Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 14, 1-23, http://www.math.u-szeged.hu/ejqtde/

On the unique continuation property for a nonlinear dispersive system

Alice Kozakevicius^{∗} & Octavio Vera ^{†}

Abstract

We solve the following problem: If (u, v) = (u(x, t), v(x, t)) is a solution of the Dispersive Coupled System with t_{1} < t_{2} which are sufficiently smooth and such that: supp u( . , t_{j}) ⊂ (a, b) and supp v( . , t_{j}) ⊂ (a, b), − ∞ < a < b < ∞, j = 1, 2. Then u ≡ 0 and v ≡ 0.

\underline{Keywords and phrases}: Dispersive coupled system, evolution equations, unique continuation property.

1 Introduction

This paper is concerned with unique continuation results for some system of nonlinear evolution equation. Indeed, a partial differential equation Lu = 0 in some open, connected domain Ω of ℝ^{n} is said to have the weak unique continuation property (UCP) if every solution u of Lu = 0 (in a suitable function space), which vanishes on some nonempty open subset of Ω vanishes in Ω. We study the UCP of the system of nonlinear evolution equations

(1.1) \{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + ∂_{x}(u v^{2}) = 0 & (P_{1}) \\ ∂_{t}v + ∂_{x}^{3}v + ∂_{x}(u^{2} v) + ∂_{x}v = 0 & (P_{2}) \end{matrix}

with 0 ≤ x ≤ 1, t ≥ 0 and where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t. The general system is

(1.2) \{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + ∂_{x}(u^{p} v^{p+1}) = 0 \\ ∂_{t}v + ∂_{x}^{3}v + ∂_{x}(u^{p+1} v^{p}) = 0 \end{matrix}

with domain −∞ < x < ∞, t ≥ 0 and where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t. The power p is an integer greater than or equal to one. This system appears as a special case of a broad class of nonlinear evolution equations studied by Ablowitz et al. [1] which can be solved by the inverse scattering method. It has the structure of a pair of Korteweg - de Vries(KdV) equations coupled through both dispersive and nonlinear effects. A system of the form (1.2) is of interest because it models the physical problem of describing the strong interaction of two-dimensional long internal gravity waves propagating on neighboring pynoclines in a stratified fluid, as in the derived model by Gear and Grimshaw [8]. Indeed,

(1.3) \{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + a_{3} ∂_{x}^{3}v + u ∂_{x}u + a_{1} v ∂_{x}v + a_{2} ∂_{x}(u v) = 0 in x ∈ ℝ, t ≥ 0 \\ b_{1} ∂_{t}v + ∂_{x}^{3}v + b_{2} a_{3} ∂_{x}^{3}u + v ∂_{x}v + b_{2} a_{2} u ∂_{x}u + b_{2} a_{1} ∂_{x}(u v) = 0 \\ u(x, 0) = φ(x) \\ v(x, 0) = ψ(x) \end{matrix}

where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t and a_{1}, a_{2}, a_{3}, b_{1}, b_{2} are real constants with b_{1} > 0 and b_{2} > 0. Mathematical results on (1.3) were given by J. Bona et al. [4].

∗Departamento de Matemática, CCNE, Universidade Federal de Santa Maria, Faixa de Camobi, Km 9, Santa Maria, RS, Brasil, CEP 97105-900. E-mail: alicek@smail.ufsm.br: This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics (Project No. 15000001).

†Departamento de Matemática, Universidad del Bío-Bío, Collao 1202, Casilla 5-C, Concepción, Chile. E-mail:

overa@ubiobio.cl ; octavipaulov@yahoo.com

They proved that the coupled system is globally well posed in H^{m}(ℝ) × H^{m}(ℝ), for any m ≥ 1 provided |a_{3}| < 1/\sqrt{b_{2}}. Recently, this result was improved by F. Linares and M. Panthee [19]. Indeed, they proved the following:

Theorem 1.1. For any (φ, ψ) ∈ H^{m}(ℝ) × H^{m}(ℝ), with m ≥ −3/4 and any b ∈ (1/2, 1), there exist T = T (||φ||_{H^{m}}, ||ψ||_{H^{m}}) and a unique solution of (1.3) in the time interval [−T, T] satisfying

u, v ∈ C([−T, T]; H^{m}(ℝ)),
u, v ∈ X_{m, b} ⊂ L_{x Loc}^{p}(ℝ; L_{t}^{2}(ℝ)) for 1 ≤ p ≤ ∞,
(u^{2})_{x}, (v^{2})_{x} ∈ X_{m, b−1},
u_{t}, v_{t} ∈ X_{m−3, b−1}

Moreover, given t ∈ (0, T), the map (φ, ψ) ↦ (u(t), v(t)) is smooth from H^{m}(ℝ) × H^{m}(ℝ) to C([−T, T]; H^{m}(ℝ)) × C([−T, T]; H^{m}(ℝ)).

Similar results in weighted Sobolev spaces were given by [29, 30] and references therein. In 1999, Alarcón-Angulo-Montenegro [2] showed that the system (1.2) is global well-posedness in the classical Sobolev space H^{m}(ℝ)×H^{m}(ℝ), m ≥ 1. For the UCP the first results are due to Saut and Scheurer [24]. They considered some dispersive operators in one space dimension of the type L = i D_{t}+α i^{2k+1} D^{2k+1}+R(x, t, D) where α ≠ 0, D = \frac{1}{i} \frac{∂}{∂x}, D_{t} = \frac{1}{i} \frac{∂}{∂t} and R(x, t, D) = ∑_{j=0}^{2k} r_{j}(x, t) D^{j}, r_{j} ∈ L_{loc}^{∞}(ℝ; L_{loc}^{2}(ℝ)). They proved that if u ∈ L_{loc}^{2}(ℝ; H_{loc}^{2k+1}(ℝ)) is a solution of Lu = 0, which vanishes in some open set Ω_{1} of ℝ_{x} × ℝ_{t}, then u vanishes in the horizontal component of Ω_{1}. As a consequence of the uniqueness of the solutions of the KdV equation in L_{loc}^{∞}(ℝ; H^{3}(ℝ)), their result immediately yields the following:

Theorem 1.2. If u ∈ L_{loc}^{∞}(ℝ; H^{3}(ℝ)) is a solution of the KdV equation

u_{t} + u_{xxx} + u u_{x} = 0 (1.4)

and vanishes on an open set of ℝ_{x} × ℝ_{t}, then u(x, t) = 0 for x ∈ ℝ, t ∈ ℝ.

In 1992, B. Zhang [32] proved using inverse scattering transform and some results from Hardy function theory that if u ∈ L_{Loc}^{∞}(ℝ; H^{m}(ℝ)), m > 3/2 is a solution of the KdV equation (1.4), then it cannot have compact support at two different moments unless it vanishes identically. As a consequence of the Miura transformation, the above results for the KdV equation (1.4) are also true for the modified Korteweg-de Vries equation

u_{t} + u_{xxx} − u^{2} u_{x} = 0. (1.5)

A variety of techniques such as spherical harmonics [26], singular integral operators [20], inverse scattering [31], and others have been used. However the Carleman methods which consists in establishing a priori estimates containing a weight has influenced a lot the development on the subject. This paper is organized as follows: In section 2, we prove two conserved integral quantities and local existence theorem. In section 3, we prove the Carleman estimate and Unique Continuation Property. In section 4, we prove the main theorem.

2 Preliminaries

We consider the following dispersive coupled system

(P) \{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + ∂_{x}(u v^{2}) = 0 & (P_{1}) \\ ∂_{t}v + ∂_{x}^{3}v + ∂_{x}(u^{2} v) + ∂_{x}v = 0 & (P_{2}) \\ u(x, 0) = u_{0}(x) ; v(x, 0) = v_{0}(x) & (P_{3}) \\ ∂_{x}^{k}u(0, t) = ∂_{x}^{k}u(1, t), k = 0, 1, 2. & (P_{4}) \\ ∂_{x}^{k}v(0, t) = ∂_{x}^{k}v(1, t), k = 0, 1, 2. & (P_{5}) \end{matrix}

with 0 ≤ x ≤ 1, t ≥ 0 and where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t.

Notation. We write time derivative by u_{t} = \frac{∂u}{∂t} = ∂_{t}u. Spatial derivatives are denoted by u_{x} = \frac{∂u}{∂x} = ∂_{x}u, u_{xx} = \frac{∂^{2}u}{∂x^{2}} = ∂_{x}^{2}u, u_{xxx} = \frac{∂^{3}u}{∂x^{3}} = ∂_{x}^{3}u.

If 𝔼 is any Banach space, its norm is written as || · ||_{𝔼}. For 1 ≤ p ≤ + ∞, the usual class of p^{th}- power Lebesgue-integrable (essentially bounded if p = + ∞) real-valued functions defined on the open set Ω in ℝ^{n} is written by L^{p}(Ω) and its norm is abbreviated as || · ||_{p}. The Sobolev space of L^{2}-functions whose derivatives up to order m also lie in L^{2} is denoted by H^{m}. We denote [H^{m_{1}}(Ω), H^{m_{2}}(Ω)] = H^{(1−θ)m_{1}+θm_{2}}(Ω) for all m_{i} > 0(i = 1, 2), m_{2} < m_{1}, 0 < θ < 1 (with equivalent norms) the interpolation of H^{m}(Ω)-spaces. If a function belongs, locally, to L^{p} or H^{m} we write f ∈ L_{loc}^{p} or f ∈ H_{loc}^{m}. C(0, T; 𝔼) denote the class of all continuous maps u : [0, T] → 𝔼 equipped with the norm ||u||_{C(0, T; 𝔼)} = sup_{0≤t≤T} ||u||_{𝔼}. u(x, t) ∈ C^{3, 1}(ℝ^{2}) if ∂_{x}u, ∂_{x}^{2}u, ∂_{x}^{3}u, ∂_{t}u ∈ C(ℝ^{2}). u(x, t) ∈ C_{0}^{3, 1}(ℝ^{2}) if u ∈ C^{3, 1}(ℝ^{2}) and u with compact support. Throughout this paper c is a generic constant, not necessarily the same at each occasion(will change from line to line), which depends in an increasing way on the indicated quantities. The next proposition is well known and it will be used frequently

Proposition 2.1. Let K be a non empty compact set and F a close subset of ℝ such that K ∩ F = ∅. Then there is ψ ∈ C_{0}^{∞}(ℝ) such that ψ = 1 in K, ψ = 0 in F and 0 ≤ ψ(x) ≤ 1, ∀ x ∈ ℝ.

