MATHEMATICS

E. B. BYKHOVSKII

ABSENCE IN THE SPACES \(C\), \(L_p\), AND \(W_p^1\) \((1\le p<2)\) OF ANALOGUES OF THE ENERGY INEQUALITY FOR THE STRING EQUATION WITH BOUNDED LEADING COEFFICIENT

(Presented by Academician V. I. Smirnov, January 23, 1965)

As is known, for the Cauchy problem

\[ v_{tt}=[a(x)v_x]_x;\qquad v|_{t=0}=\varphi(x);\qquad v_t|_{t=0}=\psi(0) \tag{1} \]

when \(a(x)\) is, say, piecewise smooth and satisfies the inequality
\[ 0<\beta\le a(x)\le \gamma, \]
there is an estimate of
\[ \int_x [v_t^2+v_x^2]\,dx \]
by the same integral at \(t=0\) over an interval of the \(x\)-axis determined by the course of the characteristics of equation (1), with constant \(K(\beta,\gamma)\). An analogous estimate holds for
\[ \int_x v^2\,dx. \]
If \(K\) is allowed also to depend on the smoothness indices of \(a(x)\), then analogous estimates also hold for the norms \(W_p^1\) and \(\mathscr L_p\) \((1\le p<2)\).

Our purpose is to show that the latter estimates do not hold with \(K(\beta,\gamma)\). Counterexamples will be constructed for

\[ \psi(x)\equiv 0. \tag{2} \]

The functions \(\varphi(x)\) and \(a(x)\), and consequently also \(v(x,t)\), will henceforth be assumed \(l\)-periodic in \(x\).

By \(\|f(x)\|\) we shall mean any one of the norms:

\[ \max_{0\le x\le l}|f(x)|,\qquad \left\{\int_0^l |f(x)|^p\,dx\right\}^{1/p},\qquad \left\{\int_0^l |f'(x)|^p\,dx\right\}^{1/p}. \tag{3} \]

Lemma 1. If in problem (1), (2), for one of the indicated norms, for some \(t_0\) the estimate

\[ \|v(x,t_0)\|\le K(\beta,\gamma)\|v(x,0)\| \]

holds, then this estimate holds uniformly in \(t\in[A_{t_0},\infty)\), \(A_{t_0}>0\).

The proof is based on the fact that the function \(v_\alpha(x,t)=v(\alpha x,\alpha t)\) is a solution of (1), (2) with \(a(\alpha x)\) and \(\varphi(\alpha x)\), while \(0<\beta\le a(\alpha x)\le\gamma\), and on the fact that for periodic \(f(x)\)
\[ \|f(\alpha x)\|=\|f(x)\|\,[\alpha^m+O(1/\alpha)]\quad(\alpha\ge1), \]
where \(O(1/\alpha)\) is written for large \(\alpha\) and is independent of \(f(x)\).

Thus, counterexamples will be constructed if one can find a sequence \(\varphi_n(x)\) with bounded \(\|\varphi_n(x)\|\) and such an \(a(x)\) that for \(v_n(x,t)\) one has

\[ \sup_{\substack{1\le n<\infty\\ 0\le t<\infty}}\|v_n(x,t)\|=\infty. \]

In what follows we shall take \(\varphi(x)\) to be odd \(2\pi\)-periodic functions, and \(a(x)\) to be an even \(2\pi\)-periodic function. Then \(v(x,t)\), for \(0\le x\le\pi\), will be the solution of the initial-boundary-value problem with boundary conditions \(v(0,t)=v(\pi,t)=0\).

For \(v(x,t)\) we have:
\[ v(x,t)=\sum_{k=1}^{\infty} c_k \cos \lambda_k t\, y_k(x), \]
where \(y_k(x)\) are the eigenfunctions of the problem
\[ \mathcal L y=[a(x)y']'=-\lambda y,\qquad y(0)=y(\pi)=0; \tag{4} \]
\(c_k\) is the coefficient in the expansion
\[ \varphi(x)=\sum_{k=1}^{\infty} c_k y_k(x). \]
Using the asymptotics of \(y_k(x)\) and adding and subtracting
\[ \sqrt{\frac{2}{\pi}}\sum_{k=1}^{\infty} c_k \cos kt \sin k\xi, \]
we obtain
\[ \begin{aligned} v[x(\xi),t] &=\sqrt{\frac{2}{\pi}}\sum_{k=1}^{\infty} c_k(\cos \lambda_k t-\cos kt)\sin k\xi \\ &\quad +\frac12 a^{-1/4}[x(\xi)]\, a^{1/4}[x(\xi+t)]\,\varphi[x(\xi+t)] \\ &\quad +\frac12 a^{-1/4}[x(\xi)]\, a^{1/4}[x(\xi-t)]\,\varphi[x(\xi-t)] +\sum_{k=1}^{\infty}\frac{c_k}{k}\eta_k(\xi), \end{aligned} \tag{5} \]
where
\[ \xi=\frac{\pi}{h}\int_{0}^{x} a^{-1/2}(z)\,dz,\qquad h=\int_{0}^{\pi} a^{-1/2}(z)\,dz, \]
and the \(\eta_k\) are uniformly bounded with respect to \(k\) and \(\xi\).

An analogous expression can be written for \(v_x[x(\xi),t]\).

