Semilinear NLW

Semilinear wave equations

[Note: Many references needed here!]

Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form

$\Box f=F(f),\Box f=f+F(f)$

respectively where $F$ is a function only of $f$ and not of its derivatives, which vanishes to more than first order.

Typically $F$ grows like $|f|^{p}$ for some power $p$. If $F$ is the gradient of some function $V$, then we have a conserved Hamiltonian

$\int |f_{t}|^{2}/2+|\nabla f|^{2}/2+V(f)\ dx.$

For NLKG there is an additional term of $|f|^{2}/2$ in the integrand, which is useful for controlling the low frequencies of $f$ . If V is positive definite then we call the NLW defocussing; if $V$ is negative definite we call the NLW focussing. The term "coercive" does not have a standard definition, but generally denotes a potential $V$ which is positive for large values of $f$ .

semilinear NLW

To analyze these equations in $H^{s}$ we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that $F$ is smooth, or that $F$ is a p^th-power type non-linearity with $p>[s]+1$.

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

Dimension d	Strauss exponent (NLKG)	$L^{2}$-critical exponent	Strauss exponent (NLW)	H^{1/2}-critical exponent	H^1-critical exponent
1	3.56155...	5	infinity	infinity	N/A
2	2.41421...	3	3.56155...	5	infinity
3	2	2.33333...	2.41421...	3	5
4	1.78078...	2	2	2.33333...	3

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

$s_{conf}=(d+1)/4-1/(p-1)$

in the focusing case; the defocusing case is still open. In the $H^{1/2}$-critical power or below, this condition is stronger than the scaling requirement.

• When $d\geq 2$ 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 $1/2-s<1/p$ to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.

Finally, in three dimensions one has ill-posedness when $p=2$ and $s=s_{conf}=0$ Lb1993.

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

$p(d/4-s)\leq 1/2((d+3)/2-s)$ (*)

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

$p((d+1)/4-s)\leq (d+1)/2d((d+3)/2-s)$; (**)

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 $H^{s_{c}}$ norm when $p$ is at or above the $H^{1/2}$-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 $H^{1}$ in the defocussing case when p is at or below the $H^{1}$-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 $H^{s}$, $s<1$, is known in the following cases:

• $d=3,p=3,s>3/4$ references:KnPoVe-p2 KnPoVe-p2
• $d=3,3\leq p<5,s>[4(p-1)+(5-p)(3p-3-4)]/[2(p-1)(7-p)]$ [MiaZgFg-p]
• $d=3,2<p<3,orn\geq 4,(d+1)^{2}/((d-1)^{2}+4)\leq p<(d-1)/(d-3)$, and

$s>[2(p-1)^{2}-(d+2-p(d-2))(d+1-p(d-1))]/[2(p-1)(d+1-p(d-3))]$

[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition $s_{conf}>s_{c}$ and the condition (**).

• $d=2,3\leq p\leq 5,s>(p-2)/(p-1)$ [Fo-p]; this is for the NLW instead of NLKG.
• $d=2,p>5,s>(p-1)/p$ [Fo-p]; this is for the NLW instead of NLKG.

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

$\Box u=(t^{2}+(1-(t^{2}-x^{2})/4)^{2})^{-(d-1)p/4+(d+3)/4}|u|^{p}$;

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

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Scattering theory for semilinear NLW

[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 $p<p_{0}(d-1)$ Si1984 (see also Rm1987, JiZz2003
  • For $d=4$ this is in Gs1981b
  • For $d=3$ this is in Jo1979
• At the critical power $p=p_{0}(d-1)$ 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 $H^{1}$ data for arbitrary NLKG:

• Decay estimates are known when $p_{0}(d)<p\leq p_{L^{2}}$MsSrWa1980, Br1984, Sr1981, Pe1985.
• Known when $p_{L^{2}}\leq p\leq p_{H^{1}}$ Na1999c, Na1999d, [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.

Scattering for large $H^{1}$ data for defocussing NLKG:

• In this case one has an a priori $L^{2}$ bound and one does not need decay at spatial infinity.
• Scattering is known for $p_{L^{2}}<p\leq p_{H^{1}}$ Na1999c, Na1999d, [Na-p5]
  • For $d>2$ and $p$ 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

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Non-relativistic limit of NLKG

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

$u_{tt}/c^{2}-Du+c^{2}u+f(u)=0$.

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

$t=\pm c{\sqrt {c^{2}+x^{2}}}$.

In the non-relativistic limit this becomes two paraboloids

$t=\pm (c^{2}+x^{2}/2)$

and so one expects $u$ to resolve as

$u=exp(ic^{2}t)v_{+}+exp(-ic^{2}t)v_{-}$ $u_{t}=ic^{2}exp(ic^{2}t)v_{+}-ic^{2}exp(ic^{2}t)v_{-}$

where $v_{+}$, $v_{-}$ solve some suitable NLS.

A special case arises if one assumes $(u_{t}-ic^{2}u)$ to be small at time zero (say $o(c)$ in some Sobolev norm). Then one expects $v_{-}$ 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 $v_{+}$ and $v_{-}$ to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency $\ll c$ portion of the evolution should evolve in a Schrodinger-type manner, while the frequency $\gg c$ 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

$iv_{t}^{+}+Dv/2=uv^{+}$ $iv_{t}^{-}-Dv/2=uv^{-}$ $Du=-|v^{+}|^{2}+|v^{-}|^{2}$

under reasonable $H^{1}$ hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields $f$ , $A$, $A_{0}$ and the Schrodinger-Poisson fields u, v^+, v^- are

$A_{0}\sim u$ $f\sim exp(ic^{2}t)v^{+}+exp(-ic^{2}t)v^{-}$ $f_{t}\sim iMexp(ic^{2})v^{+}-iMexp(-ic^{2}t)v^{-}$

where $M=sqrt{c^{4}-c^{2}D}$ (a variant of $c^{2}$).

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Specific semilinear wave equations

Sine-Gordon

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