Semilinear NLW

Semilinear wave equations

[Note: Many references needed here!]

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

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

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

Typically $F$ is a power type nonlinearity. If $F$ is the gradient of some function $V$, then we have a conserved Hamiltonian

$\int |{\frac {\phi _{t}|^{2}}{2}}+|{\frac {\nabla \phi |^{2}}{2}}+V(\phi )\ dx.$

For NLKG there is an additional term of $|\phi |^{2}/2$ in the integrand, which is useful for controlling the low frequencies of $f$ . If V is positive definite then we call the NLW defocusing; if $V$ is negative definite we call the NLW focusing.

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

Necessary conditions for LWP

The following necessary conditions for LWP are known.

1. 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.

1. 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 Kp1993.

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:

• $d=3,p=3,s>3/4$ 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.

A substantial scattering theory for NLW and NLKG is known.

The non-relativistic limit of NLKG has attracted a fair amount of research.

Specific semilinear wave equations

• Sine-Gordon
• Quadratic NLW/NLKG
• Cubic NLW/NLKG (on R, on R^2, on R^3, and on R^4)
• Quartic NLW/NLKG
• Quintic NLW/NLKG (on R, on R^2, and on R^3)
• Septic NLW/NLKG (on R, on R^2, and on R^3)