On the unique continuation property of solutions of the three-dimensional Zakharov-Kuznetsov equation

TL;DR

This paper establishes a unique continuation property for solutions of the three-dimensional Zakharov-Kuznetsov equation, showing that solutions with sufficiently fast decay at two different times must be identical.

Contribution

It proves a new unique continuation result for the 3D Zakharov-Kuznetsov equation based on decay conditions at two time points.

Findings

Solutions with exponential decay at two times are identical.

The decay condition ensures the solutions coincide.

The result applies to sufficiently smooth solutions.

Abstract

We prove that if the difference of two sufficiently smooth solutions of the three-dimensional Zakharov-Kuznetsov equation $$\partial_{t}u+\partial_{x}\triangle u+u\partial_{x}u=0 \text{,}\quad (x,y,z)\in\mathbb R^3, \;t\in[0,1],$$ decays as $e^{-a(x^2+y^2+z^2)^{3/4}}$ at two different times, for some $a>0$ large enough, then both solutions coincide.

Equations

$${\partial }_{t}u+{\partial }_x△u+u{\partial }_{x}u=0\text{,}(x,y,z)\in {\mathbb{R}}^3,t\in [0,1],$$

$$u_1,u_2\in C([0,1];H^4({\mathbb{R}}^3)\cap L^2((1+x^2+y^2+z^2{)}^{8/5+ϵ}dxdydz))\cap C^1([0,1];L^2({\mathbb{R}}^3)),$$

$$u_1(0)-u_2(0),u_1(1)-u_2(1)\in L^2(e^{a(x^2+y^2+z^2{)}^{3/4}}dxdydz),$$

$$\Vert e^{\lambda |x|}{e}^{\beta |y|}{e}^{\beta |z|}w{\Vert }_{L_t^{\infty }{L}_{xyz}^2(D)}\le$$

$$+\Vert e^{\lambda |x|}{e}^{\beta |y|}{e}^{\beta |z|}({\partial }_t+{\partial }_x\Delta )w{\Vert }_{L_t^1{L}_{xyz}^2(D)};$$

$$\Vert e^{\lambda |x|}{e}^{\beta |y|}{e}^{\beta |z|}Lw{\Vert }_{L_x^{\infty }{L}_{tyz}^2(D)}\le$$

$$+C_1\Vert e^{\lambda |x|}{e}^{\beta |y|}{e}^{\beta |z|}({\partial }_t+{\partial }_x\Delta )w{\Vert }_{L_x^1{L}_{tyz}^2(D)};$$

$$\{(x,y,z):R-1\le \sqrt{x^2+y^2+z^2}\le R\}\times [0,1],$$

$$\displaystyle A_{R}(v):=\big{(}\int_{0}^{1}\int\limits_{R-1\leq\sqrt{x^{2}+y^{2}+z^{2}}\leq R}\sum_{0\leq k+l+n\leq 2}|\partial_{x}^{k}\partial_{y}^{l}\partial_{z}^{n}v(t)(x,y,z)|^{2}dx\,dy\,dz\,dt\big{)}^{\frac{1}{2}}\,.$$

$$\Vert v{\Vert }_{L^2(Q)}\le C{e}^{14\overline{C}{R}^{\frac{3}{2}}}{A}_R(v)\forall R\ge R_0.$$

$$g\mapsto e^{\lambda x}{e}^{\beta y}{e}^{\beta z}({\partial }_t+{\partial }_x\Delta )e^{-\beta z}{e}^{-\beta y}{e}^{-\lambda x}g.$$

$$m_0(\xi ,\eta ,\zeta ,\tau ):=\frac{1}{i\tau +(i\xi -\lambda {)}^3+(i\xi -\lambda )(i\eta -\beta {)}^2+(i\xi -\lambda )(i\zeta -\beta {)}^2}.$$

$$\Vert [m_0\hat{h}]\stackrel{ˇ}{}({\cdot }_x,{\cdot }_y,{\cdot }_z,t){\Vert }_{L_{xyz}^2({\mathbb{R}}^3)}\le \Vert h{\Vert }_{L_t^1{L}_{xyz}^2({\mathbb{R}}^4)}\forall t\in \mathbb{R},$$

$$m_{k,l,n}(\xi ,\eta ,\zeta ,\tau ):=(i\xi -\lambda {)}^k(i\eta -\beta {)}^l(i\zeta -\beta {)}^n{m}_0(\xi ,\eta ,\zeta ,\tau ),$$

$$m_0(\xi ,\eta ,\zeta ,\tau ):=\frac{1}{i\tau +(i\xi -\lambda {)}^3+(i\xi -\lambda )(i\eta -\beta {)}^2+(i\xi -\lambda )(i\zeta -\beta {)}^2}.$$

$$\Vert [m_{k,l,n}\hat{h}]\stackrel{ˇ}{}{\Vert }_{L_x^{\infty }{L}_{tyz}^2({\mathbb{R}}^4)}\le C_1\Vert h{\Vert }_{L_x^1{L}_{tyz}^2({\mathbb{R}}^4)}.$$

