## 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
Full Text: