论文标题
存在非线性klein-gordon方程的对数距离的两阶波
Existence of two-solitary waves with logarithmic distance for the nonlinear Klein-Gordon equation
论文作者
论文摘要
$ \ newCommand \ normt [1] {\ left \ lvert＃1 \ right \ rvert \ rvert_ {l^2}}} \ newCommand \ normo \ normo [1] {\ left \ left \ lvert＃1 \ right \ rert \ rerver \ rerver \ rerver \ rvert \ rvert \ rvert \ newCommand \ normPro [1] {\ left \ lvert＃1 \ right \ rvert \ rvert_ {e}} $我们考虑焦点非线性klein -klein -gordon（nlkg）方程\ begin \ begin {equination*} \ partial_ partial_ partial_ {ttt} {tt} \ Mathbb {r} \ times \ Mathbb {r}^d \ end {equation*}对于$ 1 \ leq d \ leq 5 $和$ p> 2 $ p> 2 $ subcritial to $ \ dot h^1 $ norm。在本文中，我们显示了方程的解决方案$ u（t）$的存在，以使\ begin {equination*} \ normo {u（t） - \ sum_ {k = 1,2} q_k（t）} + normt {\ normt {\ partial_t} \ end {equation*}其中$ q_k（t，x）$是两个方程式的孤独波，带有translations $ z_k：\ mathbb {r} \ to \ mathbb {r}^d $ saffectigy \ ableans painking \ begin {equination {equination {equination {equination {equination {qore {equient {equient {equient {equiation*} | z_1（t） - Z_______________2（t） - Z_2（T） - Z_2（t）| \ sim 2 \ log（t）\ quad \ text {as} t \ to +\ infty。 \ end {equation*}
$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}} \newcommand\normo[1]{\left\lVert#1\right\rVert_{H^1}} \newcommand\normpro[1]{\left\lVert#1\right\rVert_{E}}$ We consider the focusing nonlinear Klein-Gordon (NLKG) equation \begin{equation*} \partial_{tt}u - Δu + u - |u|^{p-1}u = 0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^d \end{equation*} for $1\leq d\leq 5$ and $p>2$ subcritical for the $\dot H^1$ norm. In this paper we show the existence of a solution $u(t)$ of the equation such that \begin{equation*} \normo{u(t) - \sum_{k=1,2}Q_k(t)} + \normt{\partial_t u(t)} \to 0\quad \mbox{as $t\to +\infty$,} \end{equation*} where $Q_k(t,x)$ are two solitary waves of the equation with translations $z_k:\mathbb{R}\to \mathbb{R}^d$ satisfying \begin{equation*} |z_1(t) - z_2(t)| \sim 2\log(t)\quad \text{as } t\to +\infty. \end{equation*}