$$m_{2,0,0}(\xi ,\eta ,\zeta ,\tau )=\frac{-i{v}^2}{v^3+wv-\tau },$$

$$\frac{w^3}{{\tau }^2}+\frac{27}{4}=0.$$

$$T_{\beta }(\eta ,\zeta ,\tau ):=\frac{w^3}{{\tau }^2}+\frac{27}{4}.$$

$$T_{\beta }(\eta ,\zeta ,\tau )=T_1(\frac{\eta }{\beta },\frac{\zeta }{\beta },\frac{\tau }{{\beta }^3}).$$

$$|T_1(\eta ,\zeta ,\tau )|\ge \frac{27}{8}.$$

$$|T_1(\eta ,\zeta ,\tau ){|}^2$$

$$\displaystyle=\Big{\{}\frac{(\eta^{2}+\zeta^{2}-2)[(\eta^{2}+\zeta^{2})^{2}-28(\eta^{2}+\zeta^{2})+12(\eta-\zeta)^{2}+4]}{\tau^{2}}+\frac{27}{4}\Big{\}}^{2}\,.$$

$$({\eta }^2+{\zeta }^2-2)\ge \frac{1}{2}({\eta }^2+{\zeta }^2)\text{and}({\eta }^2+{\zeta }^2{)}^2-28({\eta }^2+{\zeta }^2)+12(\eta -\zeta {)}^2+4\ge \frac{1}{2}({\eta }^2+{\zeta }^2{)}^2$$

$$|T_{1}(\eta,\zeta,\tau)|^{2}\geq\Big{\{}\frac{1}{4}\frac{(\eta^{2}+\zeta^{2})^{3}}{\tau^{2}}+\frac{27}{4}\Big{\}}^{2}\geq\Big{(}\frac{27}{8}\Big{)}^{2}\,.$$

$$\Big{|}\frac{(\eta^{2}+\zeta^{2}-2)[(\eta^{2}+\zeta^{2})^{2}-28(\eta^{2}+\zeta^{2})+12(\eta-\zeta)^{2}+4]}{\tau^{2}}\Big{|}\leq\frac{27}{8}\,.$$

$$|T_{1}(\eta,\zeta,\tau)|^{2}\geq\Big{(}\frac{27}{8}\Big{)}^{2}\,.$$

$$\forall (\eta ,\zeta ,\tau )\in K_{\beta }|\eta |\le C\beta ,|\zeta |\le C\beta ,|\tau |\le C{\beta }^3.$$

$$m_{2,0,0}=\sum _{j=1}^3\frac{A_j}{v-v_j}=\sum _{j=1}^3\frac{A_j}{\xi -\mathrm{Re}(v_j)-i[\mathrm{Im}(v_j)-\lambda ]},$$

$$A_{j}:=\lim_{v\to v_{j}}\frac{(v-v_{j})(-iv^{2})}{P(v)}=-\frac{iv_{j}^{2}}{3v_{j}^{2}+w}=-i\big{[}\frac{1}{3}-\frac{1}{3}\frac{w}{3v_{j}^{2}+w}\big{]}\,.$$

$$|3v_j^2+w|\ge \delta |w|.$$

Full text

[TABLE]

decays as $e^{-a(x^{2}+y^{2}+z^{2})^{3/4}}$ at two different times, for some $a>0$ large enough, then both solutions coincide.

1. introduction

In this paper we consider the three-dimensional Zakharov-Kuznetsov (ZK) equation

[TABLE]

where $\triangle$ is the three-dimensional spatial Laplace operator.

Equation (1.1) is a three-dimensional generalization of the Korteweg-de Vries (KdV) equation, and was introduced by Zakharov and Kuznetsov in [24] to describe unidirectional propagation of ionic-acoustic waves in magnetized plasma. The rigorous derivation of the Zakharov-Kuznetsov equation from the Euler-Poisson system with magnetic field was done by Lannes, Linares and Saut in [17].

The Cauchy problem associated to the equation (1.1), in the context of classical Sobolev spaces, has been recently studied. In [19] Linares and Saut proved that the Cauchy problem is locally well-posed for initial data in $H^{s}(\mathbb{R}^{3})$ with $s>9/8$, by adapting to the ZK equation the techniques developed in [15] for the Benjamin-Ono equation. In [22] Ribaud and Vento obtained the local well-posedness for initial data in $H^{s}(\mathbb{R}^{3})$ for $s>1$. The proof of this result is based on a sharp maximal function estimate in time-weighted spaces. In [20] Molinet and Pilod established the global well-posedness in $H^{s}(\mathbb{R}^{3})$ with $s>1$. In order to obtain the global result the authors work in the framework of the functional spaces introduced by Koch and Tataru in [16]. Other results concerning the well-posedness of the modified ZK equation in three space dimensions can be found in [8] and [9].

It is important to point out that the ZK equation in two-space variables has also been studied intensively in the last decades and that many of the results obtained for it have been tried to be generalized to the three-dimensional version. With respect to the Cauchy problem associated to the two-dimensional ZK equation we refer, among others, to the works of Faminskii [6], Linares and Pastor [18], Grünrock and Herr [10], and Molinet and Pilod [20].

