## Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases.(English)Zbl 1187.35238

The authors study the initial value problem in statistical physics: $$i\partial_t u(t,x)= [-\Delta+x+ V(x)] u+ g(u)$$, $$t\in\mathbb{R}$$, $$x\in\mathbb{R}^3$$, $$u(0,x)= u_0(x)$$. Here $$g:\mathbb{R}\to\mathbb{R}$$ is a real-valued odd $$C^2$$ function satisfying $$g(0)= g'(0)= 0$$, $$|g''(s)|\leq C(|s|^{\alpha_1}+ |x|^{\alpha_2})$$, $$0<\alpha_1\leq\alpha_2< 3$$, which is extended to a complex function via $$g(e^{i\theta} s)= e^{i\theta}g(s)$$.
The center manifold is formed by the collection of solutions:
$u_E(t, x)= e^{-iEt}\psi_E(x),\quad (-\Delta+ V)\psi_E+ g(\psi_E)= E\psi_E.$
Let $$\psi_E= a\psi_0+ h$$, $$a=\langle\psi_0, \psi_E\rangle$$, $$h= P_c\psi_E$$, $$P_c$$: projection onto the continuous part of the spectrum of $$H=-\Delta+ V$$. $$H_a= \{-i\partial\psi_E/\partial a_2, i\partial\psi_E/\partial a_1\}^\perp$$, $$a= a_1+ ia_2$$.
Theorem. Let $$p_1= 3+\alpha_1$$, $$p_2= 3+\alpha_2$$. One finds an $$\varepsilon_0> 0$$ such that, for $$u_0(x)$$ satisfying
$\max\{\| u_0\|_{L^{p_2'}},\,\| u_0\|_{H^1}\}\leq\varepsilon_0,$
there exists a $$C^1$$ function $$a(t):\mathbb{R}\to C$$ such that, for all $$t\in\mathbb{R}$$, $u(t,x)= \psi_E(x,t)+ \eta(t,x)\equiv a(t)\psi_0(x)+ h(a(t))+ \eta(t,x)$ holds. $$\psi_E(x,t)$$, $$t\in\mathbb{R}$$, are on the center manifold, and $$\eta(t,x)\in H_{a(t)}$$.
There exist $$\Psi_{E\pm}(x)$$ and $$\theta(t)\in C^1(\mathbb{R})$$ such that $$\lim_{t\to\pm\infty}\theta(t)= 0$$ and
$\lim_{t\to\pm\infty} \|\psi_E(x, t)- e^{-it(E\pm-\theta(t))}\psi_{E\pm}(x)\|_{H^2\cap L_\sigma^2}= 0$
hold. Radiative part $$\eta(t,x)$$ satisfies the decay estimates.
Finally, the authors try to extend the estimates of $$\| e^{-iHt}P_c u_0\|_{L^p}$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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