Definition 2.2. Let L be an evolution operator acting on functions defined on some connected open set Ω of ℝ^{2} = ℝ_{x} × ℝ_{t}. L is said to have the horizontal unique continuation property if every solution u of Lu = 0 that vanishes on some nonempty open set Ω_{1} ⊂ Ω vanishes in the horizontal component of Ω_{1} in Ω, i. e., in Ω_{h} = {(x, t) ∈ Ω/ ∃ x_{1}, (x_{1}, t) ∈ Ω_{1}}.

Lemma 2.3. The equation (P) has the following conserved integral quantities, i. e.,

\frac{d}{dt} ∫_{0}^{1}(u^{2} + v^{2}) dx = 0,  (2.1)
\frac{d}{dt} ∫_{0}^{1} [u^{2} v^{2} − (u_{x}^{2} + v_{x}^{2}) + v^{2}] dx = 0.  (2.2)

Proof. (2.1) Straightforward. We show (2.2). Multiplying (P_{1}) by (u v^{2} + u_{xx}) and integrating over x ∈ (0, 1) we have

\frac{1}{2} ∫_{0}^{1}(u^{2})_{t} v^{2} dx + ∫_{0}^{1} u_{xx} u_{t} dx + ∫_{0}^{1}(u v^{2}) u_{xxx} dx

+\frac{1}{2} ∫_{0}^{1}(u_{xx}^{2})_{x} dx + \frac{1}{2} ∫_{0}^{1}[(u v^{2})^{2}]_{x} dx + ∫_{0}^{1}(u v^{2})_{x} u_{xx} dx = 0.

Each term is treated separately integrating by parts

\frac{1}{2} ∫_{0}^{1}(u^{2})_{t} v^{2} dx − ∫_{0}^{1} u_{x} u_{xt} dx − ∫_{0}^{1}(u v^{2})_{x} u_{xx} dx

+\frac{1}{2} ∫_{0}^{1}(u_{xx}^{2})_{x} dx + \frac{1}{2} ∫_{0}^{1}[(u v^{2})^{2}]_{x} dx + ∫_{0}^{1}(u v^{2})_{x} u_{xx} dx = 0

where

\frac{1}{2} ∫_{0}^{1}(u^{2})_{t}v^{2}dx − \frac{1}{2} ∫_{0}^{1}(u_{x}^{2})_{t}dx = 0.  (2.3)

Similarly, multiplying (P_{2}) by (u^{2} v + v_{xx} + v) and integrating over x ∈ (0, 1) we have

\frac{1}{2} ∫_{0}^{1} u^{2} (v^{2})_{t} dx + ∫_{0}^{1} v_{xx} v_{t} dx + \frac{1}{2} ∫_{0}^{1}(v^{2})_{t} dx + ∫_{0}^{1}(u^{2} v) v_{xxx} dx

+\frac{1}{2} ∫_{0}^{1}(v_{xx}^{2})_{x} dx + ∫_{0}^{1} v v_{xxx} dx + \frac{1}{2} ∫_{0}^{1}[(u^{2} v)^{2}]_{x} dx + ∫_{0}^{1}(u^{2} v)_{x} v_{xx} dx

+ ∫_{0}^{1}(u^{2} v)_{x} v dx + ∫_{0}^{1}(u^{2} v)_{x} v_{x} dx + \frac{1}{2} ∫_{0}^{1}(v_{x}^{2})_{x} dx + \frac{1}{2} ∫_{0}^{1}(v^{2})_{x} dx = 0.

Each term is treated separately, integrating by parts

\frac{1}{2} ∫_{0}^{1} u^{2} (v^{2})_{t} dx − ∫_{0}^{1} v_{x} v_{xt} dx + \frac{1}{2} ∫_{0}^{1}(v^{2})_{t} dx − ∫_{0}^{1}(u^{2} v)_{x} v_{xx} dx + \frac{1}{2} ∫_{0}^{1}(v_{xx}^{2})_{x} dx

+ ∫_{0}^{1} v v_{xxx} dx + \frac{1}{2} ∫_{0}^{1}[(u^{2} v)^{2}]_{x} dx + ∫_{0}^{1}(u^{2} v)_{x} v_{xx} dx + ∫_{0}^{1}(u^{2} v)_{x} v dx − ∫_{0}^{1}(u^{2} v)_{x} v dx

+\frac{1}{2} ∫_{0}^{1}(v_{x}^{2})_{x} dx + \frac{1}{2} ∫_{0}^{1}(v^{2})_{x} dx = 0

where

\frac{1}{2} ∫_{0}^{1} u^{2} (v^{2})_{t} dx − \frac{1}{2} ∫_{0}^{1}(v_{x}^{2})_{t} dx + \frac{1}{2} ∫_{0}^{1}(v^{2})_{t} dx = 0  (2.4)

adding (2.3) and (2.4), we have

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1} [u^{2} v^{2} − (u_{x}^{2} + v_{x}^{2}) + v^{2}] dx = 0

where

\frac{d}{dt} ∫_{0}^{1} [u^{2}v^{2} − (u_{x}^{2} + v_{x}^{2}) + v^{2}] dx = 0.  (2.5)

Lemma 2.4. For all u ∈ H^{1}(Ω)

||u||_{L^{∞}(Ω)} ≤ c ||u||_{L^{2}(Ω)}^{1/2} (||u||_{L^{2}(Ω)} + ||\frac{du}{dx}||_{L^{2}(Ω)})^{1/2}  (2.6)

and for all u ∈ H^{3}(Ω)

||u||_{L^{4}(Ω)} ≤ c ||u||_{L^{2}(Ω)}^{11/12} (||u||_{L^{2}(Ω)} + ||\frac{d^{3}u}{dx^{3}} ||_{L^{2}(Ω)})^{1/12}  (2.7)
||\frac{du}{dx}||_{L^{4}(Ω)} ≤ c ||u||_{L^{2}(Ω)}^{7/12} (||u||_{L^{2}(Ω)} + ||\frac{d^{3}u}{dx^{3}} ||_{L^{2}(Ω)})^{5/12}  (2.8)

Proof. See [28]. Theorem 2.5 (Local Existence). Let (u_{0}, v_{0}) ∈ H^{1}(0, 1)×H^{1}(0, 1) with u_{0}(0) = u_{0}(1) and v_{0}(0) = v_{0}(1). Then there exist T > 0 and (u, v) such that (u, v) is a solution of (P). (u, v) ∈ L^{∞}(0, T; H^{1}(0, 1)) × L^{∞}(0, T; H^{1}(0, 1)) and the initial data u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x) are satisfied. Proof. For ϵ > 0, we approximate the system (P) by the parabolic system

(R) \{ \begin{matrix}∂_{t}u_{ϵ} + ∂_{x}^{3}u_{ϵ} + ∂_{x}(u_{ϵ} v_{ϵ}^{2}) + ϵ ∂_{x}^{4}u_{ϵ} = 0 & (R_{1}) \\ ∂_{t}v_{ϵ} + ∂_{x}^{3}v_{ϵ} + ∂_{x}(u_{ϵ}^{2} v_{ϵ}) + ∂_{x}v_{ϵ} + ϵ ∂_{x}^{4}v_{ϵ} = 0 & (R_{2}) \\ u_{ϵ}(x, 0) = u_{0}(x) ; v_{ϵ}(x, 0) = v_{0}(x) & (R_{3}) \\ ∂_{x}^{k}u_{ϵ}(0, t) = ∂_{x}^{k}u_{ϵ}(1, t) ; ∂_{x}^{k}v_{ϵ}(0, t) = ∂_{x}^{k}v_{ϵ}(1, t), k = 0, 1, 2, 3. & (R_{4}) \end{matrix}

We rewrite the above equations in a more friendly way as

u_{t} + u_{xxx} + (u v^{2})_{x} + ϵ u_{xxxx} = 0  (2.9)
v_{t} + v_{xxx} + (u^{2} v)_{x} + v_{x} + ϵ v_{xxxx} = 0  (2.10)

We multiply (2.9) by u, integrate over x ∈ Ω = (0, 1), to have

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1} u^{2} dx − \frac{1}{2} ∫_{0}^{1}(u^{2})_{x} v^{2} dx + ϵ ∫_{0}^{1} u_{xx}^{2} dx = 0.  (2.11)

Similarly, we multiply (2.10) by v, integrate over x ∈ Ω = (0, 1), and we have

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1} v^{2} dx − \frac{1}{2} ∫_{0}^{1} u^{2}(v^{2})_{x} dx + ϵ ∫_{0}^{1} v_{xx}^{2} dx = 0.  (2.12)

Adding (2.11) and (2.12) we obtain

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1}(u^{2} + v^{2}) dx + ϵ ∫_{0}^{1}(u_{xx}^{2} + v_{xx}^{2}) dx = 0

where

\frac{d}{dt} ∫_{0}^{1}(u^{2} + v^{2}) dx + 2 ϵ ∫_{0}^{1}(u_{xx}^{2} + v_{xx}^{2}) dx = 0  (2.13)

then

||u_{ϵ}||_{L^{2}(0, 1)}^{2} + ||v_{ϵ}||_{L^{2}(0, 1)}^{2} + 2 ϵ ||u_{ϵ}||_{L^{2}(0, T; H^{2}(0, 1))}^{2} + 2 ϵ ||v_{ϵ}||_{L^{2}(0, T; H^{2}(0, 1))}^{2} = c.

