Equipartition of energy for an illposed system with cross friction (Q1983213)

Let $H$ be a complex Hilbert space, $S:D(S)\subset H\to H$ be an unbounded selfadjoint operator such that $S=S^*\geq\varepsilon I, \varepsilon>0$, $\sigma(S)$ denote the spectrum of $S$, and $F$ is a real, nonnegative Borel function on $\sigma(S)$ and $F(S)$ is the corresponding selfadjoint operator defined by the functional calculus for $S$. Using previous results of the authors, the properties of the coupled system of differential-operator equations \[ \begin{cases} u_{tt}+2F(S)v_t+S^2u=0\\ v_{tt}+2F(S)u_t+S^2v=0 \end{cases} \] are investigated. Sufficient conditions have been established for the parameters of the problem at which the total energy of the system $E(t)=K(t)+P(t)$ has a finite boundary $\lim_{t\to\infty}E(t)=E_\infty$ and $\lim_{t\to\infty}K(t)=\lim_{t\to\infty}P(t)=\frac12 E_\infty$, where $K(t)$ - kinetic and $P(t)$ - potential energies of the system.