Semilinear NLW

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Semilinear wave equations

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[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 Hs 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 L2-critical power p_{L^2} = 1 + 4/d, the H1 / 2-critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1), and the H1-critical power p_{H^1} = 1 + 4/{d-2}.

Dimension d

Strauss exponent (NLKG)

L2-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
    sconf = (d + 1) / 4 − 1 / (p − 1)
    in the focusing case; the defocusing case is still open. In the H1 / 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 = sconf = 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 H1 / 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 H1 in the defocussing case when p is at or below the H1-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 Hs, s < 1, is known in the following cases:

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 sconf > sc and the condition (**).

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

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