Semilinear NLW

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Semilinear wave equations

1 Semilinear wave equations
- 1.1 Necessary conditions for LWP
- 1.2 Specific 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 $| φ | 2 / 2$ 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

$s_c = \frac{d}{2} - \frac{2}{(p-1)}$ . 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

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Necessary conditions for LWP

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
$s c o n f = (d + 1) / 4 - 1 / (p - 1)$
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 $s = s c o n f = 0$ 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
$d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)$ , 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 $s c o n f > s c$ 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.

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