Where in particular

||u_{ϵ}||_{L^{∞}(0, T; L^{2}(0, 1))} ≤ c ; ||v_{ϵ}||_{L^{∞}(0, T; L^{2}(0, 1))} ≤ c

\sqrt{ϵ} ||\frac{∂^{2}u_{ϵ}}{∂x^{2}} ||_{L^{2}(0, T; L^{2}(0, 1))} ≤ c ; \sqrt{ϵ} ||\frac{∂^{2}v_{ϵ}}{∂x^{2}} ||_{L^{2}(0, T; L^{2}(0, 1))} ≤ c

if and only if

\sqrt{ϵ} ||\frac{∂^{2}u_{ϵ}}{∂x^{2}} ||_{L^{2}(Q)} ≤ c ; \sqrt{ϵ} ||\frac{∂^{2}v_{ϵ}}{∂x^{2}} ||_{L^{2}(Q)} ≤ c

or

u_{ϵ}, v_{ϵ} ∈ L^{∞}(0, T; L^{2}(0, 1))  (2.14)
\sqrt{ϵ} u_{ϵ}, \sqrt{ϵ} v_{ϵ} ∈ L^{∞}(0, T; H^{2}(0, 1))  (2.15)

On the other hand, we multiply the equations in (R) by (u v^{2} + u_{xx}) and (u^{2} v + v_{xx} + v), respectively, and integrating over x ∈ Ω = (0, 1) and using Lemma 2.3, we obtain

\frac{d}{dt} ∫_{0}^{1} [\frac{1}{2} u^{2} v^{2} − \frac{1}{2} (u_{x}^{2} + v_{x}^{2}) + \frac{1}{2} v^{2}] dx + \frac{1}{2} ∫_{0}^{1} u^{2} (v^{2})_{x} dx

−ϵ ∫_{0}^{1}(u v^{2})_{x} u_{xxx} dx − ϵ ∫_{0}^{1}(u^{2} v)_{x} v_{xxx} dx − ϵ ∫_{0}^{1} u_{xxx}^{2} dx − ϵ ∫_{0}^{1} u_{xxx}^{2} dx − ∫_{0}^{1} v_{xx}^{2} dx = 0

then

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1} u_{x}^{2} dx + \frac{1}{2} \frac{d}{dt} ∫_{0}^{1} v_{x}^{2} dx + ϵ ∫_{0}^{1} u_{xxx} dx + ϵ ∫_{0}^{1} v_{xxx} dx + ∫_{0}^{1} v_{xx}^{2} dx = \frac{1}{2} \frac{d}{dt} ∫_{0}^{1} u^{2} v^{2} dx

+ \frac{1}{2} ∫_{0}^{1} v^{2} dx + \frac{1}{2} \frac{d}{dt} ∫_{0}^{1} u^{2} (v^{2})_{x} dx − ϵ ∫_{0}^{1}(u v^{2})_{x} u_{xxx} dx − ϵ ∫_{0}^{1}(u^{2} v)_{x} v_{xxx} dx

hence

\frac{d}{dt}||u_{x}||_{L^{2}(0, 1)}^{2} + \frac{d}{dt}||v_{x}||_{L^{2}(0, 1)}^{2} + 2 ϵ ||u_{xxx}||_{L^{2}(0, 1)}^{2} + 2 ϵ ||v_{xxx}||_{L^{2}(0, 1)}^{2} + 2 ||v_{xx}||_{L^{2}(0, 1)}^{2}

= \frac{d}{dt} ∫_{0}^{1} u^{2} v^{2} dx + \frac{d}{dt}||v||_{L^{2}(0, 1)}^{2} + ∫_{0}^{1} u^{2}(v^{2})_{x} dx − 2 ϵ ∫_{0}^{1}(u v^{2})_{x} u_{xxx} dx − 2 ϵ ∫_{0}^{1}(u^{2} v)_{x} v_{xxx} dx.

Integrating over t ∈ [0, T] we have

||u_{x}||_{L^{2}(0, 1)}^{2} + ||v_{x}||_{L^{2}(0, 1)}^{2} + 2 ϵ ∫_{0}^{t} ||u_{xxx}||_{L^{2}(0, 1)}^{2} dσ + 2 ϵ ∫_{0}^{t} ||v_{xxx}||_{L^{2}(0, 1)}^{2} dσ + 2 ∫_{0}^{t} ||v_{xx}||_{L^{2}(0, 1)}^{2} dσ
= ||\frac{du_{0}}{dx} ||_{L^{2}(0, 1)}^{2} + ||\frac{dv_{0}}{dx} ||_{L^{2}(0, 1)}^{2} + ∫_{0}^{1} u^{2}v^{2}dx − ∫_{0}^{1} u_{0}^{2}v_{0}^{2} dx + ||v||_{L^{2}(0, 1)}^{2} − ||v_{0}||_{L^{2}(0, 1)}^{2}

− 2 ϵ ∫_{0}^{t} ∫_{0}^{1}(u v^{2})_{x} u_{xxx} dx dσ − 2 ϵ ∫_{0}^{t} ∫_{0}^{1}(u^{2} v)_{x} v_{xxx}dxdσ + ∫_{0}^{t} ∫_{0}^{1} u^{2} (v^{2})_{x} dx dσ

or

||u_{x}||_{L^{2}(0, 1)}^{2} + ||v_{x}||_{L^{2}(0, 1)}^{2} + 2 ϵ ∫_{0}^{t} ||u_{xxx}||_{L^{2}(0, 1)}^{2} dσ + 2 ϵ ∫_{0}^{t} ||v_{xxx}||_{L^{2}(0, 1)}^{2} dσ + 2 ∫_{0}^{t} ||v_{xx}||_{L^{2}(0, 1)}^{2} dσ

= ||\frac{du_{0}}{dx} ||_{L^{2}(0, 1)}^{2} + ||\frac{dv_{0}}{dx} ||_{L^{2}(0, 1)}^{2} + ∫_{0}^{1} u^{2} v^{2} dx − ∫_{0}^{1} u_{0}^{2} v_{0}^{2} dx + ||v||_{L^{2}(0, 1)}^{2} − ||v_{0}||_{L^{2}(0, 1)}^{2}

− 2 ϵ ∫_{0}^{t} ∫_{0}^{1} v^{2} u_{x} u_{xxx} dx dσ − 4 ϵ ∫_{0}^{t} ∫_{0}^{1} u v v_{x} u_{xxx} dx dσ − 4 ϵ ∫_{0}^{t} ∫_{0}^{1} u v u_{x} v_{xxx} dx dσ
− 2 ϵ ∫_{0}^{t} ∫_{0}^{1} u^{2} v_{x} v_{xxx} dx dσ + 2 ∫_{0}^{t} ∫_{0}^{1} u^{2}v v_{x} dx dσ.  (2.16)

On the other hand, using the Lemma 2.4 and performing appropriate calculations we obtain

|∫_{0}^{1} u^{2} v^{2} dx| ≤ ||u||_{L^{∞}(0, 1)} ||v||_{L^{∞}(0, 1)} ∫_{0}^{1} |u| |v| dx
≤ ||u||_{L^{∞}(0, 1)} ||v||_{L^{∞}(0, 1)} ||u||_{L^{2}(0, 1)} ||v||_{L^{2}(0, 1)}
≤ c ||u||_{L^{∞}(0, 1)} ||v||_{L^{∞}(0, 1)}
≤ c ||u||_{L^{2}(0, 1)}^{1/2} (||u||_{L^{2}(0, 1)} + ||\frac{du}{dx}||_{L^{2}(0, 1)})^{1/2} ||v||_{L^{2}(0, 1)}^{1/2} (||v||_{L^{2}(0, 1)} + ||\frac{dv}{dx}||_{L^{2}(0, 1)})^{1/2}

≤ c + \frac{1}{2} ||\frac{du}{dx}||_{L^{2}(0, 1)}^{2} + \frac{1}{2} ||\frac{du}{dx}||_{L^{2}(0, 1)}^{2}
= c + \frac{1}{2} ||u_{x}||_{L^{2}(0, 1)}^{2} + \frac{1}{2} ||v_{x}||_{L^{2}(0, 1)}^{2}

hence

|∫_{0}^{1} u^{2} v^{2}dx| ≤ c + \frac{1}{2} ||u_{x}||_{L^{2}(0, 1)}^{2} + \frac{1}{2} ||v_{x}||_{L^{2}(0, 1)}^{2}.  (2.17)

We also have

|∫_{0}^{1} v^{2} u_{x} u_{xxx} dx| ≤ ∫_{0}^{1} |v|^{2} |u_{x}| |u_{xxx}| dx ≤ ||v||_{L^{∞}(0, 1)} ∫_{0}^{1} |v| |u_{x}| |u_{xxx}| dx
≤ ||v||_{L^{∞}(0, 1)} (∫_{Ω} |v|^{4} dx)^{1/4} (∫_{Ω} |\frac{du}{dx}|^{4} dx)^{1/4} (∫_{Ω} |\frac{d^{3}u}{dx^{3}} |^{2} dx)^{1/2}
≤ ||v||_{L^{∞}(0, 1)} ||v||_{L^{4}(0, 1)} ||\frac{du}{dx}||_{L^{4}(0, 1)} ||\frac{d^{3}u}{dx^{3}} ||_{L^{4}(0, 1)}
≤ c ||v||_{L^{2}(0, 1)}^{11/12} (||v||_{L^{2}(0, 1)} + ||\frac{d^{3}v}{dx^{3}} ||_{L^{2}(0, 1)}^{2})^{1/12} ×

||u||_{L^{2}(0, 1)}^{7/12} (||u||_{L^{2}(0, 1)} + ||\frac{d^{3}u}{dx^{3}} ||_{L^{2}(0, 1)}^{2})^{5/12} ||u_{xxx}||_{L^{2}(0, 1)}

≤ c + \frac{1}{4} ||\frac{d^{3}u}{dx^{3}} ||_{L^{2}(0, 1)}^{2} + \frac{1}{4} ||\frac{d^{3}v}{dx^{3}} ||_{L^{2}(0, 1)}^{2} .

Hence,

|∫_{0}^{1} v^{2} u_{x} u_{xxx} dx| ≤ c + \frac{1}{4} ||u_{xxx}||_{L^{2}(0, 1)}^{2} + \frac{1}{4} ||v_{xxx}||_{L^{2}(0, 1)}^{2}  (2.18)

We calculate in similar form the terms

|∫_{0}^{1} u v v_{x} u_{xxx} dx| , |∫_{0}^{1} u v u_{x} v_{xxx} dx| , |∫_{0}^{1} u^{2} v_{x} v_{xxx} dx| , |∫_{0}^{1} u^{2} v v_{x} dx| . This way we have ||u_{x}||_{L^{2}(0, 1)}^{2} + ||v_{x}||_{L^{2}(0, 1)}^{2} + ϵ ∫_{0}^{t} ||u_{xxx}||_{L^{2}(0, 1)}^{2} dσ + ϵ ∫_{0}^{t} ||v_{xxx}||_{L^{2}(0, 1)}^{2} dσ + ∫_{0}^{t} ||v_{xx}||_{L^{2}(0, 1)}^{2} dσ ≤ c or

||\frac{du_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} + ||\frac{dv_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} + ϵ ∫_{0}^{t} ||\frac{d^{3}u_{ϵ}}{dx^{3}} ||_{L^{2}(0, 1)}^{2} dσ + ϵ ∫_{0}^{t} ||\frac{d^{3}v_{ϵ}}{dx^{3}} ||_{L^{2}(0, 1)}^{2} dσ + ∫_{0}^{t} ||\frac{d^{2}v_{ϵ}}{dx^{2}} ||_{L^{2}(0, 1)}^{2} dσ ≤ c.

