UTMS 2003–12

March 5, 2003

One unique continuation for a linearized
Benjamin-Bona-Mahony equation
byMasahiro Yamamoto

TUNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANONE UNIQUE CONTINUATION FOR A LINEARIZEDBENJAMIN-BONA-MAHONY EQUATION

Masahiro Yamamoto
Department of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba Meguro Tokyo 153 Japan
tel: +81-3-5465-8328
fax: +81-3-5465-7017
e-mail : myama@ms.u-tokyo.ac.jp
Abstract.

For a linearized Benjamin-Bona-Mahony equation:∂t u − ∂x2 ∂t u = p(x, t)∂x u + q(x, t)u,x ∈ (0, 1), 0 < t < T,we prove a unique continuation property by a Carleman estimate. The main result is:
if u(1, t) = ∂x u(1, t) = 0 for t ∈ (0, T ) and u(x, 0) = 0 for x ∈ (0, 1), then u(x, t) = 0for (x, t) ∈ (0, 1) × (0, T ).

§1. Introduction.
We consider a linearized Benjamin-Bona-Mahony equation:∂t u(x, t) − ∂x2 ∂t u(x, t) = p(x, t)∂xu(x, t) + q(x, t)u(x, t),0 < x < 1, 0 < t < T.
(1.1)

Here we assume thatp ∈ L∞ ((0, 1) × (0, T )),q ∈ L∞ (0, T ; L2(0, 1)).

(1.2)

Here and henceforth, we set∂x =∂∂2∂, ∂x2 =.
,
∂=t
∂x
∂x2
∂t

1991 Mathematics Subject Classification. 35B60, 35Q35, 35R25.
Key words and phrases. unique continuation, Benjamin-Bona-Mahony equation, Carlemanestimate.
Typeset by AMS-TEX
1

2M. YAMAMOTO

We discuss a unique continuation property: for some open subset ω ⊂ (0, 1), canwe conclude thatu(x, t) = 0 for (x, t) ∈ ω × (0, T ) or u(1, t) = ∂x u(1, t) = 0 for t ∈ (0, T )implies that u(x, t) = 0 for (x, t) ∈ (0, 1) × (0, T )?

(1.3)

However, this is not necessarily true, as is pointed out in Zhang and Zuazua [11]:let p = q ≡ 0, and let us take u(x, t) = u0 (x) ≡ 0, a t-independent function suchthat u0 ∈ C0∞ (0, 1) and supp u0 ⊂ (0, 1) \ ω. Then u0 satisﬁes (1.1), but u0 does
not vanish identically over (0, 1).
This counterexample demonstrates one diﬃculty in the unique continuation
property which is diﬀerent from other cases such as the KdV equation. We aresuggested that other boundary condition may guarantee the unique continuation,
as long as we assume that u vanishes in a cylindrical subdomain ω × (0, T ) or thatthe lateral Cauchy data (i.e. u(1, t) = ∂x u(1, t) = 0 for t ∈ (0, T )) vanish. In fact,
Zhang and Zuazua [11] proves that (1.3) implies u ≡ 0 in (0, 1) × (0, ∞) provided
that u(0, t) = u(1, t) = 0, 0 < t < T and p, q are independent of t.
In this paper, assuming that u(x, 0) = 0, 0 < x < 1, we prove that u(1, t) =∂x u(1, t) = 0, 0 < t < T , implies u(x, t) = 0, 0 < x < 1, 0 < t < T .
Theorem. Let ∂xj ∂tk u ∈ C([0, 1] × [0, T ]) with j = 0, 1, 2 and k = 0, 1, satisfy (1.1).
Assume (1.2). Ifu(1, t) = ∂x u(1, t) = 0,0<t<T(1.4)

andu(x, 0) = 0,0 < x < 1,(1.5)

UNIQUE CONTINUATION FOR BBM EQUATION

3then u(x, t) = 0, 0 < x < 1, 0 < t < T .
Our proof is based on a Carleman estimate, so that we can prove conditional
stability in the continuation by a usual method (e.g., Isakov [Chapter 3, 5]) and
our argument is valid for a multidimensional analogue∂t u − ∂t ∆u =n

pi (x, t)i=1∂u+ q(x, t)u∂xifor x = (x1 , ..., xn) ∈ Rn and t ∈ (0, T ). Furthermore we can prove the theorem
under a weaker regularity assumption on u, but, for simplicity, we mainly consider
the classical solution u.
For the Benjamin-Bona-Mahony equation, we can prove the unique continuation
not across the characteristics (e.g., Theorems 3.1 - 3.2 in Davila and Menzala [3]).
For the unique continuation for other dispersive equations such as the KdV equation, we refer to Isakov [6]. In [3] and [6], the main tool is a Carleman estimate. As
for applications of the Carleman estimate to the unique continuation, we furtherrefer to Hörmander [4], Isakov [7], Saut and Scheurer [9], Tataru [10].
This paper is composed of three sections. Section 2 is preliminaries where we establish a simple Carleman estimate and an integral inequality. Section 3 is devoted
to the proof of our main theorem.
§2. Preliminaries.
Let Q ⊂ (0, 1) × (−T, T ) ≡ {(x, t); 0 < x < 1, −T < t < T } be a subdomain suchthat the boundary ∂Q is of piecewise C 2 . Then we haveLemma 1. Let ϕ ∈ C(R2 ), and for any T ∈ (−T, T ) let ϕ = ϕ(·, t) ∈ C ∞ (R)satisfy(∂x2 ϕ)(x, t) > 0,(x, t) ∈ Q(2.1)

4

M. YAMAMOTOand|(∂x ϕ)(x, t)| > 0,(x, t) ∈ Q.

(2.2)

Then there exist constants C = C(ϕ, Q) > 0 and S = S(ϕ, Q) > 0 such that

Q

2

322sϕ

(s|∂x u| + s |u| )edxdt ≤ C

Q|∂x2 u|2 e2sϕ dxdt

(2.3)

for all s > S and u ∈ C 2 (Q) such that u = ∂x u = 0 on ∂Q.
Proof. We will apply the argument in Bukhgeim [1]. By a usual density argument, it suﬃces to prove the lemma for u ∈ C0∞ (Q). We set v = esϕ u andLv = −esϕ ∂x2 (e−sϕ v). Then [the right-hand side of (2.3)] = C

Q|Lv|2 dxdt, andfor the proof, it is suﬃcient to prove that
Q

23



2(s|∂x v| + s |v| )dxdt ≤ C

Q|Lv|2 dxdt.

We directly calculate:
Lv = −∂x2 v + 2s(∂x ϕ)(∂x v) + (s(∂x2 ϕ) − s2 (∂x ϕ)2 )v.
Formally we calculate the adjoint operator L∗ of L to obtainL∗ v = −∂x2 v − 2s(∂x ϕ)(∂x v) − (s(∂x2 ϕ) + s2 (∂x ϕ)2 )v.
Setting L+ = 12 (L + L∗ ) and L− = 12 (L − L∗ ), we have L = L+ + L− , so that
Lv2L2 (Q) = L+ v2L2 (Q) + L− v2L2 (Q) + (L+ v, L− v)L2 (Q) + (L− v, L+ v)L2 (Q)≥(L+ v, L− v)L2 (Q) + (L− v, L+ v)L2 (Q) = ((L+ L− − L− L+ )v, v)L2 (Q)by integration by parts and v = ∂x v = 0 on ∂Q. Here and henceforth we set(u, v)L2 (Q) =Q1uvdxdt and uL2 (Q) = (u, u)L2 2 (Q) .

UNIQUE CONTINUATION FOR BBM EQUATION

5Direct calculations yield(L+ L− − L− L+ )v = −s(∂x4 ϕ)v − 4s(∂x3 ϕ)(∂x v)−4s(∂x2 ϕ)(∂x2 v) + 2s3 (∂x ϕ)∂x (|∂x ϕ|2 )v.
Henceforth C > 0 denotes a generic constant which is dependent on ϕ, Q, but
independent of s. By integration by parts and the Schwarz inequality, we haveLv2L2 (Q) ≥ 4s3
+4sQ2Q(∂x ϕ) (∂x2 ϕ)v 2 dxdt − Cs(∂x2 ϕ)|∂x v|2 dxdt − CQQv 2 dxdt|∂x v|2 dxdt.

Therefore under assumptions (2.1) and (2.2), if we take S > 0 suﬃciently large,
then the proof of Lemma 1 is complete.
We arbitrarily choose µ ∈ (0, 1) and δ > 0, and set


|t|2

 ψ(x, t) = 1 − (x − (1 + δ)) −T
ϕ(x, t) = eλψ(x,t) − 1.

µ,(2.4)

If we ﬁx λ > 0 suﬃciently large, then ϕ satisﬁes (2.1) and (2.2) in Q ⊂ (0, 1) ×(−T, T ). We further set
Q(ε) = {(x, t) ∈ (0, 1) × R; ϕ(x, t) > ε}

(2.5)

for ε ≥ 0.
We sett1hε (t) = 1 + δ − 1 − log(1 + ε) −λTµ12,t > 0.

Then, for suﬃciently large λ > 0 and small ε ≥ 0, we haveQ(ε) = {(x, t) ∈ (0, 1) × R; hε (|t|) < x < 1}.

(2.6)

6

M. YAMAMOTO

We can directly verify that (x, t) ∈ Q(ε) implies
1|t| < T 1 − δ 2 − log(1 + ε)
λ1µ< T.