Lemma 2. Let an infinite sequence \(\sigma\) of zeros and ones be given arbitrarily. Then there exists a strictly positive \(a(x)\in C_2[0,\pi]\) such that the spectrum \(\lambda_k\) of problem (4) has the following property: for every natural \(n\) there is a \(t_n>0\) such that the numbers
\[ -\frac12(\cos \lambda_{4k}t_n-\cos 4kt_n)\qquad (k=1,2,\ldots,n) \]
are equal, respectively, to the first \(n\) terms of the sequence \(\sigma\).

In the proof, first the \(\lambda_k\) are chosen in a suitable way according to \(\sigma\), and then \(a(x)\) is constructed as the solution of the inverse Sturm–Liouville problem \((^1)\). It is not hard to see that \(t_n\to+\infty\) as \(n\to\infty\).

Let us consider in detail the case of the space \(C\). Let
\[ S[f]=\sum_{k=1}^{\infty} c_k \sin k\xi \]
be a series converging uniformly but not absolutely. We shall consider the series
\[ S[f(4\xi)]=\sum_{k=1}^{\infty} c_k \sin 4k\xi, \]
whose convergence character is the same, and denote its sum by \(S(\xi)\), and by \(\sigma S_n(\xi)\) the partial sum thinned out by the sequence \(\sigma\), respectively. One may assert that \(|S_n(\xi)|\) are uniformly bounded, but if \(\xi_0\) is a point of nonabsolute convergence, then there exists a \(\sigma\) for which
\[ \sigma S_n[f(4\xi_0)]\to\infty. \]
Using the indicated \(\sigma\), we construct \(a(x)\) as stated in Lemma 2, taking the \(\lambda_k\) with indices not divisible by \(4\) arbitrarily, provided only that the inverse problem can be solved (for example, \(\lambda_k=k\)). In problem (1), (2) we take
\[ \varphi_n(x)=\sum_{k=1}^{n} c_k y_{4k}(x). \]
Then \(|\varphi_n(x)|\) are uniformly bounded with respect to \(x\) and \(n\) (by virtue of the asymptotics of \(y_k(x)\)).

Let \(v_n(x,t)\) be the solution of problem (1), (2) with the indicated \(a(x)\) and \(\varphi_n(x)\). Using (5), it is not hard to verify that
\[ \sup_{n,x,t\in[0,\infty)} |v_n(x,t)|=\infty, \]
which completes the construction of the example. Examples for the remaining norms are constructed analogously, but more complicated properties of trigonometric-

series. For \(\mathscr L_p\) one takes the series
\[ S[f(4\xi)]=\sum_{k=1}^{\infty} c_k \sin 4k\xi \]
such that: a) \(S_n(\xi)\to f(4\xi)\) in \(\mathscr L_p\), and b) there exists a \(\sigma\) such that
\[ \|\sigma S_n\|_{\mathscr L_1}\underset{n\to\infty}{\longrightarrow}\infty. \]
The existence of such a series is guaranteed by

Lemma 3. Let
\[ S[f]=\frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx);\quad f\in\mathscr L_p\ (1<p) \]
be such that
\[ \sum_{k=1}^{\infty}(a_k^2+b_k^2)=\infty \]
(i.e. \(p<2\)).

There exists such a \(\sigma\) that
\[ \|\sigma S_n\|_{\mathscr L_1}\underset{n\to\infty}{\longrightarrow}\infty. \]

The proof is based on theorems from (²) (vol. 1, pp. 214, 148, 266).

For the space \(W_p^1\) one takes the series
\[ \sum_{k=1}^{\infty}\frac{1}{k^{1+\beta}}\sin kx, \]
where
\[ 1-\frac{1}{p}<\beta<\frac{1}{2}. \]
It converges absolutely and uniformly to some \(f(x)\), while the differentiated series
\[ \sum_{k=1}^{\infty}\frac{1}{k^\beta}\cos kx \]
converges uniformly outside any neighborhood of \(x=0\) ((²), vol. 1, p. 4). Its sum \(f'(x)\) in a neighborhood of \(x=0\) behaves like \(1/x^{1-\beta}\) ((²), vol. 1, p. 70). Consequently, \(f'(x)\in\mathscr L_p\) (\(p<2\)); the series converges to it in \(\mathscr L_p\), and
\[ \sum_{k=1}^{\infty}\frac{1}{k^{2\beta}}=\infty, \]
after which Lemma 3 is used.

Thus, the following has been proved.

Theorem. Whatever \(t_0>0\) is prescribed, in the Cauchy problem (1), (2) one can choose a sequence of coefficients \(a_{(n)}(x)\) and initial data \(\varphi_{(n)}(x)\), satisfying uniformly in \(n\) the condition
\[ 0<\beta\le a_{(n)}(x)\le \gamma;\qquad \|\varphi_{(n)}(x)\|\le \operatorname{const}\quad (1\le p<2), \]
so that, for the corresponding solutions \(v_{(n)}(x,t)\) of problem (1), (2), one has not only
\[ \|v_{(n)}(x,t_0)\|_p\underset{n\to\infty}{\longrightarrow}\infty, \]
but even
\[ \|v_{(n)}(x,t_0)\|_1\underset{n\to\infty}{\longrightarrow}\infty. \]

All functions involved here are \(2\pi\)-periodic in \(x\), and by \(\|f(x)\|_p\) is meant one of the norms (3); moreover, in the case of \(C\) the subscript is superfluous.

Leningrad State University
named after M. V. Lomonosov

Received
21 I 1965

References

¹ I. M. Gelfand, B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat., 15, No. 4, 309 (1951).
² A. Zygmund, Trigonometric Series, 2nd ed., 1, 2, Cambridge, 1959.