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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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