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
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
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
.
Notable powers of p include the L^{2}-critical power , the H^{1 / 2}-critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1), and the H^{1}-critical power .
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 |
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_{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 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 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 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:
[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition s_{conf} > s_{c} and the condition (**).
- 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
Specific semilinear wave equations
- Sine-Gordon
- Liouville's equation
- 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)