In addition to the Cauchy problem, another important aspect in the study of certain evolution equations is related to the unique continuation principles. Unique continuation principles consist in the determination of sufficient conditions, in both spatial and time variables, which guarantee that two solutions to the same equation are equal. For nonlinear dispersive equations, unique continuation principles have been obtained for several models including the KdV equation, the nonlinear Schrödinger equation and the Benjamin-Ono equation. See for example [1], [2], [5], [7], [12], [13], [14], [23] and references therein.

Our aim in this work is to extend to the three-dimensional case the unique continuation principle obtained in [3] for the two-dimensional ZK equation. In order to achieve this generalization we follow the scheme developed in [3], which uses the ideas of [5] for the KdV equation. This principle is the main result of our work, and we state it as follows:

Theorem 1.1**.**

Suppose that for some small $\epsilon>0$

[TABLE]

are solutions of equation (1.1). Then there exists a universal constant $a_{0}>0$, such that if for some $a>a_{0}$

[TABLE]

then $u_{1}\equiv u_{2}$.

With the same techniques used in [4] for the two-dimensional ZK equation, it can be proved that the Cauchy problem associated to the equation (1.1) is globally well-posed in the weighted Sobolev space $Z_{4,r}:=H^{4}(\mathbb{R}^{3})\cap L^{2}((1+x^{2}+y^{2}+z^{2})^{r}\,dx\,dy\,dz)$ for $r\leq 2$. This shows that hypothesis (1.2) in Theorem 1.1 can be changed by this one: $u_{1},u_{2}\in C([0,1];H^{4}(\mathbb{R}^{3}))\cap C^{1}([0,1];L^{2}(\mathbb{R}^{3}))$ are solutions of equation (1.1) such that $u_{1}(0)$ and $u_{2}(0)$ belong to the class $L^{2}((1+x^{2}+y^{2}+z^{2})^{8/5+\epsilon}\,dx\,dy\,dz)$ for some $0<\epsilon\leq 2/5$.

The proof of Theorem 1.1 is based upon the comparison of three-dimensional spatial Carleman type estimates and a lower estimate. To establish the Carleman estimates we used simple procedures which involve partial fraction decompositions and Fourier transform methods.

The Carleman type estimates are stated in the next theorem.

Theorem 1.2**.**

Let $D:=\mathbb{R}^{3}\times[0,1]=\{(x,y,z,t)\mid(x,y,z)\in\mathbb{R}^{3},t\in[0,1]\}$ and let $w\in C([0,1];H^{4}(\mathbb{R}^{3})\cap C^{1}([0,1];L^{2}(\mathbb{R}^{3}))$ such that for all $t\in[0,1]$ supp$\,w(t)\subseteq K$, where $K$ is a compact subset of $\mathbb{R}^{3}$.Then

(i) For $\lambda>0$ and $\beta>0$,

[TABLE]

(ii) There exist $C_{1}>0$ and $C_{2}>1$, independent of the set $K$, such that for $\beta\geq 1$ and $\lambda\geq C_{2}\beta$

[TABLE]

where $L$ denotes any operator in the set $\{\partial_{x},\partial_{y},\partial_{z},\partial_{x}^{2},\partial_{xy},\partial_{xz},\partial_{y}^{2},\partial_{yz},\partial_{z}^{2}\}$.

In order to bound the $H^{2}$-norm of the difference $v=u_{1}-u_{2}$ in the region

[TABLE]

with an exponential of the form $e^{-cR^{3/2}}$, we use a lower estimate, which will be established in the following theorem:

Theorem 1.3**.**

Let $u_{1},u_{2}\in C([0,1];H^{3}(\mathbb{R}^{3}))\cap C^{1}([0,1];L^{2}(\mathbb{R}^{3}))$ be solutions of (1.1). Define $v:=u_{1}-u_{2}$. Let $\delta>0$, $r\in(0,\frac{1}{2})$, $Q:=\{(x,y,z,t)\mid\sqrt{x^{2}+y^{2}+z^{2}}\leq 1,t\in[r,1-r]\}$ and suppose that $\|v\|_{L^{2}(Q)}\geq\delta$. For $R>0$ let us define

[TABLE]

Then there are constants $\overline{C}=\overline{C}(r)>0$, $C>0$ and $R_{0}\geq 2$ such that

[TABLE]

The central part of the article is the proof of Lemmas 2.2 and 3.1, stated below. The proof of Lemma 2.2 is an improved version of the proof of the corresponding Lemma in [3], while the proof of Lemma 3.1, although it follows the same course of the proof of the corresponding Lemma in [3], is technically more complicated due to the change of two to three dimensions.

This article is organized in the same way as [3]: in section 2 we prove the Carleman estimates stated in Theorem 1.2. Section 3 contains the proof of Theorem 1.3. The proof of the main result, Theorem 1.1, is given in section 4.

Throughout the paper the letter $C$ will denote diverse constants, which may change from line to line, and whose dependence on certain parameters is clearly established in all cases. Sometimes, for a parameter $a$, we will use the notation $C_{a}$ to make emphasis in the fact that the constant $C$ depends upon $a$. We frequently write $f(\cdot_{s})$ to denote a function $s\mapsto f(s)$.

