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Compact uniform attractors for dissipative lattice dynamical systems with delays. (English) Zbl 1153.37037

One is given a Banach space \(E\), a space of symbols \(\Sigma\) and an indexed process of mappings.
\[ U_\sigma(t,\tau):E\to E,\quad t\geq \tau\in\mathbb R,\;\sigma\in \Sigma\tag{1} \]
such that
\[ U_\sigma(t,s)U_\sigma(s,t)=U_\sigma(t,\tau),\quad U(\tau,\tau)=\tau d \]
where \(t\geq s\geq\tau\), \(\tau\in\mathbb R\). In several definitions, various notions such as that of an attractor for the process (1), uniform w.r.t. \(\sigma\in\Sigma\), are introduced. The space \(\Sigma\) of symbols is coupled to the process (1) via a translation semigroup \(T(h)\), \(h\geq 0\) which satisfies
\[ T(h)\Sigma=\Sigma,\quad U_\sigma(t+h,\tau+h)=U_{T(h)\sigma}(t,\tau) \]
for \(t\geq \tau\), \(\tau\in\mathbb R\), \(h\geq 0\) and \(\sigma\in\Sigma\). One now takes for \(E\) a Hilbert space \(H\) of sequences:
\[ H=\{u=(u_m)_{m\in\mathbb Z},\;u_m\in K_,\;\Sigma|u_m|^2<\infty\}. \]
Here, \(K\) is \(\mathbb C\) or \(\mathbb R\). A scalar product is given by
\[ (u,v)=\Sigma u_m\overline u_m,\;\|u\|^2=(u,u). \]
Based on this frame, the authors study the longtime behaviour of dissipative lattice dynamical systems with delay (LDS), defined on the sequence space \(H\). Their main result in this connection (Theorem 3.1) gives necessary and sufficient conditions for the existence of a compact attractor, uniform w.r.t. \(\sigma\in \Sigma\). The authors then consider a special LDS with delay, namely the Zakharov equations:
\[ \begin{aligned} & i\psi_t+\psi_{xx}+i\alpha\psi-\psi\phi=f(x,t)+g(\psi_t)\\ & \phi_{tt}+\beta\phi_t-\phi_{xx}+\gamma\phi-(|\psi|^2)_{xx}=\widetilde f(x,t)+\widetilde g(\phi_t).\end{aligned}\tag{2} \]
In the continuous case, (2) is properly interpreted in a suitable function space, whereby \(\psi_t, \phi_t\) stand for the delay, which \(g\), \(\widetilde g\) functionals acting on the delay. As to the delay, one is given \(v>0\), \(\tau<T\) and one associates with \(u:[\tau-\nu,T]\to\mathbb C\) and \(t\in[\tau,T]\) the delay \(u_t\) given by \(u_t(s)=u(t+s)\), \(s\in[-\gamma,0]\). The authors then interpret (2) in the space \(H\) of sequences. Their main theorem (Theorem 4.1) asserts that under suitable assumptions on the delay functionals \(g\), \(\widehat g\), the discretised version of (2) admits a compact attractor, uniform w.r.t. \(\sigma\in\Sigma\).
The proofs of the various theorems and lemmas require difficult computations and subtle estimates. The paper relies strongly on earlier papers by various authors.

MSC:

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems
35Q55 NLS equations (nonlinear Schrödinger equations)
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