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in BchNic1993, Nic1995, BluSf2003

Non-relativistic limit of NLKG

By inserting a parameter (the speed of light), one can rewrite NLKG as


One can then ask for what happens in the non-relativistic limit (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, should be localized near the double hyperboloid


In the non-relativistic limit this becomes two paraboloids

and so one expects to resolve as

where , solve some suitable NLS.

A special case arises if one assumes to be small at time zero (say in some Sobolev norm). Then one expects to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], Nj1990, Ts1984, [MacNaOz-p], [Na-p]. In more general situations one expects and to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency portion of the evolution should evolve in a Schrodinger-type manner, while the frequency portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).

A similar non-relativistic limit result holds for the [#mkg Maxwell-Klein-Gordon] system (in the Coulomb gauge), where the limiting equation is the coupled
Schrodinger-Poisson system

under reasonable hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields , , and the Schrodinger-Poisson fields u, v^+, v^- are

where (a variant of ).

Specific semilinear wave equations


Quadratic NLW/NLKG

Cubic NLW/NLKG on R

Cubic NLW/NLKG on R2

Cubic NLW/NLKG on R3

Cubic NLW/NLKG on R4

Quartic NLW/NLKG

Quintic NLW/NLKG on R

Quintic NLW/NLKG on R2

Quintic NLW/NLKG on R3

Septic NLW/NLKG on R

Septic NLW/NLKG on R2

Septic NLW/NLKG on R3