2. Carleman Type Estimates (Proof of Theorem 1.2)

The proof of Theorem 1.2, as in [3], is based on two lemmas. These lemmas express boundedness properties of the operators $T_{0}$ and $(\partial_{x}-\lambda)^{k}(\partial_{y}-\beta)^{l}(\partial_{z}-\beta)^{n}T_{0}$, where $k$, $l$ and $n$ are nonnegative integers with $0<k+l+n\leq 2$, and $T_{0}$ is the inverse operator of

[TABLE]

The operator $T_{0}$ is defined through the space-time Fourier transform by the multiplier

[TABLE]

The first lemma, whose proof is analogous to the one of Lemma 2.1 in [3], is the following.

Lemma 2.1**.**

Let $h\in L^{1}(\mathbb{R}^{4})$ with $\|h\|_{L_{t}^{1}L_{xyz}^{2}(\mathbb{R}^{4})}<\infty$. Then for all $(\lambda,\beta)\neq(0,0)$, $m_{0}\widehat{h}\in S^{\prime}(\mathbb{R}^{4})$ and $[m_{0}\widehat{h}]{\,\widecheck{\;}\,}$ defines a bounded function from $\mathbb{R}_{t}$ with values in $L_{xyz}^{2}$. Besides,

[TABLE]

where $\widehat{\;}$ and ${\,\widecheck{\;}\,}$ denote, respectively, the Fourier transform and its inverse in $S^{\prime}(\mathbb{R}^{4})$.

The second lemma is similar to Lemma 2.2 in [3]. Its proof also follows the ideas of the proof of Lemma 2.2 in [3] but it involves some significant changes and simplifications, and by this reason we present it in detail.

Lemma 2.2**.**

Let $h\in L^{1}(\mathbb{R}^{4})$ with $\|h\|_{L_{x}^{1}L_{tyz}^{2}(\mathbb{R}^{4})}<\infty$. For $\beta\geq 1$, $\lambda\in\mathbb{R}$, $k,l,n\in\{0,1,2\}$, and $0<k+l+n\leq 2$, let

[TABLE]

where $m_{0}$ is defined by

[TABLE]

Then $m_{k,l,n}\widehat{h}\in S^{\prime}(\mathbb{R}^{4})$ and there are constants $C_{1}>0$ and $C_{2}>1$, independent from $h$ and $\beta$, such that if $\lambda\geq C_{2}\beta$ then

[TABLE]

Proof.

We will consider only the case $k=2$ and $l=n=0$. The proof of the other cases is similar. Let us observe that

[TABLE]

where $v:=\xi+i\lambda$ and $w:=(\eta+i\beta)^{2}+(\zeta+i\beta)^{2}$. If the polynomial $P(v):=v^{3}+wv-\tau$ has multiple roots then

[TABLE]

Let us define the function $T_{\beta}:\mathbb{R}^{3}\longrightarrow\mathbb{C}$ by

[TABLE]

Let us observe that if $P$ has multiple roots then $T_{\beta}(\eta,\zeta,\tau)=0$ and that

[TABLE]

Now we will prove that there exists a compact set $K$, $K\subset\mathbb{R}^{3}$, such that for $(\eta,\zeta,\tau)\in\mathbb{R}^{3}-K$ we have

[TABLE]

From an easy calculation it follows that

[TABLE]

Let us take $R>0$ such that

[TABLE]

for $(\eta^{2}+\zeta^{2})>R^{2}$. Then for $(\eta,\zeta,\tau)\in\mathbb{R}^{3}$ with $(\eta^{2}+\zeta^{2})>R^{2}$ we have

[TABLE]

Now suppose $(\eta^{2}+\zeta^{2})\leq R^{2}$. There exists $R^{\prime}>0$ such that if $|\tau|\geq R^{\prime}$ then

[TABLE]

In consequence for $(\eta,\zeta,\tau)\in\mathbb{R}^{3}$ with $(\eta^{2}+\zeta^{2})\leq R^{2}$ and $|\tau|\geq R^{\prime}$ we have

[TABLE]

We define the compact $K$ to be the set $\{(\eta,\zeta,\tau)\mid(\eta^{2}+\zeta^{2})\leq R^{2}\;\text{and}\;|\tau|\leq R^{\prime}\}$.

Now we define the compact set $K_{\beta}$ to be the set $\{(\eta,\zeta,\tau)\mid(\frac{\eta}{\beta},\frac{\zeta}{\beta},\frac{\tau}{\beta^{3}})\in K\}$. Let $C>1$ such that, for every $(\eta^{\prime},\zeta^{\prime},\tau^{\prime})\in K$, $\max\{|\eta^{\prime}|,|\zeta^{\prime}|,|\tau^{\prime}|\}\leq C$. Then

[TABLE]

i) If $(\eta,\zeta,\tau)\in\mathbb{R}^{3}-K_{\beta}$ then $|T_{\beta}(\eta,\zeta,\tau)|=|T_{1}(\frac{\eta}{\beta},\frac{\zeta}{\beta},\frac{\tau}{\beta^{3}})|\geq\frac{27}{8}$ and in consequence the polynomial $P$ does not have multiple roots, and we can use decomposition in partial fractions to obtain