In particular,

||\frac{du_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} ≤ c, ||\frac{dv_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} ≤ c, \sqrt{ϵ} ||\frac{d^{3}u_{ϵ}}{dx^{3}} ||_{L^{2}(0, T; L^{2}(0, 1))}^{2} ≤ c, \sqrt{ϵ} ||\frac{d^{3}v_{ϵ}}{dx^{3}} ||_{L^{2}(0, T; L^{2}(0, 1))}^{2} ≤ c

then

||\frac{du_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} ≤ c, ||\frac{dv_{ϵ}}{dx} ||_{L^{2}(0, 1)}^{2} ≤ c, \sqrt{ϵ} ||\frac{d^{3}u_{ϵ}}{dx^{3}} ||_{L^{2}(Q)}^{2} ≤ c, \sqrt{ϵ} ||\frac{d^{3}v_{ϵ}}{dx^{3}} ||_{L^{2}(Q)}^{2} ≤ c

or

u_{ϵ}, v_{ϵ} ∈ L^{∞}(0, T; H^{1}(0, 1)) ⋂ L^{2}(0, T; H^{3}(0, 1))  (2.19)
v_{ϵ} ∈ L^{2}(0, T; H^{2}(0, 1))  (2.20)

EJQTDE, 2005, No. 14, p. 7 Hence from (2.14)-(2.15) and (2.19)-(2.20) we have the existence of subsequences u_{ϵ_{j}} =^{def} u_{ϵ} and v_{ϵ_{j}} =^{def} v_{ϵ} such that

u_{ϵ} ⇀^{∗} u weakly in L^{∞}(0, T; L^{2}(0, 1)) ↪ L^{2}(0, T; L^{2}(0, 1)) = L^{2}(Q)

v_{ϵ} ⇀^{∗} v weakly in L^{∞}(0, T; L^{2}(0, 1)) ↪ L^{2}(0, T; L^{2}(0, 1)) = L^{2}(Q)

\frac{∂u_{ϵ}}{∂x} ⇀^{∗} \frac{∂u}{∂x} weakly in L^{∞}(0, T; L^{2}(0, 1)) ↪ L^{2}(0, T; L^{2}(0, 1)) = L^{2}(Q)

\frac{∂v_{ϵ}}{∂x} ⇀^{∗} \frac{∂v}{∂x} weakly in L^{∞}(0, T; L^{2}(0, 1)) ↪ L^{2}(0, T; L^{2}(0, 1)) = L^{2}(Q)

from the equation (R) we deduce that

\frac{∂u_{ϵ}}{∂t} ⇀^{∗} \frac{∂u}{∂t} weakly in L^{2}(0, T; H^{−2}(0, 1))

\frac{∂v_{ϵ}}{∂t} ⇀^{∗} \frac{∂v}{∂t} weakly in L^{2}(0, T; H^{−2}(0, 1)).

By other hand, we have H^{1}(0, 1) ↪^{c} L^{2}(0, 1) ↪ H^{−2}(0, 1). Using Lions-Aubin’s compactness Theorem

u_{ϵ} → u strongly in L^{2}(Q)
v_{ϵ} → v strongly in L^{2}(Q).

Then

∂_{x}(u_{ϵ} v_{ϵ}^{2}) = 2 u_{ϵ} v_{ϵ} \frac{∂v_{ϵ}}{∂x} + \frac{∂u_{ϵ}}{∂x} v_{ϵ} v_{ϵ} ⟶ 2 u v \frac{∂v}{∂x} + \frac{∂u}{∂x} v v = ∂_{x}(u v^{2}) in D′(0, 1).

The other terms are calculated in a similar way and therefore we can pass to the limit in the equation (R). Finally, u, v are solutions of the equation (P) and the theorem follows.

3 Carleman’s estimate and unique continuation property

We consider the equation (P), then

(Q) \{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + v^{2} u_{x} + 2 u v v_{x} = 0 \\ ∂_{t}v + ∂_{x}^{3}v + 2 u v u_{x} + u^{2} v_{x} + v_{x} = 0 \end{matrix}

We rewrite the above equations as

∂_{t} [ \begin{matrix}u \\ v \end{matrix} ] + ∂_{x}^{3} [ \begin{matrix}u \\ v \end{matrix} ] + [ \begin{matrix}v^{2} & 2 u v \\ 2 u v & u^{2} \end{matrix} ] ∂_{x} [ \begin{matrix}u \\ v \end{matrix} ] + [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ] ∂_{x} [ \begin{matrix}u \\ v \end{matrix} ] = [ \begin{matrix}0 \\ 0 \end{matrix} ]

Let U = U(x, t),

U = [ \begin{matrix}u \\ v \end{matrix} ] ; B(U) = [ \begin{matrix}v^{2} & 2 u v \\ 2 u v & u^{2} \end{matrix} ] = [ \begin{matrix}f_{1} & f_{2} \\ f_{3} & f_{4} \end{matrix} ] ; C = [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

Hence in (Q) we obtain

U_{t} + U_{xxx} + (B(U) + C) U_{x} = 0,  0 ≤ x ≤ 1, t ≥ 0  (3.1)
U(x, 0) = U_{0}(x) ; ∂_{x}^{k}U(0, t) = ∂_{x}^{k}U(1, t),  k = 0, 1, 2.  (3.2)

Then

LU = [I∂_{t} + I ∂_{x}^{3} + B(U) ∂_{x}] U.  (3.3)

System (3.1) may be written as

LU = 0  (3.4)

with

L = I∂_{t} + I ∂_{x}^{3} + B(U) ∂_{x}.  (3.5)

We see in (3.1) and (3.4) that the system (3.1) may be written as LU = 0 where the operator L is given in (3.5). It has the form:

L = [ \begin{matrix}L_{1} & f_{2} ∂_{x} \\ f_{3} ∂_{x} & L_{2} \end{matrix} ] (α)

with

L_{1} = ∂_{t} + ∂_{x}^{3} + f_{1} ∂_{x}  (3.6)
L_{2} = ∂_{t} + ∂_{x}^{3} + f_{4} ∂_{x}.  (3.7)

Proposition 3.1 (Carleman’s Estimate). Let δ > 0, 𝔹_{δ} = {(x, t) ∈ ℝ^{2}/ x^{2} + t^{2} < δ^{2}}, φ(x, t) = (x − δ)^{2} + δ^{2} t^{2} and the differential operator L defined by (3.5). Assume that f_{k} ∈ L^{∞}(𝔹_{δ}), k = 1, 2, 3, 4. Then

3 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 12 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ}dx dt ≤ 2 ∫_{𝔹_{δ}} |LΦ|^{2} e^{2 τ φ} dx dt  (3.8)

for any Φ ∈ C_{0}^{∞}(𝔹_{δ}) × C_{0}^{∞}(𝔹_{δ}) and τ > 0 large enough.

Proof. We consider the operator P_{1} = ∂_{t} + ∂_{x}^{3}, then using the Treve inequality

96 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2τ φ} dx dt ≤ ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt  (3.9)
384 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ} dx dt ≤ ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt  (3.10)

whenever Φ ∈ C_{0}^{∞}(𝔹_{δ}) and τ > 0. Adding up the inequalities (3.9) and (3.10), we obtain

96 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 384 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2}e^{2 τ φ} dx dt ≤ 2 ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt

for any Φ ∈ C_{0}^{∞}(𝔹_{δ}) and τ > 0. Then

12 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ} dx dt ≤ \frac{1}{4} ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt  (3.11)

for any Φ ∈ C_{0}^{∞}(𝔹_{δ}) and τ > 0. Similarly, we have for the operator P_{1} = ∂_{t} + ∂_{x}^{3}.

12 τ^{2} ∫_{𝔹_{δ}} |Ψ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Ψ|^{2} e^{2 τ φ} dx dt ≤ \frac{1}{4} ∫_{𝔹_{δ}} |P_{1}Ψ|^{2} e^{2 τ φ} dx dt  (3.12)

for any Ψ ∈ C_{0}^{∞}(𝔹_{δ}) and τ > 0. On the other hand,

∫_{𝔹_{δ}} |f_{1}Φ_{x}|^{2}e^{2 τ φ} dx dt ≤ ||f_{1}||_{L^{∞}(𝔹_{δ})}^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt.  (3.13)

Let τ ≥ \frac{\sqrt{6}}{12} ||f_{1}||_{L^{∞}(𝔹_{δ})}, then τ^{2} ≥ \frac{1}{24} ||f_{1}||_{L^{∞}(𝔹_{δ})}^{2}. This way in (3.9) we have

∫_{𝔹_{δ}} |P_{1}Φ|^{2}e^{2 τ φ} dx dt ≥ 96 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt ≥ 96 \frac{1}{24} ||f_{1}||_{L^{∞}(𝔹_{δ})}^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt
= 4 ||f_{1}||_{L^{∞}(𝔹_{δ})}^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt ≥ 4 ∫_{𝔹_{δ}} |f_{1}Φ_{x}|^{2}e^{2 τ φ} dx dt (using (3.13))

hence

∫_{𝔹_{δ}} |f_{1}Φ_{x}|^{2} e^{2 τ φ} dx dt ≤ \frac{1}{4} ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt.  (3.14)

Then adding (3.12) and (3.14) we have

∫_{𝔹_{δ}} |f_{1}Φ_{x}|^{2} e^{2 τ φ} dx dt + 12 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ} dx dt
≤ \frac{1}{2} ∫_{𝔹_{δ}} |P_{1}Φ|^{2} e^{2 τ φ} dx dt.  (3.15)

But L_{1} = ∂_{t} + ∂_{x}^{3} + f_{1} ∂_{x} = P_{1} + f_{1} ∂_{x}. Then P_{1}Φ = L_{1}Φ − f_{1}Φ_{x} and |P_{1}Φ|^{2} ≤ 2 |L_{1}Φ|^{2} + 2 |f_{1}Φ_{x}|^{2}. Hence, in (3.15)

∫_{𝔹_{δ}} |f_{1}Φ_{x}|^{2} e^{2 τ φ} dx dt + 12 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ} dx dt
≤ ∫_{𝔹_{δ}} |L_{1}Φ|^{2} e^{2 τ φ} dx dt + ∫_{𝔹_{δ}} |f_{1} Φ_{x}|^{2}e^{2 τ φ} dx dt

then

12 τ^{2} ∫_{𝔹_{δ}} |Φ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Φ|^{2} e^{2 τ φ} dx dt ≤ ∫_{𝔹_{δ}} |L_{1}Φ|^{2} e^{2 τ φ} dx dt.  (3.16)

for any Φ ∈ C_{0}^{∞}(𝔹_{δ}) and τ ≥ \frac{\sqrt{6}}{12} ||f_{1}||_{L^{∞}(𝔹_{δ})}. Performing similar calculations with (3.12) and the operator L_{2}, we obtain

