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 u vtt tt+ + 2 2 F F ((S S ) )v u t t + + S S22u v= = 0 0 for (u, v) : [0, ∞) → H H with selfadjoint S : D(S) → H and operator-valued damping F ≥ 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.

Publication Title

Rocky Mountain Journal of Mathematics

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