Scattering for NLW/NLKG

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

The Strauss exponent

$p_{0}(d)=[d+2+{\sqrt {d^{2}+12d+4}}]/2d$

plays a key role in the GWP and scattering theory. We have $p_{0}(1)=[3+{\sqrt {17}}]/2$; $p_{0}(2)=1+sqrt(2);p_{0}(3)=2$; note that $p_{0}(d-1)$ is always between the $L^{2}$ and $H^{1/2}$ critical powers, and $p_{0}(d)$ is always between the $H^{1/2}$ and $H^{1}$ critical powers.

Another key power is

$p_{*}(d)=[d+2+sqrt(d^{2}+8d)]/2(d-1)$

which lies between the $L^{2}$ critical power and $p_{0}(d-1)$.

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 $H^{1}$-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the $L^{2}$ norm (even with conservation of energy).

Scattering for small $H^{1}$ data for arbitrary NLW:

• Known for $p_{*}(d)<p\leq p_{H^{1/2}}$ Sr1981.
• For $p<p_{0}(d-1)$ one has blow-up Si1984.
• When $d=3$ this is extended to $5/2<p\leq p_{H^{1/2}}$, but scattering fails for $p<5/2$ [Hi-p3]
• When $d=4$ this is extended to $p_{0}(d-1)=2<p<5/2$, 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 $H^{1}$ data for defocussing NLW:

• Known for $p_{H^{1/2}}<p\leq p_{H^{1}}$ BaSa1998, BaGd1997 (GWP was established earlier in GiVl1987).
• Known for $p=p_{H^{1/2}}$, $d=3$ BaeSgZz1990
• When $d=3$ this is extended to $p_{*}(3)<p\leq p_{H^{1/2}}$ [Hi-p3]
• When $d=4$ this is extended to $p_{*}(4)<p<5/2$ [Hi-p3]
• For $d>4$ one expects scattering when $p_{0}(d-1)<p\leq p_{H^{1/2}}$, but this is not known.

Scattering for small smooth compactly supported data for arbitrary NLW:

• GWP and scattering when $p>p_{0}(d-1)$ GeLbSo1997
  • For $d=3$ this is in Jo1979
• Blow-up for arbitrary nonzero data when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p < p_0(d-1)} Si1984 (see also Rm1987, JiZz2003
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=4} this is in Gs1981b
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=3} this is in Jo1979
• At the critical power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = p_0(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} data for arbitrary NLKG:

• Decay estimates are known when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_0(d) < p \leq p_{L^2}} MsSrWa1980, Br1984, Sr1981, Pe1985.
• Known when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{L^2} \leq p \leq p_{H^1}} Na1999c, Na1999d, [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.

Scattering for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} data for defocussing NLKG:

• In this case one has an a priori Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} bound and one does not need decay at spatial infinity.
• Scattering is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{L^2} < p \leq p_{H^1}} Na1999c, Na1999d, [Na-p5]
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d>2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} not $H^{1}$-critical this is in Br1985 GiVl1985b
  • The $L^{2}$-critical case $p=p_{L^{2}}$ 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 $p\leq 1+2/d$ 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 $R^{3}$] 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