Some Recent Works to One Dimensional Klein-Gordon Equations (Single Case)

Edited by Leun Kim

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-Segur1 (1981)
$3uu_t^2 – 3uu_x^2 – u^3$ O O O Yagi2 (1994)
$u_t^2 u_x + u^2 u_x + u_x^3$ X X X Yordanov3 (1995)
$au^3 + O(u^4),\; a\neq 0$ O O X Georgiev-Yordanov4 (1996)
$\sum_{j=1}^7 c_j F_j\;(c_j\in\R)$ O O O Moriyama5 (1997)
$\sum_{j=1}^{10} c_j G_j\;(c_j\in\R)$ O O O Katayama6 (1999)
$-b(\pa_t u)^3\;(b>0)$ O O X Sunagawa7 (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.

  1. M. Ablowitz and H. Segur, Solitions and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
  2. K. Yagi, Normal forms and nonlinear Klein-Gordon equations in one space dimension, Master thesis, Waseda University, March (1994).
  3. B. Yordanov, Blow-up for the one-dimensional Klein-Gordon equation with a cubic nonlinearity (1995).
  4. V. Georgiev and B. Yordanov, Asymptotic behaviour of the one-dimensional Klein-Gordon equation with a cubic nonlinearity (1996).
  5. 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.
  6. 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.
  7. H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400.
  8. 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.
  9. Leun Kim, 근황 : Klein-Gordon Equation 연구 (2013년 4월 13일).
  10. Leun Kim, History of Some Major Works to the Klein-Gordon equations.
  11. Leun Kim, 1차원 클라인 골든 방정식에 대한 고찰 (Time Decay 의 관점에서).

 

 
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.



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