On the Unique Continuation Property for a Nonlinear Dispersive System

We solve the following problem: If (u, v) = (u(x, t), v(x, t)) is a solution of the Dispersive Coupled System with t1 < t2 which are sufficiently smooth and such that: supp u(.


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) 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) x v + ∂ x (u p+1 v p ) = 0 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, 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].
They proved that the coupled system is globally well posed in H m (R) × H m (R), for any m ≥ 1 provided 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 Moreover, given t ∈ (0, T ), the map (ϕ, ψ) → (u(t), v(t)) is smooth from 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 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: ) is a solution of the KdV equation 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 (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.

Preliminaries
We consider the following dispersive coupled system (P ) If E is any Banach space, its norm is written as || • || E .For 1 ≤ p ≤ + ∞, the usual class of p thpower 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 ) 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 Definition 2.2.Let L be an evolution operator acting on functions defined on some connected open set L is said to have the horizontal unique continuation property if every solution u of Lu = 0 that vanishes on some nonempty open set Lemma 2.3.The equation (P ) has the following conserved integral quantities, i. e., d dt 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 Each term is treated separately integrating by parts Each term is treated separately, integrating by parts 1 2 3) and (2.4), we have and for all u ∈ H 3 (Ω) (2.7) (2.8) Proof.See [28].
Proof.For > 0, we approximate the system (P ) by the parabolic system (R) We rewrite the above equations in a more friendly way as We multiply (2.9) by u, integrate over x ∈ Ω = (0, 1), to have Similarly, we multiply (2.10) by v, integrate over x ∈ Ω = (0, 1), and we have Adding (2.11) and (2.12) we obtain 1 2 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 Integrating over t ∈ [0, T ] we have On the other hand, using the Lemma 2.4 and performing appropriate calculations we obtain (2.17) EJQTDE, 2005, No. 14, p. 6 We also have . Hence, We calculate in similar form the terms This way we have In particular, from the equation (R) we deduce that By other hand, we have Then 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.

Carleman's estimate and unique continuation property
We consider the equation (P ), then We rewrite the above equations as Hence in (Q) we obtain with 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: with and the differential operator L defined by (3.5).Assume that Proof.We consider the operator for any Φ ∈ C ∞ 0 (B δ ) and τ > 0. Similarly, we have for the operator for any Ψ ∈ C ∞ 0 (B δ ) and τ > 0. On the other hand, and where ϕ(0, 0) = δ 2 .Therefore, if (x, t) ∈ supp LV there is an > 0 such that ϕ(x, t) ≤ δ 2 − .We can choose a neighborhood N of (0, 0) at which ϕ(x, t) > δ 2 − and obtain from (3.20) 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 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 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 and 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. 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.

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].
Multiplying the equation (4.2) by u ϕ n , and integrating by parts we get Integrating by parts we obtain 1 2 Similarly, multiplying the equation by v ϕ n and integrating by parts, we obtain Hence, from (4.3) and (4.5) (using (4.1)) Using that ϕ n ≥ 0 and the Holder inequality where Integrating over t ∈ [0, 1] we have This way if with We have the following extension to higher derivatives.
)) be a solution of the coupled system equations (P ) such that with for all λ ∈ R, with c independent of λ.
Then, on one hand we have that and on the other hand, and Then 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 For each M we consider the sequence {φ M } in C ∞ 0 (R) defined by φ M (x) = φ( x M ), then φ M ≡ 1 in a neighborhood of 0 and supp using Lemma 4.4 to u M (x, t) we get 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 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.
We will show that there exists a large number R > 0 such that supp u( .Then the result will follow from Theorem 3.7. Let µ ∈ C ∞ 0 (R) be a nondecreasing function such that 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 Then, by using Lemma 4.5 We estimate the term ||e λ x µ R • V 1 • v|| L 8/7 (R×[0, 1]) .We define ) with q ∈ [0, ∞).Now, we fix R so large such that 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

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

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