论文标题
Schrödinger和Klein-Gordon方程的准线性哈密顿扰动的长期解决方案
Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori
论文作者
论文摘要
我们考虑了CubicSchrödinger和Cubic（衍生）Klein-Gordon方程的准线性，Hamiltonian的扰动。如果$ \ varepsilon \ ll1 $是初始基准的大小，我们证明解决方案的寿命严格大于本地存在时间$ \ varepsilon^{ - 2} $。更准确地说，在schrödinger方程式中，我们表明寿命至少是$ o（\ varepsilon^{ - 4}）$的顺序，在klein-gordon案例中，我们证明了解决方案至少存在$ o（\ varepsilon^egepsilon^geeq^geeq geq { - {8/3}^$ a as $ o o（\ v varepsilon^a as a $）关于klein-gordon方程，在半线性扰动的情况下，我们的结果也提出了新颖性：我们表明，寿命至少处于$ o（\ varepsilon^{ - {10/3}^ - }^ - }^ - }^ - }）$，改进的情况下，对于立方非网状和$ d \ geq4 $，一般$，一般结果，
We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time $\varepsilon^{-2}$. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order $O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and $d\geq4$, the general results in [17,24].
