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A (concentration-)compact attractor for high-dimensional nonlinear Schrödinger equations. (English) Zbl 1142.35088

Asymptotic behaviour of large data solutions to nonlinear Schrödinger (NLS) equation
\[ i\partial u/\partial t+\Delta u= F(u)\quad (u= u(x,t);\;t\geq 0,\;x\in\mathbb{R}^d) \]
is thoroughly analyzed, in particular the existence of an attractor is established.
Hypotheses: \(d\geq 5\), \(F(z)= G'(|z|^2)z\), where \(G(0)= 0\); power type inequalities
\[ |F(z)|\leq|z|^p,\quad |F'(z)|\leq|z|^{p-1},\quad |F'(z)- F'(w)|\leq C|z- w|^\vartheta(|z|+|w|)^{p- 1-\vartheta}, \]
where \(0<\vartheta\leq \min(p- 1,1)\) and \(p> 1\); the mass-supercriticality and energy subcriticality \(1+ 4/d< p< 1+ 4/(d- 2)\); and the globally bounded energy norm \[ E_x(u)= \int(|u|^2+ |\nabla u|^2)\,dx \] on every compact interval of time variable.
In the spherically symmetrical case, there exists a compact set \(K\) which is invariant under the NLS flow and such that for every spherically symetric forward-global solution \(u\) of energy at most \(E\)
\[ \lim_{t\to\infty} \text{dist}(u(t)- e^{it\Delta}u_+, K)= 0 \]
for a unique radiation state \(u_+\). Alternatively this is expressed by
\[ u(t)= e^{it\Delta} u_++ w(t)+ 0, \]
i.e., the sum of a radiation term satisfying the linear Schrödinger equation and the remainder \(w(t)\) which is pseudoperiodically involved in \(K\). More complicated result with certain precompact translation-invariant attractor \(K\) is obtained even for the non-spherically symmetric case, namely, it is shownthat
\[ u(t_n)= e^{it_n\Delta} u_++ \sum \tau_{j,n} w_j+ 0\quad(\text{finite sum}) \]
valid for a given sequence \(t_n\to\infty\), where \(\tau_{j,n}\) are shift operators and \(w_j\in K\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35B15 Almost and pseudo-almost periodic solutions to PDEs
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