[TABLE]

where $v_{j}$, $j=1,2,3$, are the different roots of $P$ and

[TABLE]

Now let us see that there is $\delta\in(0,1)$ such that if $(\eta,\zeta,\tau)\in\mathbb{R}^{3}-K_{\beta}$ and $v_{j}\in\mathbb{C}$ is a root of polynomial $P$ then

[TABLE]

In other words, let us prove that there is $\delta>0$ such that if $|3v_{j}^{2}+w|<\delta|w|$ and $v_{j}\in\mathbb{C}$ is a root of polynomial $P$ then $(\eta,\zeta,\tau)\in K_{\beta}$. Taking into account that $v_{j}$ is a root of $P$ if and only if

[TABLE]

let us prove that there is $\delta>0$ such that if $|\frac{v_{j}^{2}}{w}+\frac{1}{3}|<\frac{\delta}{3}$ and \frac{1}{\frac{v_{j}^{2}}{w}\big{(}\frac{v_{j}^{2}}{w}+1\big{)}^{2}}=\frac{w^{3}}{\tau^{2}}, then $(\eta,\zeta,\tau)\in K_{\beta}$.

By the continuity of the complex-valued function $\frac{1}{z(z+1)^{2}}$ in $z=-\frac{1}{3}$, there is $\delta\in(0,1)$ such that if $|z+\frac{1}{3}|<\frac{\delta}{3}$ then $|\frac{1}{z(z+1)^{2}}+\frac{27}{4}|<\frac{27}{8}$. Hence if \big{|}\frac{v_{j}^{2}}{w}+\frac{1}{3}\big{|}<\frac{\delta}{3}, we have that

[TABLE]

and by (2.5) we conclude that

[TABLE]

i.e. $|T_{\beta}(\eta,\zeta,\tau)|<\frac{27}{8}$ and, in consequence, $(\eta,\zeta,\tau)\in K_{\beta}$.

From (2.3) and (2.4) we conclude that if $(\eta,\zeta,\tau)\in\mathbb{R}^{3}-K_{\beta}$, then

[TABLE]

On the other hand, let us observe that the set $A:=\cup_{j=1}^{3}\{(\eta,\zeta,\tau)\mid\operatorname{Im}(v_{j})-\lambda=0\}$ has three-dimensional measure [math].

For $(\eta,\zeta,\tau)\in(\mathbb{R}^{3}-K_{\beta})-A$, $x\in\mathbb{R}$ and $j=1,2,3$ let us define

[TABLE]

Then from (2.2) and (2.6) it is clear that

[TABLE]

ii) If $(\eta,\zeta,\tau)\in K_{\beta}$ , using (2.1) and the fact that $C>1$, for $\lambda\geq 3C\beta$ it follows that

[TABLE]

Then,

[TABLE]

where

[TABLE]

In consequence,

[TABLE]

From (2.7) and (2.8) we conclude that for $\lambda\geq 3C\beta$ and $\beta\geq 1$ and a.e. $(\eta,\zeta,\tau)\in\mathbb{R}^{3}$:

[TABLE]

Finally, from estimate (2.9), by applying Plancherel’s formula in $L_{tyz}^{2}(\mathbb{R}^{3})$ and Minkowski’s integral inequality, it follows that

[TABLE]

∎

Remark 1. The statement of Lemma 2.2 is also true if in the definition of the multipliers $m_{0}$ and $m_{k,l,n}$, instead of $(i\xi-\lambda)$, $(i\eta-\beta)$ and $(i\zeta-\beta)$ we consider the other seven possible triplets of the type $(i\xi\pm\lambda)$, $(i\eta\pm\beta)$ and $(i\zeta\pm\beta)$.

Proof of Theorem 1.2

The proof of Theorem 1.2 follows from Lemmas 2.1 and 2.2 in a similar way as it was done in the proof of Theorem 1.2 in [3].∎

3. A lower estimate (proof of Theorem 1.3)

In the proof of Lemma 3.1, basis for the proof of Theorem 1.3, we follow the ideas contained in works [11] and [5]. This Lemma refers to a boundedness property of the inverse of the operator associated to the linear part of the ZK equation. This property is expressed in $L^{2}$ spaces with an exponential weight $e^{\psi}$, where the exponent $\psi$ depends on $x,y,z,t$, and a free parameter $\alpha$.

Lemma 3.1**.**

Let $\phi:[0,1]\rightarrow\mathbb{R}$ be a $C^{\infty}$ function and let $D:=\mathbb{R}^{3}\times[0,1]$. Let us assume that $R>1$ and define

[TABLE]

Then, there is $\overline{C}=max(\|\phi^{\prime}\|_{L^{\infty}},\|\phi^{\prime\prime}\|_{L^{\infty}},1)>0$ such that the inequality

[TABLE]

holds if $\alpha\geq\overline{C}R^{3/2}$ and $g\in C([0,1];H^{3}(\mathbb{R}^{3}))\cap C^{1}([0,1];L^{2}(\mathbb{R}^{3}))$ is such that

• (i)

$g(0)=g(1)=0$;

• (ii)

there is $M>0$ such that $supp\,g(t)\subset[-M,M]\times[-M,M]\times[-M,M]$ for all $t\in[0,1]$;

• (iii)

$supp\,g(\cdot_{t})(\cdot_{x},\cdot_{y},\cdot_{z})\subset\{(x,y,z,t):|\frac{x}{R}+\phi(t)|\geq 1\}$.

