EQUIPARTITION OF ENERGY FOR NONAUTONOMOUS WAVE EQUATIONS

. Consider wave equations of the form

with A an injective selfadjoint operator on a complex Hilbert space H. The kinetic, potential, and total energies of a solution u are K(t) = u (t) 2 , P (t) = Au(t) 2 , E(t) = K(t) + P (t).
Finite energy solutions are those mild solutions for which E(t) is finite. For such solutions E(t) = E(0), that is, energy is conserved, and asymptotic equipartition of energy lim t−→±∞ K(t) = lim This can be verified by differentiating both sides with respect to t. Is is easily seen that e tM : t ∈ R is a (C o ) unitary group on H 2 , and thus the total energy is conserved for all strong solutions u ∈ C 2 (R, H) which correspond to f ∈ D(A 2 ) and g ∈ D(A) in (1). The same is true for all mild solutions which correspond to e tM Af g : t ∈ R for f ∈ D(A) and g ∈ H. Here K [resp. P ] represents the kinetic [resp. potential] energy, and E(t) = E(0) is the total energy which is conserved. Equipartition of energy, i.e. lim t−→±∞ 0)) for all finite energy mild solutions holds iff e itA h, k −→ 0 as t → ±∞ for all h, k ∈ H. A sufficient condition for this condition to hold is that A is spectrally absolutely continuous (SAC). To define this, write by the spectral theorem using a resolution of the identity. Then A is SAC means the bounded monotone function Our goal in this paper is to extend this result to the context of replacing A by A(t), a family of time dependent commuting nonnegative injective selfadjoint operators on H, such that the new result reduces to the old result when A does not depend on t and is SAC. This will be the first asymptotic equipartition of energy result for a nonautonomous system generalizing (1).
The outline of the paper is as follows. In Section 2 we review the proof of asymptotic equipartition of energy for (1). In Sections 3 and 4 we formulate the problem and prove the wellposedness theorem. In Section 5 we establish the equipartition of energy results. Section 6 contains an example.
2. The autonomous wave equation. The unique mild solution to (1) can be written as where Then The law of cosines together with unitarity of e itA implies and K(t) − P (t) = −4Re e itA AF, AG .
Thus, energy is conserved, and energy is asymptotically equipartitioned iff for all h ∈ H as t → ±∞. In the SAC case, the Riemman-Lebesgue Lemma gives the conclusion. The factorization This uses the fact the space-time operators in parenthesis are injective and commute. Here N denotes the null space. Thus, {e ±itA h : t ∈ R} gives a general mild solution of u = ±iAu with u(0) = h, as h varies. Care must be used in finding a nonautonomous version of the preceding.
3. The nonautonomous framework. Let B = B * on H. Then u(t) = e itB h is the unique mild solution of as h ∈ H varies. Moreover, by the Spectral Theorem, is a family of commuting selfadjoint operators on H, then for U o , L 2 as above and for b(t) : Ω → R, Σ-measurable for all t ∈ R. For simplicity, we will restrict to nonnegative time, t ≥ 0. Next we state our first assumption.
are both independent of t. Moreover, the commutator [A(t), A(s)] = A(t)A(s) − A(s)A(t) = 0 in the sense that the bounded operators e iτ A(t) and e iσA(s) commute for all t, s, τ, σ ∈ R + . Further, assume for all f ∈ D o , and there exists a bounded function Note that (5) implies that 0 ∈ ρ(A(t)) holds either for all t ∈ R + or for no t ∈ R + . In the former case We may view a(t) : Ω → (0, ∞) as a(t, ω) with a(·, ·) : R + × Ω → (0, ∞). By (HYP1) and old theorem of J. L. Doob [1], without loss of generality we may assume a(·, ·) is jointly measurable on R + × Ω in the (Borel sets × Σ) sense. Note that here each a(t, ·) is defined on Ω\N t where µ(N t ) = 0 for each t ∈ R + . There are uncountably many null sets N t , but Doob's theorem says this is not a problem; they can be chosen so that a(t, ω) is jointly measurable and certain integrals over R + × Ω will exist.
Another form of the Spectral Theorem says that, due to the commuting hypothesis in (HYP1), there is a function such that for each t ∈ R + , F (t, A(0)) = A(t); (9) moreover F (t, x) is an C 1 function of t ∈ R + for each fixed x. Later we shall assume more regularity on F (t, x) as a function on R + × (0, ∞). Now using (6)-(7) we have where Also, by (8) and (9), for t ∈ R + , where Now let F 1 , F 2 ∈ D 1 and t ≥ 0. Define Suppressing the argument t, we get Combining (13) and (14) yelds We can rewrite this as But in the nonautonomous case, the two components w,w of W satisfy different equations, namely Still, this reduces to a single 2 × 2 system as given by (15) and these are different second order single equations. Furthermore, Q(t) (see (15)) is normal but not selfadjoint because the imaginary part, iA 0 I I 0 , is nonzero.
So the relevant second order equation here that is wellposed is a second order system in H or, equivalently, a second order equation in H 2 . For this system, we now establish wellposedness. Asymptotic equipartition of energy can be considered here, even though the equation does not conserve energy.
4. The nonautonomous system. We consider the system where as before −M 2 is identified with −M (t) 2 I 0 0 I for convenience. As mentioned before, in the autonomous case, A ≡ 0, and both w andw satisfy u + A 2 u = 0.
But when A depends in a nontrivial way on t, neither w norw satisfy a single second order equation. Still, the pair W = w w does satisfy a second order equation governed by the evolution operator family {Z(t, s) : t ≥ s ≥ 0} which can be described as follows. Now W = Q W implies Then for t ≥ s ≥ 0 and Z(t, s) is not unitary because Q(t) is normal but not skewadjoint for each t ∈ R + .
and each component of 1 2 (iA(0) −1 H 2 + H 1 ) is F 1 , and each component of Proof. The proof is a straightfoward consequence of (HYP1), the previous calculations, and some elementary algebra showing the relationship between (H 1 , H 2 ) and (F 1 , F 2 ). The point is that solutions are built from linear combinations of which is the spirit of a nonautonomous d'Alembert's formula. This is a new feature of our first result.

Asymptotic equipartition of energy. Let
be as in the preceding theorem. We want to obtain asymptotic equipartition of energy. Define the total kinetic energy of the system as and the total potential energy as It is convenient to consider refinements of these, namely Using (18), (19) and the Law of Cosines, we get Moreover, we see that We now make this notion more precise.
HYP 2 Note that k 1 of (HYP1) satisfies for some ε 1 > 0 and all t ∈ R + . Assume that for every f ∈ D o A (·)f ∈ L 1 (R + , H).
We now state our main result.
In fact, A(∞) will automatically be SAC, provided that F (∞, x), as a function of x, is piecewise strictly monotone. This follows from Lemma 4.1 in [5].
We observe that