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Kernel sections for processes and nonautonomous lattice systems. (English) Zbl 1149.37040

The authors investigate the asymptotic behaviour of nonautonomous dynamical systems. That is, one is given a process on a Banach space \(X\), i.e. a family of mappings
\[ U(t,\tau): X\to X,\quad t\geq \tau,\;\tau\in\mathbb R \]
which satisfy \(U(\tau,\tau)\)=Id, \(U(t,s)U(s,\tau)=U(t,\tau)\) and simultaneous continuity of \(U(t,\tau)x\) in \(t,\tau,x\). A function \(u\in C(X,X)\) is a complete trajectory of the process if \(u(t)=U(t,\tau)u(\tau)\) for \(t\geq\tau\). The kernel \(K\) of the process consists of all bounded, complete trajectories, while \(K(s)=\{u(s)\mid u\in K\}\) is the kernel section of \(K\) at time \(s\in\mathbb R\). Quite a number of further definitions are introduced, which will be used below without explanations. Based on these concepts, the authors prove a number of results which extend well known results, valid in the autonomous case. Theorem 3.1 eg. asserts among others that if the process \(U(t,\tau)\), \(t\geq\tau\) is pullback bounded dissipative at \(\tau\in\mathbb R\) with a bounded absorbing set \(B_\tau\subseteq X\), and if it is pullback \(\omega\)-limit compact at \(\tau\in\mathbb R\), then: (a) the pullback \(\omega\)-limit set \(\Omega(\tau,B_\tau)\) is nonempty, compact and invariant, ie. \(U(t,\tau)\Omega(\tau,B_\tau)=\Omega(t,B_\tau)\) for \(t\geq \tau\), \(\tau\in\mathbb R\). Likewise, Theorem 3.2 asserts that if \(U(t,\tau)\), \(t\geq \tau\) has a family of compact and globally pullback attractive kernel sections \(\{K(\tau)\mid t\in\mathbb R\}\), then \(U(t,\tau)\), \(t\geq \tau\) is pullback dissipative and pullback \(\omega\)-limit compact at each \(\tau\in\mathbb R\).
Subsequently, the underlying space \(X\) is identified with the Banach space \(\ell^p_\rho\) given by
\[ \ell^p_\rho=\{u=(u_i)_{i\in \mathbb Z};\;\sum_i\rho_i|u_i|^p<\infty,\;u_i\in\mathbb R^\ell\},\;p>0,\;\ell\in\mathbb N, \]
with \(\rho(i)\), \(i\in\mathbb Z\) a positive weightfunction and with norm \(\|u\|=(\sum\rho_i\|u_i\|^p)^{1/p}\). In case \(p=2\), Theorem 4.2 describes interesting bounds for the Kolmogorov \(\varepsilon\)-entropy \(K_\varepsilon(\widetilde K_0(\tau))\) of the kernel section \(\widetilde K_0(\tau)\) of the process \(U(t,\tau)\), \(t\geq \tau\), provided the latter satisfies suitable assumptions.
In the last part of the paper, the foregoing concepts and results are applied to first-order nonautonomous lattice systems, described by a sequence of equations
\[ \dot u_i+(Au)_i+f(u_i,t)=g_i(t),\quad u_i(\tau)=u_{i,\tau},\quad i\in\mathbb Z,\;t>\tau, \]
defined on \(\ell^2_\rho\) and subject to various conditions. Eg. \(A\) is selfadjoint, and of the form \(A=DD^*\) with \(D\) of the form
\[ (Du)_i=\sum^{\ell=m_0}_{\ell=-m_0}d_\ell u_{i+\ell}. \]
Further assumptions are imposed on \(f,g,\rho\). The authors now obtain a series of interesting results relative to this lattice system.

MSC:

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
34A35 Ordinary differential equations of infinite order
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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