Scattering theory and large time asymptotics of solutions to the Hartree type equations with a long range potential
Scattering theory and large time asymptotics of solutions to the Hartree type equations with a long range potential
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长程势Hartree型方程解的散射理论和大时间渐近性
DOI：
10.14492/hokmj/1350911928
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发表时间：
2001
影响因子：
0.5
通讯作者：
P. Naumkin
中科院分区：
文献类型：
--
作者：
N. Hayashi;P. Naumkin
. We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations where the nonlinear interaction term is f( |u|^{2})=V*|u|^{2} , V(x)= \lambda|x|^{-\delta} , \lambda \in R , 0 <\delta<1 . We suppose that in the case n \geq 2 the initial data u _{0}\in H^{n+2,0}\cap H^{0,n+2} and the value \epsilon=||u0||_{H^{n+2,0}}+||u0||_{H^{0,n+2}} is sufficiently small and in one-dimensional case (n=1) we assume that e ^{\beta|x|}u_{0}\in L^{2} , \beta>0 and the value \epsilon=||e^{\beta|x|}u_{0}||_{L^{2}} is sufficiently small. Then we prove that there exists a unique final state \hat{u}+\in H^{n+2,0} such that the asymptotics <\infty\} , m , s \in R . Analogous results are obtained for the following NLS equation with cubic nonlinearity and growing with time coefficient, where 0 <\delta<1 , n \geq 1 .
. We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations where the nonlinear interaction term is f( |u|^{2})=V*|u|^{2} , V(x)= \lambda|x|^{-\delta} , \lambda \in R , 0 <\delta<1 . We suppose that in the case n \geq 2 the initial data u _{0}\in H^{n+2,0}\cap H^{0,n+2} and the value \epsilon=||u0||_{H^{n+2,0}}+||u0||_{H^{0,n+2}} is sufficiently small and in one-dimensional case (n=1) we assume that e ^{\beta|x|}u_{0}\in L^{2} , \beta>0 and the value \epsilon=||e^{\beta|x|}u_{0}||_{L^{2}} is sufficiently small. Then we prove that there exists a unique final state \hat{u}+\in H^{n+2,0} such that the asymptotics <\infty\} , m , s \in R . Analogous results are obtained for the following NLS equation with cubic nonlinearity and growing with time coefficient, where 0 <\delta<1 , n \geq 1 .