12 τ^{2} ∫_{𝔹_{δ}} |Ψ_{x}|^{2} e^{2 τ φ} dx dt + 48 τ^{3} ∫_{𝔹_{δ}} |Ψ|^{2} e^{2 τ φ} dx dt ≤ ∫_{𝔹_{δ}} |L_{2}Ψ|^{2} e^{2 τ φ} dx dt.  (3.17)

for any Ψ ∈ C_{0}^{∞}(𝔹_{δ}) and τ ≥ \frac{\sqrt{6}}{12} ||f_{4}||_{L^{∞}(𝔹_{δ})}. Summing up (3.16) and (3.17), we have

3 τ^{2} ∫_{𝔹_{δ}} |Θ_{x}|^{2} e^{2 τ φ} dx dt + 12 τ^{3} ∫_{𝔹_{δ}} |Θ|^{2} e^{2 τ φ} dx dt ≤ \frac{1}{4} ∫_{𝔹_{δ}}( |L_{1}Φ|^{2} + |L_{2}Ψ|^{2} ) e^{2 τ φ} dx dt. (3.18)

Whenever

Θ = [ \begin{matrix}Φ \\ Ψ \end{matrix} ] ∈ C_{0}^{∞}(𝔹_{δ}) × C_{0}^{∞}(𝔹_{δ})  and  τ ≥ Max \{\frac{\sqrt{6}}{12} ||f_{1}||_{L^{∞}(𝔹_{δ})} ; \frac{\sqrt{6}}{12} ||f_{4}||_{L^{∞}(𝔹_{δ})}\} .

Similarly, since f_{2}, f_{3} ∈ L^{∞}(𝔹_{δ}) and according to L and (3.6), (3.7) we have

|LΘ|^{2} = |L_{1}Φ + f_{2}Ψ_{x}|^{2} + |L_{2}Ψ + f_{3}Φ_{x}|^{2}

Then we can add f_{2}Ψ_{x} to L_{1}Φ and f_{3}Φ_{x} to L_{2}Ψ in (3.18), and obtain Carleman’s estimate (3.8) when

Θ = [ \begin{matrix}Φ \\ Ψ \end{matrix} ] ∈ C_{0}^{∞}(𝔹_{δ}) × C_{0}^{∞}(𝔹_{δ})

and τ satisfying τ ≥ Max \{\frac{\sqrt{6}}{12} ||f_{1}||_{L^{∞}(𝔹_{δ})} ; \frac{\sqrt{6}}{12} ||f_{4}||_{L^{∞}(𝔹_{δ})}, \frac{\sqrt{6}}{6} ||f_{2}||_{L^{∞}(𝔹_{δ})} ; \frac{\sqrt{6}}{6} ||f_{3}||_{L^{∞}(𝔹_{δ})}\} . Remark 3.2. The estimate (3.8) is invariant under changes of signs of any term in L_{1} or L_{2}. Corollary 3.3. Assume that, in addition to the hypotheses of Proposition 3.1, we have for any T > 0 and a > 0 that

V = [ \begin{matrix}ξ \\ η \end{matrix} ] ∈ L^{2}(−T, T; H^{3}(−a, a) × H^{3}(−a, a)).

V_{t} = [ \begin{matrix}ξ_{t} \\ η_{t} \end{matrix} ] ∈ L^{2}(−T, T; H^{3}(−a, a) × L^{2}(−a, a)).

and that supp ξ and supp η are compact sets in 𝔹_{δ}. Then, the estimate (3.8) holds with V instead of

Θ = [ \begin{matrix}Φ \\ Ψ \end{matrix} ]

Indeed,

3 τ^{2} ∫_{𝔹_{δ}} |V_{x}|^{2} e^{2 τ φ} dx dt + 12 τ^{3} ∫_{𝔹_{δ}} |V |^{2} e^{2 τ φ} dx dt ≤ 2 ∫_{𝔹_{δ}} |LV |^{2} e^{2 τ φ} dx dt  (3.19)

for τ > 0 sufficiently large. Proof. Choose a regularization sequence {ρ_{ϵ}(x, t)}_{ϵ>0}. Consider the functions

V^{ϵ} = ρ_{ϵ} ∗ V = [ \begin{matrix}ρ_{ϵ} ∗ ξ \\ ρ_{ϵ} ∗ η \end{matrix} ] (∗ denote the usual convolution)

Hence, the inequality (3.8) is valid replacing

Θ = [ \begin{matrix}Φ \\ Ψ \end{matrix} ] by V^{ϵ} = [ \begin{matrix}ρ_{ϵ} ∗ ξ \\ ρ_{ϵ} ∗ η \end{matrix} ] if ϵ > 0 is sufficiently small.

Taking the limit as ϵ → 0^{+} the result follows. Theorem 3.4. Let T > 0, a > 0 and R = (−a, a) × (−T, T). Let L be the operator defined in (α) and let

U = [ \begin{matrix}u \\ v \end{matrix} ] ∈ L^{2}(−T, T ; H^{3}(−a, a) × H^{3}(−a, a))

be a solution of the differential equation (3.4), LU = 0. Assume that f_{k} ∈ L^{∞}(ℝ), k = 1, 2, 3, 4 and U ≡ 0 when x < t^{2} in a neighborhood of (0, 0). Then, there exists a neighborhood of (0, 0) for which U ≡ 0. Proof. We choose a positive number δ < 1 such that 𝔹_{δ} lies in the neighborhood where U ≡ 0 when x < t^{2}. Let χ ∈ C_{0}^{∞}(𝔹_{δ}), χ ≡ 1 on a neighborhood N of (0, 0) and set

V = χU = [ \begin{matrix}χ u \\ χ v \end{matrix} ] .

The function V satisfies the conditions of Corollary 3.3, and L = 0 on N because V = U on N . Hence by (3.19)

6 τ^{3} ∫_{𝔹_{δ}} |V |^{2} e^{2 τ φ} dx dt ≤ ∫_{𝔹_{δ}−N} |LV |^{2} e^{2 τ φ} dx dt  (3.20)

when τ > 0 is sufficiently large. On the other hand, if (x, t) ∈ supp V one has 0 ≤ t^{2} ≤ x < δ < 1 and

φ(x, t) = (x − δ)^{2} + δ^{2} t^{2} = (δ − x)^{2} + δ^{2} t^{2} = (t^{2} − δ)^{2} + δ^{2} t^{2}
= t^{4} − 2 t^{2} δ + δ^{2} + δ^{2} t^{2} = t^{2} [ (t^{2} − 2 δ + δ^{2}) + δ^{2} ]
< δ^{2} [ (δ^{2} − 2 δ + δ^{2}) + δ^{2} ] < δ^{2}

EJQTDE, 2005, No. 14, p. 11 and where φ(0, 0) = δ^{2}. Therefore, if (x, t) ∈ supp LV there is an ϵ > 0 such that φ(x, t) ≤ δ^{2} − ϵ. We can choose a neighborhood N′ of (0, 0) at which φ(x, t) > δ^{2} − ϵ and obtain from (3.20)

∫_{N′} |V |^{2} e^{2 τ φ} dx dt ≤ \frac{1}{6 τ^{3}} ∫_{IB_{δ}−N} |LV |^{2} e^{2 τ φ} dx dt  (3.21)

Taking limit in (3.21) as τ → + ∞ one deduces that U = V ≡ 0 on N′.

Definition 3.5. By a Holmgren’s transformation we mean a transformation which is defined by ξ = t, η = x + t^{2} and which maps the half-space x ≥ 0 into the domain Ω = {(η, ξ) ∈ ℝ^{2} : η − ξ^{2} ≥ 0}.

Corollary 3.6. Under the assumptions of Theorem 3.4, consider the curve x = μ_{0}(t), μ_{0}(0) = 0, μ_{0} a continuously differentiable function in a neighborhood of (0, 0). Suppose that U ≡ 0 in the region x < μ_{0}(t) in a neighborhood of (0, 0). Then, there exists a neighborhood of (0, 0) where U ≡ 0.

Proof. We consider the Holmgren transformation

(x, t) ⟶ (η, ξ) , η = x − μ_{0}(t) + t^{2} , ξ = t.

With this variables the function U = U(η, ξ) satisfies U ≡ 0 when η < ξ^{2} in a neighborhood of (0, 0) and FU = 0, where

F = [ \begin{matrix}∂_{ξ} + ∂_{η}^{3} + F_{1} ∂_{η} & f_{2} ∂_{η} \\ f_{2} ∂_{η} & ∂_{ξ} + ∂_{η}^{3} + F_{4} ∂_{η} \end{matrix} ]

where

F_{1}(η, ξ) = f_{1}(x, t) + (2 ξ − μ′_{0}(ξ))  and  F_{4}(η, ξ) = f_{4}(x, t) + (2 ξ − μ′_{0}(ξ))

Thus by using Theorem 3.4, and Holmgren’s transformation we conclude that there exists a neighborhood of (0, 0) in the x t− plane where U ≡ 0.

Theorem 3.7. Let T > 0 and Ω = (0, 1) × (0, T). Assume that f_{k} ∈ L_{Loc}^{∞}(Ω), k = 1, 2, 3, 4. Let

U = [ \begin{matrix}u \\ v \end{matrix} ] ∈ L^{2}(0, T; H^{3}(0, 1) × H^{3}(0, 1))

be a solution of the equation (3.1). If U ≡ 0 in an open subset Ω_{1} of Ω, then U ≡ 0 in the horizontal component Ω_{h} of Ω_{1} in Ω.

Proof. We prove the theorem for the equation (3.1) or the equivalent equation LU = 0, where L is the operator defined in (2.4). The proof follows as in [21] applying Corollary 3.6 and considering Remark 3.2.

Corollary 3.8. Let T > 0, Ω = (0, 1) × (0, T) and let

U = [ \begin{matrix}u \\ v \end{matrix} ] ∈ C(0, T; H_{p}^{3}(0, 1) × H_{p}^{3}(0, 1))

be a solution of equation (3.1). Suppose that U vanishes in an open subset Ω_{1} of Ω. Then U ≡ 0 in Ω.

4 The Main Theorem

The first result is concerned with the decay properties of solutions to the coupled system. The idea goes back to T. Kato [11].