Proof.

Let us define $f:=e^{\psi}g$. Since $e^{\psi}\partial_{x}e^{-\psi}f=(\partial_{x}-\psi_{x})f$, to establish (3.1) it is sufficient to prove that

[TABLE]

where $Tf:=e^{\psi}(\partial_{t}+\partial_{x}\Delta)e^{-\psi}f$.

Using the fact that $\psi_{xxx}=\psi_{yyx}=\psi_{yx}=\psi_{zzx}=\psi_{zx}=0$, it can be seen that

[TABLE]

We write the operator $T$ as the sum of a symmetric operator $S$ and an antisymmetric operator $A$, as follows:

[TABLE]

If we denote by $\langle\cdotp,\cdotp\rangle$ the inner product in the real Hilbert space $L^{2}(D)\equiv L^{2}$, then we have that

[TABLE]

From (3.3) and (3.4) we have

[TABLE]

Taking into account $(3.7)$ in [3] and the symmetry between $y$ and $z$ in the ZK equation it is clear that

[TABLE]

In order to have the complete expression for $2\langle Sf,Af\rangle$ we still must calculate $2\langle II_{y}f,IV_{z}f\rangle+\langle II_{z}f,IV_{y}f\rangle$. For that it is enough to calculate the first term and, by symmetry, the expression for the second one will be obtained from the expression for the the first term by replacing $y$ by $z$ and $z$ by $y$.

Applying integration by parts we obtain

[TABLE]

By symmetry

[TABLE]

Taking into account that

[TABLE]

from (3.8) and (3.9) follows that

[TABLE]

From (3.6), (3.7) and (3.10) follows that

[TABLE]

Let us observe that $\psi_{xx}=\psi_{yy}=\psi_{zz}=\dfrac{2\alpha}{R^{2}}\geq 0$. In order to bound $2\langle Sf,Af\rangle$ from below we rewrite the expression in (3.11) in an adequate manner as follows:

[TABLE]

Now we will bound the integrals $\int\limits_{D}IN_{j}$, $j=1,2,3$, from below. Taking into account that

[TABLE]

and using integration by parts it follows that

[TABLE]

For the integral $\int\limits_{D}IN_{2}$ we have, by applying integration by parts and rewriting in an appropriate manner, that

[TABLE]

With respect to the integral $\int\limits_{D}IN_{3}$, taking into account that

[TABLE]

we have that

[TABLE]

From (3.5), (3.12), (3.13), (3.14), and (3.15), since $\psi_{xx}=\psi_{yy}=\psi_{zz}=\dfrac{2\alpha}{R^{2}}$, it follows that

[TABLE]

We observe now that

[TABLE]

and define $\overline{C}:=\max\{\|\phi^{\prime}\|_{L^{\infty}},\|\phi^{\prime\prime}\|_{L^{\infty}},1\}$. If we take $\alpha\geq\overline{C}R^{3/2}>1$, for the points $(x,y,z,t)$ where $|\dfrac{x}{R}+\phi(t)|\geq 1$, we can see, since $\max\{\dfrac{\alpha^{3}}{R^{6}},\dfrac{\alpha^{3}\overline{C}}{R^{3}},\alpha\overline{C},\dfrac{\alpha^{4}}{R^{6}}\}\leq\dfrac{\alpha^{5}}{R^{6}}$ and $\dfrac{\alpha\overline{C}}{R}\leq\dfrac{\alpha^{3}}{R^{4}}$, that

[TABLE]

[TABLE]

[TABLE]

Therefore, since the functions $f$ and $f_{x}$ are supported in $\{(x,y,z,t):|\dfrac{x}{R}+\phi(t)|\geq 1\}$, from (3.16) to (3.19), it follows, for $\alpha\geq\overline{C}R^{3/2}$, that

[TABLE]

which proves (3.2). ∎

Proof of Theorem 1.3

The proof of this theorem is analogous to the Theorem 1.3 in [3] and is based upon Lemma 3.1, and in consequence we omit it.∎

4. Proof of Theorem 1.1

We will follow the same lines of the proof of Theorem 1.1 in [3]. Let $v:=u_{1}-u_{2}$. As it was shown in [3] it is sufficient to prove that if $v(0),v(1)\in L^{2}(e^{a(x^{2}+y^{2}+z^{2})^{\frac{3}{4}}}dx\,dy\,dz)$ for some $a$ greater than a certain universal positive constant $a_{0}$ then $v\equiv 0$ in the set $D_{0}:=\{(x,y,z,t)\mid(x,y,z)\in\mathbb{R}^{3},t\in[\frac{1}{3},\frac{2}{3}]\}$.

