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 $f$ .

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

• 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, p = 3, s > 3/4} KnPoVe-p2
• 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, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]} [MiaZgFg-p]
• 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, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)} , 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 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 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 s_{conf} > s_c} and the condition (**).

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

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 \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)