Lemma 4.1. Let (u, v) = (u(x, t), v(x, t)) be a solution of the coupled system equations (P) such that

sup_{t∈[0, 1]} ||u( . , t)||_{H^{1}(ℝ)} < +∞ ; sup_{t∈[0, 1]} ||v( . , t)||_{H^{1}(ℝ)} < +∞

and e^{βx} u_{0} ∈ L^{2}(ℝ); e^{βx} v_{0} ∈ L^{2}(ℝ), ∀ β > 0. Then e^{βx}u ∈ C([0, 1]; L^{2}(ℝ)); e^{βx}v ∈ C([0, 1]; L^{2}(ℝ)).

Proof. Let φ_{n} ∈ C^{∞}(ℝ) be defined by

φ_{n}(x) = \begin{cases}e^{β x} & , for x ≤ n \\ e^{2 β x} & , for x > 10 n \end{cases}

with

φ_{n}(x) ≤ e^{2 β x} ; 0 ≤ φ′_{n}(x) ≤ β φ_{n}(x) ; |φ_{n}^{(j)}(x)| ≤ β^{j} φ_{n}(x)  j = 2, 3.  (4.1)

Examples.

Figure 1: These are sample figures for different values of n.

Figure 2: This is a figure comparing two functions with different values of n.

Let

u_{t} + u_{xxx} + u_{x} v^{2} + u (v^{2})_{x} = 0.  (4.2)

Multiplying the equation (4.2) by u φ_{n}, and integrating by parts we get

∫_{ℝ} u u_{t} φ_{n} dx + ∫_{ℝ} u u_{xxx} φ_{n} dx + ∫_{ℝ} u u_{x} v^{2} φ_{n} dx + ∫_{ℝ} u^{2} (v^{2})_{x} φ_{n} dx = 0

hence

\frac{1}{2} \frac{d}{dt} ∫_{ℝ} u^{2} φ_{n} dx + ∫_{ℝ} u u_{xxx} φ_{n} dx + \frac{1}{2} ∫_{ℝ}(u^{2})_{x} v^{2} φ_{n} dx + ∫_{ℝ} u^{2} (v^{2})_{x} φ_{n} dx = 0.

Integrating by parts we obtain

\frac{1}{2} \frac{d}{dt} ∫_{ℝ} u^{2} φ_{n} dx + \frac{3}{2} ∫_{ℝ} u_{x}^{2} φ′_{n} dx − \frac{1}{2} ∫_{ℝ} u^{2} φ′′′_{n} dx + \frac{1}{2} ∫_{ℝ}(u^{2})_{x} v^{2} φ_{n} dx

− ∫_{ℝ}(u^{2})_{x} v^{2} φ_{n} dx − ∫_{ℝ} u^{2}v^{2} φ′_{n} dx = 0

then

\frac{d}{dt} ∫_{ℝ} u^{2} φ_{n} dx + 3 ∫_{ℝ} u_{x}^{2} φ′_{n} dx − ∫_{ℝ} u^{2} φ′′′_{n} dx − ∫_{ℝ}(u^{2})_{x} v^{2} φ_{n} dx

−2 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0.  (4.3)

Similarly, multiplying the equation

v_{t} + v_{xxx} + (u^{2})_{x} v + v_{x} u^{2} = 0  (4.4)

by v φ_{n} and integrating by parts, we obtain

\frac{d}{dt} ∫_{ℝ} v^{2} φ_{n} dx + 3 ∫_{ℝ} v_{x}^{2} φ′_{n} dx − ∫_{ℝ} v^{2} φ′′′_{n} dx − ∫_{ℝ} u^{2} (v^{2})_{x} φ_{n} dx

− 2 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0.  (4.5)

Hence, from (4.3) and (4.5)

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 ∫_{ℝ}[u_{x}^{2} + v_{x}^{2}] φ′_{n} dx − ∫_{ℝ}[u^{2} + v^{2}] φ′′′_{n} dx

− ∫_{ℝ}[(u^{2})_{x} v^{2} + u^{2} (v^{2})_{x}] φ_{n}dx − 4 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0

then

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n}dx + 3 ∫_{ℝ}[u_{x}^{2} + v_{x}^{2}] φ′_{n} dx − ∫_{ℝ}[u^{2} + v^{2}] φ′′′_{n} dx

− ∫_{ℝ}(u^{2} v^{2})_{x} φ_{n} dx − 4 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0

hence,

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 ∫_{ℝ}[u_{x}^{2} + v_{x}^{2} ] φ′_{n} dx − ∫_{ℝ}[u^{2} + v^{2}] φ′′′_{n} dx

+ ∫_{ℝ} u^{2} v^{2} φ′_{n} dx − 4 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 ∫_{ℝ}[u_{x}^{2} + v_{x}^{2}] φ′_{n} dx − ∫_{ℝ}[u^{2} + v^{2}] φ′′′_{n} dx

−3 ∫_{ℝ} u^{2} v^{2} φ′_{n} dx = 0

hence,

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 ∫_{ℝ}[u_{x}^{2} + v_{x}^{2}] φ′_{n} dx = ∫_{ℝ}[u^{2} + v^{2}] φ′_{n} dx + 3 ∫_{ℝ} u u v v φ′_{n} dx

≤ β^{3} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 β ∫_{ℝ} u u v v φ_{n} dx. (using (4.1))

Using that φ_{n} ≥ 0 and the Holder inequality

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx ≤ β^{3} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + 3 β ||u||_{L^{∞}(ℝ)} ||v||_{L^{∞}(ℝ)} ∫_{ℝ} u v φ_{n} dx
≤ β^{3} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx

+ 3 β ||u||_{L^{∞}(ℝ)} ||v||_{L^{∞}(IR)} (∫_{ℝ} u^{2} φ_{n} dx)^{1/2} (∫_{ℝ} v^{2} φ_{n} dx)^{1/2}
≤ β^{3} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx + \frac{3}{2} β ||u||_{L^{∞}(ℝ)} ||v||_{L^{∞}(IR)} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx
≤ [β^{3} + \frac{3}{2} β ||u||_{L^{∞}(ℝ)} ||v||_{L^{∞}(ℝ)}] ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx

then

\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx ≤ c_{1} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx

where c_{1} = β^{3} + \frac{3}{2} β ||u||_{L^{∞}(ℝ)} ||v||_{L^{∞}(ℝ)}. Thus,

\frac{\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx}{∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx} ≤ c_{1}.

Integrating over t ∈ [0, 1] we have

∫_{0}^{t} \frac{\frac{d}{dt} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx}{∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx} ds ≤ c_{1} t

then

Ln [\frac{∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx}{∫_{ℝ}[u_{0}^{2} + v_{0}^{2}] φ_{n} dx}] ≤ c_{1} t

hence

∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx ≤ [∫_{ℝ}[u_{0}^{2} + v_{0}^{2}] φ_{n} dx] e^{c_{1} t}

thus

sup_{t∈[0, 1]} ∫_{ℝ}[u^{2} + v^{2}] φ_{n} dx ≤ [∫_{ℝ}[u_{0}^{2} + v_{0}^{2}] φ_{n} dx] e^{c_{1}} ≤ [∫_{ℝ}[u_{0}^{2} + v_{0}^{2}] e^{2 β x} dx] e^{c_{1}}

Now, taking n → +∞ we obtain

sup_{t∈[0, 1]} ∫_{ℝ}[u^{2} + v^{2}] e^{2 β x} dx ≤ [∫_{ℝ}[u_{0}^{2} + v_{0}^{2}] e^{2 β x} dx] e^{c_{1}}

This way if

sup_{t∈[0, 1]} ||u( . , t)||_{H^{1}(ℝ)} < ∞ ; sup_{t∈[0, 1]} ||v( . , t)||_{H^{1}(ℝ)} < +∞

and

e^{β x} u_{0} ∈ L^{2}(ℝ) ; e^{β x} v_{0} ∈ L^{2}(ℝ), ∀ β > 0

then

e^{β x} u ∈ C([0, 1]; L^{2}(ℝ)) ; e^{β x} v ∈ C([0, 1]; L^{2}(ℝ))

with

c_{1} = β^{3} + \frac{3}{2} β ||u||_{L^{∞}(ℝ×[0, 1])} ||v||_{L^{∞}(ℝ×[0, 1])}.

We have the following extension to higher derivatives. Lemma 4.2. Let j ∈ ℕ. Let (u, v) = (u(x, t), v(x, t)) be a solution of the coupled system equations (P) such that

sup_{t∈[0, 1]} ||u( . , t)||_{H^{j+1}(ℝ)} < +∞ ; sup_{t∈[0, 1]} ||v( . , t)||_{H^{j+1}(ℝ)} < +∞

and

e^{β x} u_{0}, …, e^{β x} ∂_{x}^{j}u_{0} ∈ L^{2}(ℝ) ; e^{β x} v_{0}, …, e^{β x} ∂_{x}^{j}v_{0} ∈ L^{2}(ℝ), ∀ β > 0

then

sup_{t∈[0, 1]} ||u(t)||_{C^{j−1}(ℝ)} ≤ c_{j} = c_{j}(u_{0}, c_{1}) ; sup_{t∈[0, 1]} ||v(t)||_{C^{j−1}(ℝ)} ≤ c_{j} = c_{j}(v_{0}, c_{1})

with c_{1} = β^{3} + \frac{3}{2} β ||u||_{L^{∞}(ℝ×[0, 1])} ||v||_{L^{∞}(ℝ×[0, 1])}. Lemma 4.3. If (u, v) ∈ C_{0}^{3, 1}(ℝ^{2}) × C_{0}^{3, 1}(ℝ^{2}), then

||e^{λ x} u||_{L^{8}(ℝ^{2})} ≤ ||e^{λ x}{∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ^{2})} ; ||e^{λ x} v||_{L^{8}(ℝ^{2})} ≤ ||e^{λ x}{∂_{t} + ∂_{x}^{3}} v||_{L^{8/7}(ℝ^{2})} for all λ ∈ ℝ, with c independent of λ. Proof. Similar to those in [15]. Lemma 4.4. If (u, v) ∈ C^{3, 1}(ℝ^{2}) × C^{3, 1}(ℝ^{2}), such that

supp u ⊆ [−M, M] × [0, 1] ; supp v ⊆ [−M, M] × [0, 1]  (4.6)

and

u(x, 0) = u(x, 1) = 0 and v(x, 0) = v(x, 1) = 0, ∀ x ∈ ℝ, (4.7)

then

||e^{λ x} u||_{L^{8}(ℝ×[0, 1])} ≤ c ||e^{λ x}{∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ×[0, 1])}  (4.8)
||e^{λ x} v||_{L^{8}(ℝ×[0, 1])} ≤ c ||e^{λ x}{∂_{t} + ∂_{x}^{3}} v||_{L^{8/7}(ℝ×[0, 1])}  (4.9)

EJQTDE, 2005, No. 14, p. 16 for all λ ∈ ℝ, with c independent of λ. Proof. Let φ_{ϵ} ∈ C_{0}^{∞}(ℝ) with

supp φ_{ϵ} ⊆ [0, 1]

φ_{ϵ}(t) = 1 for t ∈ (ϵ, 1 − ϵ),
0 ≤ φ_{ϵ}(t) ≤ 1 and |φ′_{ϵ}(t)| ≤ c/ϵ, c constant positive.

