View source for Scattering for NLW/NLKG

The ''Strauss exponent''

<center><math>p_0(d) = [d + 2 + \sqrt{d^2 + 12d + 4}]/2d</math></center>

plays a key role in the GWP and scattering theory. We have <math>p_0(1) = [3+\sqrt{17}]/2</math>; <math>p_0(2) = 1+sqrt(2); p_0(3) = 2</math>; note that <math>p_0(d-1)</math> is always between the <math>L^2</math> and <math>H^{1/2}</math> critical powers, and <math>p_0(d)</math> is always between the <math>H^{1/2}</math> and <math>H^1</math> critical powers.

Another key power is

<center><math>p_*(d) = [d+2 + sqrt(d^2 + 8d)]/2(d-1)</math></center>

which lies between the <math>L^2</math> critical power and <math>p_0(d-1)</math>.

'''Caveats''': the <math>d=1,2</math> 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 <math>H^1</math>-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the <math>L^2</math> norm (even with conservation of energy).

Scattering for small <math>H^1</math> data for arbitrary NLW:

* Known for <math>p_*(d) < p \leq p_{H^{1/2}}</math> [[Sr1981]].
* For <math>p < p_0(d-1)</math> one has blow-up [[Si1984]].
* When <math>d=3</math> this is extended to <math>5/2 < p \leq p_{H^{1/2}}</math>, but scattering fails for <math>p<5/2</math> [[Hi-p3]]
* When <math>d=4</math> this is extended to <math>p_0(d-1) = 2 < p < 5/2</math>, but scattering fails for <math>p<2</math> [[Hi-p3]]
* An alternate argument based on conformal compactification but giving slightly different results are in [[BcKkZz1999]]

Scattering for large <math>H^1</math> data for [[defocusing]] NLW:

* Known for <math>p_{H^{1/2}} < p \leq p_{H^1}</math> [[BaSa1998]], [[BaGd1997]] ([[GWP]] was established earlier in [[GiVl1987]]).
* Known for <math>p = p_{H^{1/2}}</math>, <math>d=3</math> [[BaeSgZz1990]]
* When <math>d=3</math> this is extended to <math>p_*(3) < p \leq p_{H^{1/2}}</math> [[Hi-p3]]
* When <math>d=4</math> this is extended to <math>p_*(4) < p < 5/2</math> [[Hi-p3]]
* For <math>d>4</math> one expects scattering when <math>p_0(d-1) < p \leq p_{H^{1/2}}</math>, but this is not known.

Scattering for small smooth compactly supported data for arbitrary NLW:

* GWP and scattering when <math>p > p_0(d-1)</math> [[GeLbSo1997]]
** For <math>d=3</math> this is in [[Jo1979]]
* Blow-up for arbitrary nonzero data when <math>p < p_0(d-1)</math> [[Si1984]] (see also [[Rm1987]], [[JiZz2003]]
** For <math>d=4</math> this is in [[Gs1981b]]
** For <math>d=3</math> this is in [[Jo1979]]
* At the critical power <math>p = p_0(d-1)</math> there is blowup for non-negative non-trivial data [[YoZgq-p2]]
** For <math>d=2,3</math> and arbitrary nonzero data this is in [[Scf1985]]
** For large data and arbitrary <math>d</math> this is in [[Lev1990]]

Scattering for small <math>H^1</math> data for arbitrary NLKG:

* Decay estimates are known when <math>p_0(d) < p \leq p_{L^2}</math>[[MsSrWa1980]], [[Br1984]], [[Sr1981]], [[Pe1985]].
* Known when <math>p_{L^2} \leq p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [[Na2001]]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.

Scattering for large <math>H^1</math> data for defocussing NLKG:

* In this case one has an a priori <math>L^2</math> bound and one does not need decay at spatial infinity.
* Scattering is known for <math>p_{L^2} < p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [[Na2001]]
** For <math>d>2</math> and <math>p</math> not <math>H^1</math>-critical this is in [[Br1985]] [[GiVl1985b]]
** The <math>L^2</math>-critical case <math>p = p_{L^2}</math> is an interesting open problem.

Scattering for small smooth compactly supported data for arbitrary NLKG:

* GWP and scattering for <math>p > 1+2/d</math> when <math>d=1,2,3</math> [[LbSo1996]]
** When <math>d=1,2</math> 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 <math>p < 1+2/d</math> [[KeTa1999]]. The endpoint <math>p=1+2/d</math> remains open but one probably also has blow-up here.
** Failure of scattering for <math>p \leq 1+2/d</math> 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 [[Quintic_NLW/NLKG_on_R3|NLW-5 on <math>R^3</math>]] one has global regularity for convex obstacles [[SmhSo1995]], and for smooth non-linearities there is the [[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]], [[Nic1996]], [[BluSf2003]]

[[Category:wave]]