Scattering for NLW/NLKG

From DispersiveWiki

The Strauss exponent

plays a key role in the GWP and scattering theory. We have ; p₀(2) = 1 + sqrt(2);p₀(3) = 2; note that p₀(d − 1) is always between the L² and H^{1 / 2} critical powers, and p₀(d) is always between the H^{1 / 2} and H¹ critical powers.

Another key power is

p _* (d) = [d + 2 + sqrt(d² + 8d)] / 2(d − 1)

which lies between the L² critical power and p₀(d − 1).

Caveats: the d = 1,2 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 H¹-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the L² norm (even with conservation of energy).

Scattering for small H¹ data for arbitrary NLW:

• Known for Sr1981.
• For p < p₀(d − 1) one has blow-up Si1984.
• When d = 3 this is extended to , but scattering fails for p < 5 / 2 Hi-p3
• When d = 4 this is extended to p₀(d − 1) = 2 < p < 5 / 2, but scattering fails for p < 2 Hi-p3
• An alternate argument based on conformal compactification but giving slightly different results are in BcKkZz1999

Scattering for large H¹ data for defocusing NLW:

• Known for BaSa1998, BaGd1997 (GWP was established earlier in GiVl1987).
• Known for , d = 3 BaeSgZz1990
• When d = 3 this is extended to Hi-p3
• When d = 4 this is extended to p _* (4) < p < 5 / 2 Hi-p3
• For d > 4 one expects scattering when , but this is not known.

Scattering for small smooth compactly supported data for arbitrary NLW:

• GWP and scattering when p > p₀(d − 1) GeLbSo1997
  • For d = 3 this is in Jo1979
• Blow-up for arbitrary nonzero data when p < p₀(d − 1) Si1984 (see also Rm1987, JiZz2003
  • For d = 4 this is in Gs1981b
  • For d = 3 this is in Jo1979
• At the critical power p = p₀(d − 1) there is blowup for non-negative non-trivial data YoZgq-p2
  • For d = 2,3 and arbitrary nonzero data this is in Scf1985
  • For large data and arbitrary d this is in Lev1990

Scattering for small H¹ data for arbitrary NLKG:

• Decay estimates are known when MsSrWa1980, Br1984, Sr1981, Pe1985.
• Known when Na1999c, Na1999d, Na2001. Indeed, one has existence of wave operators and asymptotic completeness in these cases.

Scattering for large H¹ data for defocussing NLKG:

• In this case one has an a priori L² bound and one does not need decay at spatial infinity.
• Scattering is known for Na1999c, Na1999d, Na2001
  • For d > 2 and p not H¹-critical this is in Br1985 GiVl1985b
  • The L²-critical case is an interesting open problem.

Scattering for small smooth compactly supported data for arbitrary NLKG:

• GWP and scattering for p > 1 + 2 / d when d = 1,2,3 LbSo1996
  • When d = 1,2 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 p < 1 + 2 / d KeTa1999. The endpoint p = 1 + 2 / d 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 R³ one has global regularity for convex obstacles SmhSo1995, and for smooth non-linearities there is the 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, Nic1996, BluSf2003