REMARKS ON SCATTERING THEORY AND LARGE TIME ASYMPTOTICS OF SOLUTIONS TO HARTREE TYPE EQUATIONS WITH A LONG RANGE POTENTIAL

Abstract

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations

$$\{\begin{array}{c}i{u}_t=-\frac{1}{2}\mathrm{\Delta }u+f(|u{|}^2)u,(t,x)\in \mathbf{R}\times {\mathbf{R}}^n,\\ u(0,x)=u_0(x),x\in {\mathbf{R}}^n,n\ge 2,\end{array}$$

where the nonlinear interaction term is $f(|u{|}^2)=V\ast |u{|}^2,V(x)=\lambda |x{|}^{-\delta }$, . We suppose that the initial data $u_0\in H^{0,l}$ and the value $ϵ={\Vert u_0\Vert }_{H^{0,l}}$ is sufficiently small, where $l$ is an integer satisfying $l\ge \left[\frac{n}{2}\right]+3$, and $\left[s\right]$ denotes the largest integer less than $s$. Then we prove that there exists a unique final state $u+\in H^{0,l-2}$ such that for all

$$u(t,x)=\frac{1}{{(it)}^{\frac{n}{2}}}\widehat{u}+(\frac{x}{t})\exp (\frac{i{x}^2}{2t}-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t})+O(1+t^{1-2\delta }))+O(t^{-n/2-\delta })$$

uniformly with respect to $x\in {\mathbf{R}}^n$ with the following decay estimate ${\Vert u(t)\Vert }_{L^p}\le Cϵt^{\frac{n}{p}-\frac{n}{2}}$, for all $t\ge 1$ and for every $2\le p\le \infty$. Furthermore we show that for there exists a unique final state $u_{+}\in H^{0,l-2}$ such that for all $t\ge 1$

$$\Vert u(t)-\exp (-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t}))U(t)u_{+}{\Vert }_{L^2}=o(t^{1-2\delta })$$

and uniformly with respect to $x\in {\mathbf{R}}^n$

$$u(t,x)=\frac{1}{{(it)}^{\frac{n}{2}}}\widehat{u}+(\frac{x}{t})\exp (\frac{i{x}^2}{2t}-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t}))+O(t^{-n/2+1-2\delta }),$$

where $\widehat{\varphi }$ denotes the Fourier transform of the function . In [5] we assumed that $u_0\in H^{m,0}\cap H^{0,m},(m=n+2)$, and showed the same results as in this paper. Here we show that we do not need regularity conditions on the initial data by showing the local existence theorem in lower order Sobolev spaces.