Example.

Let u_{ϵ}(x, t) = φ_{ϵ}(t) u(x, t), hence

supp u_{ϵ}(x, t) ⊆ supp φ_{ϵ}(t) ∩ supp u(x, t) ⊆ supp u(x, t) ⊆ [−M, M] × [0, 1].

Then, on one hand we have that

||e^{λ x} u_{ϵ}||_{L^{8}(ℝ^{2})} ⟶ ||e^{λ x} u||_{L^{8}(ℝ×[0, 1])} as ϵ ↓ 0  (4.10)

and on the other hand,

{∂_{t} + ∂_{x}^{3}} u_{ϵ}(x, t) = {∂_{t} + ∂_{x}^{3}} [ φ_{ϵ}(t) u(x, t) ] = φ_{ϵ}(t) {∂_{t} + ∂_{x}^{3}} u + φ′_{ϵ}(t) u.  (4.11)

Hence,

||φ_{ϵ}(t) {∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ^{2})} ⟶ ||{∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ^{2})}

and

||φ′_{ϵ}(t) u||_{L^{8/7}(ℝ^{2})} = [∫_{IR} ∫_{ℝ} |φ′_{ϵ}(t) u(x, t)|^{8/7} dx dt]^{7/8}
= [∫_{0}^{1} ∫_{−M}^{M} |φ′_{ϵ}(t)|^{8/7} |u(x, t)|^{8/7} dx dt]^{7/8} (using (4.6))
= [∫_{0}^{ϵ} ∫_{−M}^{M} |φ′_{ϵ}(t)|^{8/7} |u(x, t)|^{8/7} dx dt + ∫_{1−ϵ}^{1} ∫_{−M}^{M} |φ′_{ϵ}(t)|^{8/7} |u(x, t)|^{8/7} dx dt]^{7/8}
≤ \frac{c}{ϵ} [∫_{0}^{ϵ} ∫_{−M}^{M} |u(x, t)|^{8/7} dx dt + ∫_{1−ϵ}^{1} ∫_{−M}^{M} |u(x, t)|^{8/7} dx dt]^{7/8} .  (4.12)

Defining

G(t) = ∫_{−M}^{M} |u(x, t)|^{8/7}dx

EJQTDE, 2005, No. 14, p. 17 then, by (4.7) we have that G(0) = G(1) = 0. G is continuous and differentiable with

G′(t) = \frac{8}{7} ∫_{−M}^{M} |u(x, t)|^{1/7} ∂_{t}u(x, t) sgn(u(x, t)) dx

hence, G′(t) is continuous,

|G′(t)| ≤ c |t| , |G′(t)| ≤ c |1 − t|

and

∫_{0}^{ϵ} G(t) dt + ∫_{1−ϵ}^{1} G(t) dt ≤ c ϵ^{2}.  (4.13)

Inserting (4.13) in (4.12) we have

||φ′_{ϵ}(t) u||_{L^{8/7}(ℝ^{2})} ≤ c \frac{1}{ϵ} ϵ^{7/4} → 0 as ϵ ↓ 0

and (4.8) follows. Similarly, we obtain (4.9). Lemma 4.5. Let (u, v) ∈ C^{3, 1}(ℝ × [0, 1]) × C^{3, 1}(ℝ × [0, 1]). Suppose that

∑_{j≤2} |∂_{x}^{j}u(x, t)| ≤ C_{β} e^{− β |x|} , ∑_{j≤2} |∂_{x}^{j}v(x, t)| ≤ C_{β} e^{− β |x|}, t ∈ [0, 1], ∀ β > 0,  (4.14)

and

u(x, 0) = u(x, 1) = 0 ; v(x, 0) = v(x, 1) = 0, ∀ x ∈ ℝ.

Then

||e^{λ x} u||_{L^{8}(ℝ×[0, 1])} ≤ c_{0} ||e^{λ x}{∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ×[0, 1])}  (4.15)
||e^{λ x} v||_{L^{8}(ℝ×[0, 1])} ≤ c_{0} ||e^{λ x}{∂_{t} + ∂_{x}^{3}} v||_{L^{8/7}(ℝ×[0, 1])}  (4.16)

for all λ ∈ ℝ, with c_{0} independent of λ. Proof. Let ϕ ∈ C_{0}^{∞}(ℝ) be an even, nonincreasing function for x > 0 with

ϕ(x) = 1, |x| ≤ 1,

supp ϕ ⊆ [−2, 2].

For each M we consider the sequence \{ϕ_{M}\} in C_{0}^{∞}(ℝ) defined by ϕ_{M}(x) = ϕ(\frac{x}{M}), then ϕ_{M} ≡ 1 in a neighborhood of 0 and supp ϕ_{M} ⊆ [−M, M]. Let u_{M}(x, t) = ϕ_{M}(x) u(x, t), then supp u_{M} ⊆ [−M, M]× [0, 1]. Hence,

{∂_{t} + ∂_{x}^{3}} u_{M}(x, t) = {∂_{t} + ∂_{x}^{3}} [ ϕ_{M}(x) u(x, t) ]
= ϕ_{M} {∂_{t} + ∂_{x}^{3}} u(x, t) + 3 ∂_{x}ϕ_{M} ∂_{x}^{2}u + 3 ∂_{x}^{2}ϕ_{M} ∂_{x}u + ∂_{x}^{3}ϕ_{M} u
= ϕ_{M} {∂_{t} + ∂_{x}^{3}} u(x, t) + E_{1} + E_{2} + E_{3}  (4.17)

using Lemma 4.4 to u_{M}(x, t) we get

||e^{λ x} u_{M}||_{L^{8}(ℝ×[0, 1])} ≤ c ||e^{λ x}{∂_{t} + ∂_{x}^{3}}u_{M}||_{L^{8/7}(ℝ×[0, 1])}
≤ c ||e^{λ x} ϕ_{M}{∂_{t} + ∂_{x}^{3}} u||_{L^{8/7}(ℝ×[0, 1])} + c ∑_{j=1}^{3} ||e^{λ x} E_{j}||_{L^{8/7}(ℝ×[0, 1])}

(4.18)

EJQTDE, 2005, No. 14, p. 18 We show that the terms involving the L^{8/7}-norm of the error E_{1}, E_{2} and E_{3} in (4.18) tend to zero as M → ∞. We consider the case x > 0 and λ > 0. From (4.14) with β > λ it follows that

||e^{λ x} E_{1}||_{L^{8/7}(ℝ×[0, 1])}^{8/7} = 3^{8/7} ∫_{0}^{1} ∫_{M}^{2M} |e^{λ x}∂_{x}ϕ_{M} ∂_{x}^{2}u|^{8/7} dx dt

≤ c ∫_{0}^{1} ∫_{M}^{2M} |\frac{e^{λ x}}{M} ∂_{x}^{2}u|^{8/7} dx dt
≤ c ∫_{0}^{1} ∫_{M}^{2M} e^{8 λ x/7}e^{− 8 β x/7} dx dt  (4.19)
= c ∫_{0}^{1} ∫_{M}^{2M} e^{− \frac{8}{7} (β−λ) x} dx dt → 0 as M → ∞.  (4.20)

Thus taking the limits as M → ∞ in (4.17) and using (4.19) we obtain (4.15). In a similar way we have (4.17) and the Lemma 4.5 follows.

Lemma 4.6. Suppose that

(u, v) ∈ C([0, 1]; H^{4}(ℝ)) ∩ C^{1}([0, 1]; H^{1}(ℝ)) × C([0, 1]; H^{4}(ℝ)) ∩ C^{1}([0, 1]; H^{1}(ℝ))

satisfies the system

\{ \begin{matrix}∂_{t}u + ∂_{x}^{3}u + ∂_{x}(u v^{2}) = 0  (x, t) ∈ ℝ × [0, 1] \\ ∂_{t}v + ∂_{x}^{3}v + ∂_{x}(u^{2} v) = 0 \end{matrix}

with

supp u(x, 0) ⊆ (−∞, b] ; supp v(x, 0) ⊆ (−∞, b].

Then for any β > 0,

∑_{j≤2} |∂_{x}^{j}u(x, t)| ≤ c_{b} e^{−β x} ;  ∑_{j≤2} |∂_{x}^{j}v(x, t)| ≤ c_{b} e^{−β x}, for x > 0, t ∈ [0, 1].

Proof. From Lemma 4.2.

Theorem 4.7. Suppose that (u(x, t), v(x, t)) is a sufficiently smooth solution of the dispersive coupled system (P). If

supp u( . , t_{j}) ⊆ (−∞, b) and supp v( . , t_{j}) ⊆ (−∞, b), j = 1, 2.

or

supp u( . , t_{j}) ⊆ (a, ∞) and supp v( . , t_{j}) ⊆ (a, ∞), j = 1, 2.

then

u(x, t) ≡ 0  and  v(x, t) ≡ 0.

Proof. Without loss of generality we assume that t_{1} = 0 and t_{2} = 1. Thus,

supp u( . , 0), supp u( . , 1) ⊆ (−∞, b) ; supp v( . , 0), supp v( . , 1) ⊆ (−∞, b).

We will show that there exists a large number R > 0 such that

supp u( . , t) ⊆ (−∞, 2R] ; supp v( . , t) ⊆ (−∞, 2R], ∀ t ∈ [0, 1].

Then the result will follow from Theorem 3.7.