Reasoning by contradiction, suppose that $v$ is not identically zero in $D_{0}$. Without loss of generality (making a translation in the variables $x$, $y$ and $z$ if necessary), we can affirm that if

[TABLE]

then $\|v\|_{L^{2}(Q)}>0$. By Theorem 1.3 there exist $\overline{C}>0$, $C>0$, and $R_{0}\geq 2$ such that

[TABLE]

For $R>3$ we take $N\in\mathbb{N}$, with $N>4C_{2}R$, where $C_{2}$ is the constant in the statement of Theorem 1.2. We construct a $C^{\infty}$ truncation function $\phi_{R,N}:[0,\infty)\longrightarrow\mathbb{R}$ supported in $(R,N+1)$, by taking a nonnegative function $\eta\in C^{\infty}(\mathbb{R})$, with unitary integral and $\text{supp}\,\eta\subset(0,1)$, and defining $\phi_{R,N}(s):=\int_{0}^{s}[\eta(s^{\prime}-{\scriptstyle{R}})-\eta(s^{\prime}-{\scriptstyle{N}})]\,ds^{\prime}$. Notice that $0\leq\phi_{R,N}\leq 1$, $\phi_{R,N}\equiv 1$ in $[R+1,N]$ and ${supp}\,\phi_{R,N}\subset(R,N+1)$. Let us define $\Phi\equiv\Phi_{R,N}:\mathbb{R}^{3}\longrightarrow\mathbb{R}$ by $\Phi(x,y,z)\equiv\Phi_{R,N}(x,y,z):=\phi_{R,N}(\sqrt{x^{2}+y^{2}+z^{2}})$ and

[TABLE]

The function $w$ satisfies the hypotheses of Theorem 1.2. Therefore, for $\beta\geq 1$ and $\lambda\geq C_{2}\beta$

[TABLE]

But

[TABLE]

where

[TABLE]

Therefore, from (4.2) and (4.3), and using the facts that $\|\cdot\|_{L^{2}(D)}\leq\|\cdot\|_{L^{\infty}_{t}L^{2}_{xyz}(D)}$ and $\|\cdot\|_{L^{1}_{t}L^{2}_{xyz}(D)}\leq C\|\cdot\|_{L^{2}(D)}$ it follows that

[TABLE]

We now estimate $I$, $II$, and $III$ separately.

Estimation of $I$: By observing that $w$ is supported in the set

[TABLE]

and applying Hölder’s inequality we see that

[TABLE]

We will now see that the norms involving $u_{1}$ and $\partial_{x}u_{2}$ on the right hand side of (4.7) tend to zero as $R\to\infty$. For that we use the following interpolation Lemma (see [21]).

Lemma 4.1**.**

Let $a,b>0$. Assume that $J^{a}f:=(1-\Delta)^{\frac{a}{2}}f\in L^{2}(\mathbb{R}^{n})$ and $(1+\sum_{i=1}^{n}x_{i}^{2})^{\frac{b}{2}}f\in L^{2}(\mathbb{R}^{n})$, where $x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$. Then for any $\theta\in(0,1)$

[TABLE]

Applying (4.8) with $a=4$, $b=8/5+\epsilon$ and $\theta=3/8+\epsilon/8$ (with $\epsilon$ as in (1.2)) we have that

[TABLE]

where $\epsilon_{1}=\frac{17}{40}\epsilon-\frac{1}{8}\epsilon^{2}>0$. Applying (4.9) with $f=u_{j}(t)$, $j=1,2$, from (1.2) and from the embbeding of $H^{3/2+\epsilon/2}(\mathbb{R}^{3})$ in $L^{\infty}(\mathbb{R}^{3})\cap C(\mathbb{R}^{3})$, we conclude that

[TABLE]

Since $3/2+\epsilon/2>1$, (4.9) is also valid for $f=u_{2}(t)$ with $J^{1}$ instead of $J^{3/2+\epsilon/2}$, and we can apply the product rule for derivatives to obtain that $\|(1+x^{2}+y^{2}+z^{2})^{\frac{1}{2}(1+\epsilon_{1})}\partial_{x}u_{2}(t)\|_{L^{2}(\mathbb{R}^{3})}$ is a bounded function of $t\in[0,1]$. Thus, applying (4.8) with $f=\partial_{x}u_{2}(t)$, $a=3$, $b=1+\epsilon_{1}$ and $\theta=1/2+\epsilon_{2}$ with $\epsilon_{2}>0$ small, we can conclude that

[TABLE]

and, in consequence, using the embedding of $H^{3/2+3\epsilon_{2}}(\mathbb{R}^{3})$ in $L^{\infty}(\mathbb{R}^{3})\cap C(\mathbb{R}^{3})$, we have

[TABLE]

From the decay properties expressed in (4.10) and (4.11) is now easy to see that the four norms in (4.7) involving $u_{1}$ and $\partial_{x}u_{2}$ tendo to zero as $R\to\infty$, and therefore the exists $R_{1}>R_{0}$ such that for $R>R_{1}$ the term $I$ can be absorbed by the terms $\|e^{\lambda|x|+\beta|y|+\beta|z|}\partial_{x}w\|_{L^{2}(D)}$ and $\|e^{\lambda|x|+\beta|y|+\beta|z|}w\|_{L^{\infty}_{x}L^{2}_{tyz}(D)}$ on the left hand side of (4.5).

