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

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

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

For NLKG there is an additional term of in the integrand, which is useful for controlling the low frequencies of . If V is positive definite then we call the NLW defocusing; if is negative definite we call the NLW focusing.


To analyze these equations in we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that is smooth, or that is a p^th-power type non-linearity with .

The scaling regularity is

.

Notable powers of include the -critical power , the -critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1), and the -critical power .

Dimension d

Strauss exponent (NLKG)

-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.

  • 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
    in the focusing case; the defocusing case is still open. In the -critical power or below, this condition is stronger than the scaling requirement.
    • When 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 to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
  • Finally, in three dimensions one has ill-posedness when and Lb1993.
  • In dimensions the above necessary conditions are also sufficient for LWP.
  • For d>4 sufficiency is only known assuming the condition
(*)

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
    ; (**)
    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 norm when is at or above the -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 in the defocussing case when p is at or below the -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 , , is known in the following cases:

  • KnPoVe-p2
  • MiaZgFg-p
  • , and

[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition and the condition (**).

  • Fo-p; this is

for the NLW instead of NLKG.

  • Fo-p; this is for the NLW

instead of NLKG. GWP and blowup has also been studied for the NLW with a conformal factor

;

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