Let μ ∈ C_{0}^{∞}(ℝ) be a nondecreasing function such that

μ(x) = \{ \begin{matrix}0  ,  x ≤ 1 \\ 1  ,  x ≥ 2 \end{matrix}

and 0 ≤ μ(x) ≤ 1, ∀ x ∈ ℝ. For each R ≠ 0 we define μ_{R}(x) = μ (\frac{x}{R}) , i. e.,

μ_{R}(x) = \{ \begin{matrix}0  ,  x ≤ R \\ 1  ,  x ≥ 2 R. \end{matrix}

Let u_{R}(x, t) = μ_{R}(x) u(x, t), then u_{R}(x, t) ∈ C_{0}^{∞}(ℝ), since (μ_{R} u) ∈ C^{∞}(ℝ) and moreover

supp u_{R} = supp (μ_{R} u) ⊆ supp μ_{R} ∩ supp u ⊆ supp μ_{R}.

From the above inequality we have that supp u_{R} ⊆ (−∞, 2 R]. Using that u(v respectively) is a sufficiently smooth function (see [19]) and Lemma 4.6 we can apply Lemma 4.5 to u_{R}(x, t) for R sufficiently larger. Thus

{∂_{t} + ∂_{x}^{3}}[ μ_{R} · u] = μ_{R} {∂_{t}u + ∂_{x}^{3}u} + 3 ∂_{x}μ_{R} · ∂_{x}^{2}u + 3 ∂_{x}^{2}μ_{R} · ∂_{x}u + ∂_{x}^{3}μ_{R} · u
= −μ_{R} {∂_{x}u · v^{2} + u ∂_{x}(v^{2})} + 3∂_{x}μ_{R} · ∂_{x}^{2}u + 3 ∂_{x}^{2}μ_{R} · ∂_{x}u + ∂_{x}^{3}μ_{R} · u
= −μ_{R} · V_{1} · v + 3 ∂_{x}μ_{R} · ∂_{x}^{2}u + 3 ∂_{x}^{2}μ_{R} · ∂_{x}u + ∂_{x}^{3}μ_{R} · u
= −μ_{R} · V_{1} · v + F_{1} + F_{2} + F_{3}
= −μ_{R} · V_{1} · v + F_{R}  (4.21)

where V_{1}(x, t) = ∂_{x}u · v + 2 u · ∂_{x}v. The similar form

{∂_{t} + ∂_{x}^{3}}μ_{R}v = −μ_{R} · V_{2} · u + 3 ∂_{x}μ_{R} · ∂_{x}^{2}v + 3 ∂_{x}^{2}μ_{R} · ∂_{x}v + ∂_{x}^{3}μ_{R} · v
= −μ_{R} · V_{2} · u + H_{1} + H_{2} + H_{3}
= −μ_{R} · V_{2} · u + H_{R}  (4.22)

where V_{2}(x, t) = ∂_{x}v · u + 2 v · ∂_{x}u. Then, by using Lemma 4.5

||e^{λ x} μ_{R} · u||_{L^{8}(ℝ×[0, 1])} ≤ c_{0} ||e^{λ x}{∂_{t} + ∂_{x}^{3}} μ_{R} · u||_{L^{8/7}(ℝ×[0, 1])}
≤ c_{0} ||e^{λ x} μ_{R} · V_{1} · v + e^{λx}F_{R}||_{L^{8/7}(ℝ×[0, 1])}  (using (4.21))
≤ c_{0} ||e^{λ x}μ_{R} · V_{1} · v||_{L^{8/7}(ℝ×[0, 1])} + c_{0} ||e^{λ x} F_{R}||_{L^{8/7}(ℝ×[0, 1])} (4.23)

We estimate the term ||e^{λ x} μ_{R} · V_{1} · v||_{L^{8/7}(ℝ×[0, 1])}.

||e^{λ x} μ_{R} · V_{1} · v||_{L^{8/7}(ℝ×[0, 1])} = (∫_{0}^{1} ∫_{ℝ} |e^{λ x} μ_{R} · V_{1} · v|^{8/7} dx dt)^{8/7}

= (∫_{0}^{1} ∫_{x>R} |e^{λ x} μ_{R} · v · V_{1}|^{8/7} dx dt)^{8/7}

≤ (∫_{0}^{1} [∫_{x≥R} |e^{λ x} μ_{R} · v|^{8} dx]^{1/7} [∫_{x≥R} |V_{1}|^{8/6} dx]^{6/7} dt)^{7/8}

= (∫_{0}^{1} [∫_{x≥R} |e^{λ x} μ_{R} · v|^{8}dx]^{1/7} [∫_{x≥R} |V_{1}|^{4/3} dx]^{6/7} dt)^{7/8}

= (∫_{0}^{1} [∫_{ℝ} |e^{λ x}μ_{R} · v|^{8} dx]^{1/7} [∫_{x≥R} |V_{1}|^{4/3} dx]^{6/7} dt)^{7/8}

≤ (∫_{0}^{1} ∫_{ℝ} |e^{λ x}μ_{R} · v|^{8} dx dt)^{1/8} (∫_{0}^{1} ∫_{x≥R} |V_{1}|^{4/3} dx dt)^{3/4}
= ||e^{λ x}μ_{R} · v||_{L^{8}(ℝ×[0, 1])} ||V_{1}||_{L^{4/3}({x≥R}×[0, 1])}

c_{0} ||e^{λ x} μ_{R} · V_{1} · v||_{L^{8/7}(ℝ×[0, 1])} ≤ c_{0} ||e^{λ x} μ_{R} · v||_{L^{8}(ℝ×[0, 1])} ||V_{1}||_{L^{4/3}({x≥R}×[0, 1])}.  (4.24)

We define V_{1}(x, t) = ∂_{x}u · v + 2 u · ∂_{x}v ∈ L^{q}(ℝ × [0, 1]) with q ∈ [0, ∞). Now, we fix R so large such that

c ||V_{1}||_{L^{4/3}(\{x≥R\}×[0, 1])} ≤ \frac{1}{2}

then

c ||e^{λ x} μ_{R} · V_{1} · v||_{L^{8/7}(ℝ×[0, 1])} ≤ \frac{1}{2} ||e^{λ x} μ_{R} · v||_{L^{8}(ℝ×[0, 1])}

this way

||e^{λ x} μ_{R} · v||_{L^{8}(ℝ×[0, 1])} ≤ \frac{1}{2} ||e^{λ x} μ_{R} · v||_{L^{8}(ℝ×[0, 1])} + c ||e^{λ x} F_{R}||_{L^{8/7}(ℝ×[0, 1])}

hence

\frac{1}{2} ||e^{λ x} μ_{R} · v||_{L^{8}(ℝ×[0, 1])} ≤ c ||e^{λ x} F_{R}||_{L^{8/7}(ℝ×[0, 1])}

thus

||e^{λ x}μ_{R} · v||_{L^{8}(ℝ×[0, 1])} ≤ 2 c ||e^{λ x} F_{R}||_{L^{8/7}(ℝ×[0, 1])}  (4.25)

To estimate the F_{R} term it suffices to consider one of the terms in F_{R}, say F_{2}, since the proofs for F_{1}, and F_{3}, are similar. We have that

F_{R} = F_{1} + F_{2} + F_{3}
= 3 ∂_{x}μ_{R} · ∂_{x}^{2}u + 3 ∂_{x}^{2}μ_{R} · ∂_{x}u + ∂_{x}^{3}μ_{R} · u

and supp F_{i}, ⊆ [R, 2R], i = 1, 2, 3. We estimate F_{2}.

2 c ||e^{λ x} F_{2}||_{L^{8/7}(ℝ×[0, 1])} = 2 c (∫_{0}^{1} ∫_{IR} |e^{λ x} F_{2}|^{8/7} dx dt)^{7/8}

= 2 c (∫_{0}^{1} ∫_{R}^{2 R} |3 e^{λ x} ∂_{x}^{2}μ_{R} · ∂_{x}u|^{8/7} dx dt)^{7/8}

= 2 \frac{c}{R^{2}} (∫_{0}^{1} ∫_{R}^{2R} e^{\frac{8}{7} λ x} |∂_{x}^{2}μ · ∂_{x}u|^{8/7} dx dt)^{7/8}

≤ 2 \frac{c}{R^{2}} (∫_{0}^{1} ∫_{R}^{2R} e^{\frac{8}{7} λ x} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8} .

Then

2 c ||e^{λ x} F_{2}||_{L^{8/7}(ℝ×[0, 1])} ≤ 2 \frac{c}{R^{2}} e^{2 λ R} (∫_{0}^{1} ∫_{R}^{2R} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8} .  (4.26)

On the other hand,

||e^{λ x}μ_{R} · u||_{L^{8}(ℝ×[0, 1])} ≥ (∫_{0}^{1} ∫_{x>2R} e^{8 λ x} |u(x, t)|^{8} dx dt)^{1/8}

(∫_{0}^{1} ∫_{x>2R} e^{8 λ x} |u(x, t)|^{8} dx dt)^{1/8} ≤ ||e^{λ x}μ_{R} · u||_{L^{8}(IR×[0, 1])}
≤ 2 c ||e^{λ x} F_{R}||_{L^{8/7}(IR×[0, 1])}
≤ 2 \frac{c}{R^{2}} e^{2 λ R} (∫_{0}^{1} ∫_{R}^{2R} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8}

hence,

(∫_{0}^{1} ∫_{x>2R} e^{8 λ x} |u(x, t)|^{8} dx dt)^{1/8} ≤ c_{0} (∫_{0}^{1} ∫_{R}^{2R} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8} .

This way we have

(∫_{0}^{1} ∫_{x>2R} e^{8 λ x} |u(x, t)|^{8} dx dt)^{1/8} ≤ ||e^{λ x} μ_{R} · u||_{L^{8}(ℝ×[0, 1])}
≤ 2 c ||e^{λ x} F_{2}||_{L^{8/7}(ℝ×[0, 1])}  (using (4.25))
≤ 2 \frac{c}{R^{2}} e^{2 λ R} (∫_{0}^{1} ∫_{R}^{2R} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8}  (using (4.26))

then

(∫_{0}^{1} ∫_{x>2R} e^{8 λ (x−2R)}|u(x, t)|^{8} dx dt)^{1/8} ≤ 2 \frac{c}{R^{2}} (∫_{0}^{1} ∫_{R}^{2R} |∂_{x}u(x, t)|^{8/7} dx dt)^{7/8}

letting λ → ∞ it follows that

u(x, t) ≡ 0  for  x > 2R,  t ∈ [0, 1]

and in a similar way we obtain that

v(x, t) ≡ 0  for  x > 2R,  t ∈ [0, 1]

which yields the proof.

Acknowledgement. The authors want to thank Prof. Carlos Picarte (Universidad del Bío-Bío) for his valuable help in the typesetting of this paper.

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(Received September 28, 2004)