Almost periodic motion of a string vibrating against a straight fixed obstacle

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u>-h inR+xQ
1 Condition (8), which is a strengthened version of energy conservation, must hold in the sense of Q'(]O, +w[ x Q). It implies that the energy integral remains unchanged through the motion. Notice that in case a shock takes place simultaneously along a set of non-zero measure in Q, we cannot have +y E C(R_, L2(Q)).
at Actually, the existence and uniqueness of a solution for (S) + (9) is obtained in the functional class (5). The solution constructed satisfies the additional property: which is used to get uniqueness, and is interpreted in [9] as equivalent to subsonic propagation of interactions.
In our case, since au/at (x, 0) = 0, we look for solutions (even as a function of t) defined on R x !i?. Our purpose is to give a precise meaning to the physical idea of 'vibrations' against the obstacle.
We prove that the solution is strongly almost-periodic as a function from R to HA(Q), generally not exactly periodic in t. We also sketch out some simple results concerning the non-harmonic Fourier series of u(x, t) with respect to t. Further computations in this direction are planned for the future.

STATEMENT OF THE RESULTS
Let us denote by (2) the system (5) + (10) with ug = 0 and R' replaced by R everywhere. We recall 0 d h < 1.
the solution of (X) is such that u(t) = u(. , t) is strongly almost-periodic as a function from R to HA(Q). Moreover, the map UO--+ u(t) is Lipschitzcontinuous from H&Q) to L"(R, H&Q)).

Remark.
In contrast with the case of equation (4), u(f) is generally not time-periodic. More precisely, we have: THEOREM 1.2. (a) h = p/q with p, q integers, the motion is periodic with p + q as a period if p + q is even, and 2(p + q) as a period if p + q is odd. In the special case when uO(-X) = LQ(X), we have always the period p + q.
In some cases the smallest period is smaller.
(b) If h 65 Q, the motion is never periodic, except in the single case LQ(X) = 1 -21x1, where the motion has the period 1 + h.
Remark. The aim of this paper is to give complete proofs of the results announced in [7], together with some more information.
Since u is almost-periodic with respect to f, for every A E R, the limit 1s called the set of exponents of U, and in general (cf. [4]) is denumerable.
Here, we have a more precise result.
where the infinite sums on the right-hand side converge uniformly and the convergence with respect to N is uniform for t E R.   Then the velocity is reversed into U+ = -u-= +2, and the motion proceeds backwards: Thus &/at (x, t) is discontinuous at t = t*, while ti(x, t) remains continuous with respect to t E R. The motion ti is periodic in t with period 1 + h.

b. The 'regular' case
In addition to (l)- In order to compute u(x, t), we generalize an idea used in [ll] by Reder. We define a function P: R + R in several steps. Step Step 2. If t E [l/2,3/2], we define (20) Step 3. We extend F on R by the condition We claim that f and F are in C*(R), because F'( -l/2) = F'(3/2) and F"( -l/2) = F"(3/2). The first condition is a consequence of (21), the second one follows from F"( -112) = 0.
Some other properties of F are summarized below.
LEMMA 3.1. F is nondecreasing: R + R. In addition (21) holds for t E R, and we have also For every x E a and t E R, we have Proof. Formula (21) means that for every 8 E [ -l/2.3/2], we have Given t E R, we choose m E 2 and 0 E [-l/2,3/2] such that t = 2m + 8. Then: Also, for t E R, we have: Let x < l/2: by (21), we have Proof. Let us introduce for convenience Then, in the sense of 9'(Q X R), we have (26)
Differentiating (25) in the distribution sense, we get after reduction: this formula making sense because Oz.? is a measure and F' is continuous. From (27), we deduce that (7) and (10) are satisfied.
To verify the initial condition, we remark that u(x, 0) =ti(F(x), 0) = @(F(x)) = 1 -2lF(x)/ and (20)  Combining these two remarks, we get (24), which will appear a convenient tool to treat the more general case where ua is only in Hi(Q). W

c. The general case
If u. satisfies conditions (l-3), we can still define F(t) as in the paragraph above. Then F(t) is non-decreasing, in H'(R) and f(t) = F(t) -t is 2-periodic. We compute u(x, t) by means of formula (23). To check that u(x, t) is the solution of (2) starting from (uo, 0), we can choose a sequence of initials ~0" satisfying (19) with ul; +UO in HA(Q). According to (24j, z/(x, t) will converge to u(x, t) in L"(R, H&Q)). A similar calculation shows that &"/at converges to du/dt in L"(H, L*(Q)). Thus all conditions (5) + (10) are checked by density. 4. PROOF OF THEOREM 1.1 Because of (24), it is enough to check the almost-periodicity when ~0 is 'regular', so that FE P(R).
a. As a first step, we prove that t-+ u(. , t) is almost-periodic in C(a), by using directly Bochner's criterion.
Let (t,J be any sequence of real numbers. We write: where m, and k, are in 2. Thus We may assume lim pnk = p and lim onl. =D , by extracting a subsequence of (t,J. Then, n-+m n-ta because of the uniform continuity of F and fi the sequence u(x, t +tJ converges uniformly on 0 x R to

u*(x, t) = ti F(x + t + p) + F(x -t -p) F(x + t + p) -F(x -t -p) +
Precompactness of the range of u in HA(Q) Thanks to conservation of energy, u(t) is bounded in HA(Q), thus weakly almost-periodic in HA(Q). According to [lo], Theorem 2.11, the strong almost-periodicity is equivalent to precompactness of the range in H&R). Even in the 'regular' case, it is not so easy to check because we cannot approach u in L"(R, H&Q)) by regular functions since U has discontinuous derivatives.
We need a technical lemma.
(for all t and almost every x) b(x, t) = $(x, t) (for all x and almost every t).
The functions a, b take only the values 0, -2 and +2, their discontinuities lie on some curves of a x R.
In the general case, we can use (24) which shows that u(x, t) is the limit in L"(R, HA(n)) of solutions associated to 'regular' initials ul;. This finishes the proof of theorem 1.1, since the statement on uo+ u(t) is actually a consequence of (24). W

PROOF OF THEOREM 1.2
(a) Since F(r) -t is 2-periodic and ti(x, r) is 1 + h-periodic in r, the first assertion is an obvious consequence of formula (23). If UO(-x) = UO(X), then b = -a and F(t) is odd. Then F(t) -t is l-periodic, and the second assertion follows.
If for instance h = l/n, then (n + 1)/n = 1 + h is a period for u(. , t), which is strictly or II 2 2) smaller than n + 1. For n = 2p, we get a period smaller than n + 1 = p + q. &IfI((.,r). p is eriodic with a period t, then the functions are also r-periodic, independently of E, cr', /.I for -t G (Y < /I d 8. As a consequence of (26)) F'*(x + 0) dx dB. By a density argument, it is sufficient to do this in the 'regular' case. Then F(t) -r = f(t) is a C' function, periodic with period 2, and we get a first development: a,[X(x, t)] = ,i, W%(X) cos knt, with (for instance) As a second step, we may write: ifk<O. (30) (c) Case uo(x) = 1 -2/x + Asin27rxl Then f(t) = A. sin 2nt, and the condition F' 2 0 is equivalent to IAl S1/2n.