# Semilinear NLW

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

### Semilinear wave equations

[Note: Many references needed here!]

Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form

${\displaystyle \Box \phi =F(\phi ),\Box \phi =\phi +F(\phi )}$

respectively where ${\displaystyle F}$ is a function only of ${\displaystyle f}$ and not of its derivatives, which vanishes to more than first order.

Typically ${\displaystyle F}$ is a power type nonlinearity. If ${\displaystyle F}$ is the gradient of some function ${\displaystyle V}$, then we have a conserved Hamiltonian

${\displaystyle \int {\frac {|\phi _{t}|^{2}}{2}}+{\frac {|\nabla \phi |^{2}}{2}}+V(\phi )\ dx.}$

For NLKG there is an additional term of ${\displaystyle |\phi |^{2}/2}$ in the integrand, which is useful for controlling the low frequencies of ${\displaystyle f}$ . If V is positive definite then we call the NLW defocusing; if ${\displaystyle V}$ is negative definite we call the NLW focusing.

To analyze these equations in ${\displaystyle H^{s}}$ we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that ${\displaystyle F}$ is smooth, or that ${\displaystyle F}$ is a p^th-power type non-linearity with ${\displaystyle p>[s]+1}$.

The scaling regularity is

${\displaystyle s_{c}={\frac {d}{2}}-{\frac {2}{(p-1)}}}$.

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

 Dimension d Strauss exponent (NLKG) ${\displaystyle L^{2}}$-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
${\displaystyle s_{conf}=(d+1)/4-1/(p-1)}$
in the focusing case; the defocusing case is still open. In the ${\displaystyle H^{1/2}}$-critical power or below, this condition is stronger than the scaling requirement.
• When ${\displaystyle d\geq 2}$ 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 ${\displaystyle 1/2-s<1/p}$ to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
• Finally, in three dimensions one has ill-posedness when ${\displaystyle p=2}$ and ${\displaystyle s=s_{conf}=0}$ Lb1993.
• In dimensions ${\displaystyle d\leq 3}$ the above necessary conditions are also sufficient for LWP.
• For d>4 sufficiency is only known assuming the condition
${\displaystyle p(d/4-s)\leq 1/2((d+3)/2-s)}$ (*)

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
${\displaystyle p((d+1)/4-s)\leq (d+1)/2d((d+3)/2-s)}$; (**)
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 ${\displaystyle H^{s_{c}}}$ norm when ${\displaystyle p}$ is at or above the ${\displaystyle H^{1/2}}$-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 ${\displaystyle H^{1}}$ in the defocussing case when p is at or below the ${\displaystyle H^{1}}$-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 ${\displaystyle H^{s}}$, ${\displaystyle s<1}$, is known in the following cases:

• ${\displaystyle d=3,p=3,s>3/4}$ KnPoVe-p2
• ${\displaystyle d=3,3\leq p<5,s>[4(p-1)+(5-p)(3p-3-4)]/[2(p-1)(7-p)]}$ MiaZgFg-p
• ${\displaystyle d=3,2, and
${\displaystyle s>[2(p-1)^{2}-(d+2-p(d-2))(d+1-p(d-1))]/[2(p-1)(d+1-p(d-3))]}$

[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition ${\displaystyle s_{conf}>s_{c}}$ and the condition (**).

• ${\displaystyle d=2,3\leq p\leq 5,s>(p-2)/(p-1)}$ Fo-p; this is

for the NLW instead of NLKG.

• ${\displaystyle d=2,p>5,s>(p-1)/p}$ Fo-p; this is for the NLW

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

${\displaystyle \Box u=(t^{2}+(1-(t^{2}-x^{2})/4)^{2})^{-(d-1)p/4+(d+3)/4}|u|^{p}}$;

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.