Estimation of $II$:

To estimate $II$ we use the expression for $F$ given in (4.4) and observe that the derivatives of $\Phi$ are supported in the sets $\{(x,y,z)\mid\sqrt{x^{2}+y^{2}+z^{2}}\in(R,R+1)\}$ and $\{(x,y,z)\mid\sqrt{x^{2}+y^{2}+z^{2}}\in(N,N+1)\}$. For the estimation in the first set we bound $e^{\lambda|x|+\beta|y|+\beta|z|}$ by $e^{(\lambda+2\beta)(R+1)}$ and apply Cauchy-Schwarz inequality. For the estimation in the second set we observe that since $v(0),\,v(1)\in L^{2}(e^{a(x^{2}+y^{2}+z^{2})^{3/4}}\,dx\,dy\,dz)$, it follows that for all $\overline{\lambda}>0$ and all $\overline{\beta}>0$ , $v(0),\,v(1)\in L^{2}(e^{2\overline{\lambda}|x|+2\overline{\beta}|y|+2\overline{\beta}|z|}\,dx\,dy\,dz)$. Applying to the equation

[TABLE]

a procedure similar to that given in Theorem 1.3 of [2] for the ZK equation, we can observe that $v$ is a bounded function from the time interval $[0,1]$ with values in $H^{3}(e^{2\overline{\lambda}|x|+2\overline{\beta}|y|+2\overline{\beta}|z|}\,dx\,dy\,dz)$ (i.e. $\partial^{\alpha}v(t)\in L^{2}(e^{2\overline{\lambda}|x|+2\overline{\beta}|y|+2\overline{\beta}|z|}\,dx\,dy\,dz)$ for all multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ with $|\alpha|\leq 3$). Thus we can take $\overline{\lambda}=\lambda+1$ and $\overline{\beta}=\beta+1$ and apply Cauchy-Schwarz inequality. In this manner we obtain

[TABLE]

where $C$ does not depend of $\lambda$, $\beta$, $R$, and $N$, and $C_{\lambda,\beta}$ does not depend of $N$ and $R$.

Estimation of $III$:

Since $\Phi_{R,N}$ and its derivatives are bounded by a constant independent of $R$ and $N$, it follows that

[TABLE]

We now consider the region $\Omega_{R}:=\{(x,y,z,t)\mid\sqrt{x^{2}+y^{2}+z^{2}}\in[6C_{2}R-1,\,6C_{2}R],\,t\in[0,1]\}$, where $C_{2}$ is the constant in the statement of Theorem 1.2. Then, from (4.6), $\Omega_{R}\subset D_{R}$, and since $N>6C_{2}R$, $v$ and $w$ coincide in $\Omega_{R}$. We now return to (4.5), replace its left hand side by a smaller amount, apply (4.13) and (4.14), and make $N\to\infty$ to obtain that

[TABLE]

For $\lambda\geq C_{2}$, let $\beta:=\lambda/C_{2}\geq 1$. Then, for $(x,y,z,t)\in\Omega_{R}$,

[TABLE]

In this way, bearing in mind the definition of $A_{6C_{2}R}(v)$ given in the statement of Theorem 1.3 and taking into account that $\lambda\geq C_{2}$ and $R>3$, from (4.15) and (4.16) we conclude that

[TABLE]

We now take $\lambda=\frac{1}{10}aR^{1/2}$ with $R$ large enough. If $x^{2}+y^{2}+z^{2}\geq R^{2}$ we see that

[TABLE]

Since $v(0),v(1)\in L^{2}(e^{a(x^{2}+y^{2}+z^{2})^{3/4}}\,dx\,dy\,dz)\cap C([0,1];H^{4}({\mathbb{R}}^{3}))$, by an interpolation argument which can be proved using smooth truncation functions and integration by parts, it can be seen that if $0\leq k+l+n\leq 3$, then $\partial_{x}^{k}\partial_{y}^{l}\partial_{z}v(0)$ and $\partial_{x}^{k}\partial_{y}^{l}\partial_{z}v(1)$ belong to the class $L^{2}(e^{\frac{1}{4}a(x^{2}+y^{2}+z^{2})^{3/4}}\,dx\,dy\,dz)$ In this way, from (4.18) and the definition of $H_{R}(\lambda,\beta)$ given in (4.14) we obtain that

[TABLE]

Hence, from (4.17) with $\lambda=\frac{1}{10}aR^{1/2}$ we have that

[TABLE]

and thus

[TABLE]

In this way, from (4.1) it follows that for $R$ large enough

[TABLE]

Thus, if $a>a_{0}:=30\times 14(6C_{2})^{3/2}\overline{C}$, then, by making $R\to\infty$, we see that $\|v\|_{L^{2}(Q)}=0$, which contradicts the original fact that $\|v\|_{L^{2}(Q)}>0$. Therefore $v\equiv 0$, and Theorem 1.1 is proved. ∎

Acknowledgments. Partially supported by Fondo Nacional de Financiamiento Para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, Project “Ecuaciones Diferenciales Dispersivas y Elípticas No Lineales”, contrato Colciencias FP44842-087-2015.