In this post, we introduce some recent works to the single nonlinear Klein-Gordon equations in one space dimension. It should be noted that cubic nonlinear Klein-Gordon equations in one space dimension are of special interest, because the large-time behavior of the solution is actually affected by the structure of the nonlinearities, even if the data are sufficiently small, smooth and localized (see [9], [10]). Now We consider the following Cauchy problem:

$$

\text{(NLKG)}\left\{

\begin{array}{l}

(\Box + 1)u = F(u,\pa u,\pa^2 u),

\qquad (t,x) \in (0,\infty) \times \R, \\

u(0,x) = \eps f(x), \;\; \pa_t u(0,x) = \eps g(x),

\qquad x \in \R,

\end{array}

\right.

$$

where $\Box = \pa_t^2 – \pa_x^2$, $u$ is a real-valued unknown function of $(t,x) \in [0,\infty)\times \R$, and the nonlinear term $F:\R^{1+2+2}\to \R$ is assumed to be smooth and cubic near the origin, i.e.,

$$

F(u,\pa u, \pa^2 u) = O((|u|+|\pa u|+|\pa^2 u|)^3)

\quad \mbox{as}\ \ (u,\pa u, \pa^2 u)\to (0,0,0).

$$

We also suppose $f$, $g \in C_0^{\infty}(\R)$ and $\eps >0$ is small enough. We are interested in large-time behavior of the solution to $\text{(NLKG)}$, such as SDGE, time-decay, and the existence of the free profile(see [11] for these terminologies). For this problem, we can arrange the recent results by the following table:

Nonlinear Term $\boldsymbol F$ |
SDGE |
$\boldsymbol{O(t^{-1/2})}$ |
Free Profile |
Reference |
---|---|---|---|---|

$u – \sin u$ | O | O | X | Ablowitz-Segur^{1} (1981) |

$3uu_t^2 – 3uu_x^2 – u^3$ | O | O | O | Yagi^{2} (1994) |

$u_t^2 u_x + u^2 u_x + u_x^3$ | X | X | X | Yordanov^{3} (1995) |

$au^3 + O(u^4),\; a\neq 0$ | O | O | X | Georgiev-Yordanov^{4} (1996) |

$\sum_{j=1}^7 c_j F_j\;(c_j\in\R)$ | O | O | O | Moriyama^{5} (1997) |

$\sum_{j=1}^{10} c_j G_j\;(c_j\in\R)$ | O | O | O | Katayama^{6} (1999) |

$-b(\pa_t u)^3\;(b>0)$ | O | O | X | Sunagawa^{7} (2006) |

? | O | X | X |

where

\begin{align*}

&F_1 = 3uu_t^2 – 3uu_x^2 -u^3

\qquad\qquad\qquad\qquad\;\;\;

G_1=(-u^2+3u_t^2-3u_x^2)u\\

&F_2=3u_t^2 u_x-u_x^3-3u^2u_x+6uu_tu_{t,x}

\qquad\quad

G_2=(-3u^2+u_t^2-u_x^2)u_t + 2(u_tu_{xx}-u_x u_{tx})u\\

&F_3=u u_x u_{xx} -u^2 u_x +u_t^2 u_x +2uu_t u_{tx}

\qquad\,

G_3=(-u^2 +u_t^2-u_x^2)u_x + 2(u_t u_{tx} – u_x u_{xx})u

\\

&F_4=(u_t^2 – u_x^2-u^2)u_{xx}-2uu_x^2

\qquad\qquad\quad\;

G_4=(-u^2+u_t^2 -u_x^2)u_{tx} -2uu_t u_x

\\

&F_5=(u_t^2 – u_x^2-u^2)u_{tx}-2uu_t u_x

\qquad\qquad\;\;

G_5=3u^2 u_t -6uu_tu_{xx} – u_t^3 -3u_t(u_{tx}^2 -u_{xx}^2)

\\

&F_6=u_t^3-3u_x^2 u_t-3u^2 u_t-6uu_x u_{tx}

\qquad\quad\,\,

G_6=-2uu_x u_{tx} -u_t u_x^2 +u_t(u_{tx}^2 -u_{xx}^2)

\\

&F_7=u_t u_x^2+uu_t u_{xx}+2uu_x u_{tx}

\qquad\qquad\quad\;\;

G_{7}=-u_x^3 +3(u_{tx}^2 – u_{xx}^2)u_x\\

&G_8=u^3 – 2u^2 u_{xx}-3uu_t^2 +2u_t^2 u_{xx}-2u_t u_x u_{tx} -(u_{tx}^2 – u_{xx}^2)u\\

&G_9=-uu_x^2 +2(u_t u_{tx} -u_x u_{xx})u_x + (u_{tx}^2 – u_{xx}^2)u\\

&G_{10}=u^2 u_x -2uu_t u_{tx} -2uu_x u_{xx}-u_t^2 u_x -u_x (u_{tx}^2 -u_{xx}^2).

\end{align*}

Like this, there are some nonlinearities $F$ that (i) SDGE holds and the solution of $\text{(NLKG)}$ behaves like the free solution as $t \to \infty$, (ii) SDGE holds and the solution satisfies the time-decay of the free solution, while it does not behave like the free solution as $t\to \infty$, (iii) the solution blows-up in finite time. But To the author’s knowledge, there is no previous paper which asserts that SDGE holds while the solution decays strictly slower than the rate $t^{-1/2}$ in $L^\infty$. It should be noted that in the case of system, there is a class of nonlinearities that SDGE holds, but the solution decays no faster than this decay rate, if we assume the mass resonance.

**References.**

- M. Ablowitz and H. Segur,
*Solitions and the Inverse Scattering Transform*, SIAM, Philadelphia, 1981. - K. Yagi,
*Normal forms and nonlinear Klein-Gordon equations in one space dimension*, Master thesis, Waseda University, March (1994). - B. Yordanov,
*Blow-up for the one-dimensional Klein-Gordon equation with a cubic nonlinearity*(1995). - V. Georgiev and B. Yordanov,
*Asymptotic behaviour of the one-dimensional Klein-Gordon equation with a cubic nonlinearity*(1996). - K. Moriyama,
*Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension*, Differential and Integral Equations, 10 (1997), 499-520. - S. Katayama,
*A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension*, J. Math. Kyoto Univ. 39 (1999), 203-213. - H. Sunagawa,
*Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms*, J. Math. Soc. Japan, 58 (2006), 379-400. - J. M. Delort,
*Existence globale et comportement asymptotique pour l’equation de Klein-Gordon quasi lineaire a donnees petites en dimension 1*, Ann. Sci. Ec. Norm. Sup. 4e Ser. 34 (2001), 1-61. - Leun Kim, 근황 : Klein-Gordon Equation 연구 (2013년 4월 13일).
- Leun Kim, History of Some Major Works to the Klein-Gordon equations.
- Leun Kim, 1차원 클라인 골든 방정식에 대한 고찰 (Time Decay 의 관점에서).

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