Semilinear NLW

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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 $H^{1/2}$ -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.