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Cesaro asymptotic equipartition of energy in the coupled case. (English) Zbl 0973.35060

For an initial-value problem of the form \[ \partial_0 u(t)= 2\pi i Au(t),\quad t\in]0,\infty[,\quad u(0+)= u_0, \] where \(A:D(A)\subseteq H\to H\) is selfadjoint in Hilbert space \(H\), we have conservation of the energy \(|u(t)|_H=|u_0|_H\) for \(t\in [0,\infty[\). If \(A\) has an interior structure as an operator matrix \(A= (A_{ij})_{i,j=1,\dots, n}\) acting in \(H= \bigoplus^n_{k=1} H_k\), then there is a long history of results giving conditions such that the canonical components \(u_j\) of a solution \(u= u_1\oplus\cdots\oplus u_n: [0,\infty[\to H\) asymptotically contribute in time average equal amounts to the (conserved) total energy (asymptotic equipartition of energy in time average). In contrast to earlier results, where selfadjointness of \(A\) was a consequence of \(A_{ij}= A^*_{ji}\) for \(i,j= 1,\dots, n\), the author considers for \(n=2\) more general cases of selfadjointness of \(A\) including coupling between the domains of the formal matrix entries \(A_{ij}\), \(i,j= 1,2\). He gives a characterization for the case when asymptotic equipartition of energy in time average (Cesàro mean) occurs. The case of \(A\) having compact resolvent yields a particularly simple characterization of the desired equipartition of energy property. The result is illustrated by a \((1+1)\)-dimensional partial differential equation example with coupled boundary condition.

MSC:

35G10 Initial value problems for linear higher-order PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
47D03 Groups and semigroups of linear operators
34D05 Asymptotic properties of solutions to ordinary differential equations
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