(2.7)

Moreover we can prove the following integral inequality:Lemma 2. There exists a constant C = C(ϕ, ε, T ) > 0 such that tQ(ε)

022sϕ|u(x, η)|dη e
dxdt ≤ CQ(ε)|u(x, t)|2 e2sϕ dxdt

for all s > 0 and u ∈ L2 (Q(ε)).
Since for any x ∈ (0, 1), the function ϕ(x, t) is decreasing in t > 0 and increasing
in t < 0, the proof is directly done (e.g., Isakov [p.153, 5], Klibanov [8]). This lemma
was used originally by Bukhgeim and Klibanov [2] for proving the uniqueness insome inverse problems by Carleman estimates. The lemma is simple but essential
for applications of Carleman estimates to inverse problems. Also see Bukhgeim [1].
§3. Proof of Theorem.
We extend u, p, q to odd functions in t ∈ (−T, T ), which are denoted by the samenotations. Then by (1.5), we see∂x2 (∂t u)(x, t) = ∂t u(x, t) − p(x, t)∂xu(x, t) − q(x, t)u(x, t),

0 < x < 1, −T < t < T,(3.1)

andu(1, t) = ∂x u(1, t) = 0,−T < t < T.

(3.2)

The proof is done along the argument towards the unique continuation by meansof a Carleman estimate (e.g., Hörmander [4], Isakov [5]) except for the applicationof Lemma 2.

UNIQUE CONTINUATION FOR BBM EQUATION

7

Let ε > 0 be suﬃciently small and let χ = χ(x, t) such thatχ ∈ C0∞ (R2 ), 0 ≤ χ ≤ 1,
1, (x, t) ∈ Q(2ε)
 χ(x, t) =0, (x, t) ∈ Q(0) \ Q(ε).

(3.3)

We set y = uχ. Then ∂xj ∂tk y ∈ C([0, 1] × [−T, T ]) for j = 0, 1, 2 and k = 0, 1, andy = ∂x y = 0on ∂Q(0)(3.4)

by (1.4), (2.6) and (2.7). Moreover we directly see∂x2 (∂t y)(x, t) = (∂t y) − p(∂x y) − qy + I,

(3.5)

whereI = ((∂x2 ∂t χ) − ∂t χ + p(∂x χ))u + 2(∂x ∂t χ)∂x u+(∂x2 χ)(∂t u) + (∂t χ)∂x2 u + 2(∂x χ)(∂x ∂t u),(x, t) ∈ Q(0).

(3.6)

Moreover by (2.6) and (2.7), we see that Q(0) ⊂ (0, 1) × (−T, T ), and (2.1) and(2.2) are true in Q(0). Consequently, in terms of (3.4), applying Lemma 1 to ∂t yin (3.5), we haveQ(0)(s|∂x ∂t y|2 + s3 |∂t y|2 )e2sϕ dxdt

≤CQ(0)2222 22sϕ(|∂t y| + p |∂x y| + q y )edxdt +Q(0)|I(x, t)|2e2sϕ dxdt.
(3.7)

Here and henceforth C > 0 denotes a generic constant which is independent ofs > 0 and m ∈ N. By deﬁntion (3.6) of I, we have I = 0 only in Q(ε) \ Q(2ε), sothatQ(0)|I(x, t)|2e2sϕ dxdt ≤ Ce4sεmax(x,t)∈[0,1]×[−T ,T ]|I(x, t)|2 ≤ Ce4sε .

(3.8)

8

M. YAMAMOTO

On the other hand, by (1.5), we have t(∂x y)(x, t) = t(∂x ∂t y)(x, η)dη,0y(x, t) =0(∂t y)(x, η)dη.

Therefore, by Lemma 2, we obtain

2 2sϕ|∂x y| eQ(0)≤Canddxdt =0Q(0)2(∂x ∂t y)(x, η)dη e2sϕ dxdt|(∂x ∂t y)(x, t)|e2sϕ dxdtQ(0)2 2sϕQ(0) t

|y(x, t)| e(3.9)

dxdt ≤ CQ(0)|(∂t y)(x, t)|2e2sϕ dxdt.

(3.10)

Noting (2.6), (2.7), the Sobolev embedding and the Schwarz inequality, we have

Q(0)

≤

 T (1−δ2 )1/µ2 2 2sϕq y e

 T (1−δ2 )1/µ
−T (1−δ 2 )1/µdxdt =

≤C

Q(0)

−T (1−δ 2 )1/µh0 (|t|)

q 2 (x, t)y 2 (x, t)e2sϕ(x,t) dx dt

q 2 (·, t)L1 (0,1) (yesϕ )(·, t)2L∞ (h0 (|t|),1) dt≤Cq2L∞ (−T,T ;L2 (0,1))
 1

 T (1−δ2 )1/µ

 1−T (1−δ 2 )1/µh0 (|t|)|∂x (yesϕ )(x, t)|2 dx dt(s2 y 2 + |∂x y|2 )e2sϕ dxdt.

