Equipartition of energy for an illposed system with cross friction

Abstract

Given a complex Hilbert space $H$, we establish a result on asymptotic energy equipartition for the abstract coupled system

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ u_{tt}+2F\left(S\right)v_t+S^{2}u=0\hfill \\ \hfill \\ \hfill \\ v_{tt}+2F\left(S\right)u_t+S^{2}v=0\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

for $\left(u,v\right):\left[0,\infty \right)\to H\oplus H$ with selfadjoint $S:D\left(S\right)\to H$ and operator-valued damping $F\ge 0$. Both the kinetic and the potential energies of solutions contain interaction terms in the general case and are conveniently weighted to account for the presence of damping. A remarkable feature of the above system is that it is wellposed if and only if $F$ is bounded, a fact that we also prove.