(3.11)

Applying (3.8) - (3.11) in (3.7), we obtainQ(0)(s|∂x ∂t y|2 + s3 |∂t y|2 )e2sϕ dxdt

≤CQ(0)((1 + s2 )|∂t y|2 + |∂x ∂t y|2 )e2sϕ dxdt + Ce4sε .

Taking s > 0 suﬃciently large, we can absorb the ﬁrst term at the right-hand side
into the left-hand side, so thatQ(0)(s|∂x ∂t y|2 + s3 |∂t y|2 )e2sϕ dxdt ≤ Ce4sε .

UNIQUE CONTINUATION FOR BBM EQUATION

9Since Q(3ε) = {(x, t) ∈ (0, 1) × R; ϕ(x, t) > 3ε} ⊂ Q(0), we have6sεe

3Q(3ε)

2s |∂t y| dxdt ≤Q(3ε)(s|∂x ∂t y|2 + s3 |∂t y|2 )e2sϕ dxdt ≤ Ce4sεfor all large s > 0. That is,Q(3ε)|∂t y|2 dxdt ≤ Cs−3 e−2sεfor all large s > 0. Taking s −→ ∞, we obtain ∂t y(x, t) = 0 for (x, t) ∈ Q(3ε).
Again, by y(x, 0) = 0, 0 < x < 1, we have y(x, t) = 0 for (x, t) ∈ Q(3ε). Since ε > 0
is arbitrary, we see that y(x, t) = 0 ift(1 + δ) − 1 −Tµ

12< x < 1,10 < t < T (1 − δ 2 ) 2 .

 µ  12Since δ > 0 is arbitrary, we have y(x, t) = 0 if 1 − 1 − Tt
< x < 1 and0 < t < T . Next, because µ ∈ (0, 1) is arbitrary, we can let µ tend to 0, so thaty(x, t) = 0 if 0 < t < T and 0 < x < 1. Thus the proof of the theorem is complete.
References
1. A.L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht,
2000.
2. A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Math. Dokl. 24 (1981), 244–247.
3. M. Davila and G.P. Menzala, Unique continuation for the Benjamin-BonaMahony and Boussinesq’s equations, Nonlinear Diﬀerential Equations and Applications NoDEA 5 (1998), 367–382.
4. L. Hörmander, Linear Partial Diﬀerential Operators, Springer-Verlag, Berlin,
1963.
5. V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990.
6. V. Isakov, Carleman type estimates in an anisotropic case and applications,
Journal of Diﬀerential Equations 105 (1993), 217–238.
7. V. Isakov, Inverse Problems for Partial Diﬀerential Equations, Springer-Verlag,
Berlin, 1998.
8. M.V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8(1992), 575–596.

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M. YAMAMOTO

9. J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations,
J. Diﬀerential Equations 66 (1987), 118–139.
10. D. Tataru, Unique continuation for solutions to PDE’s; between Hörmander’stheorem and Holmgren’s theorem, Commun. in Partial Diﬀerential Equations
20 (1995), 855–884.
11. X. Zhang and E. Zuazua, Unique continuation for the linearized BenjaminBona-Mahony equations with space-dependent potential, preprint.

Preprint Series, Graduate School of Mathematical Sciences, The University of TokyoUTMS
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2003–2 Oleg Yu. Imanuvilov and Masahiro Yamamoto: Carleman estimates for thenon-stationary Lamé system and the application to an inverse problem.
2003–3 Xu Bin: Sobolev norms of eigenfunctions on a closed Riemannian manifold.
2003–4 Xu Bin: Compact Lie group actions on compact manifolds of nonpositive curvature.
2003–5 J. Elschner, G. Schmidt and M. Yamamoto: An inverse problem in periodic
diﬀractive optics: Global nuiqueness with a single wave number.
2003–6 Naoto Kumano-go: New curvilinear integrals along paths of Feynman path integral.
2003–7 Susumu Yamazaki: Microsupport of Whitney solutions to systems with regular
singularities and its applications.
2003–8 Naoto Kumano-go and Daisuke Fujiwara: Smooth functional derivatives inFeynman path integral.
2003–9 Sungwhan Kim and Masahiro Yamamoto : Unique determination of inhomogeneity in a stationary isotropic Lamé system with varible coeﬃcients.
2003–10 Reiji Tomatsu: Amenable discrete quantum groups.
2003–11 Jin Cheng and Masahiro Yamamoto: Global uniqueness in the inverse acoustic
scattering problem within polygonal obstacles.
2003–12 Masahiro Yamamoto: One unique continuation for a linearized BenjaminBona-Mahony equation.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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