Introduction This article concerns a unique continuation principle for the equation ∂tu+ (−1)k+1∂xnu+P(u, ∂xu

Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 246, pp. 1–25.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

UNIQUE CONTINUATION PRINCIPLE FOR HIGH ORDER EQUATIONS OF KORTEWEG-DE VRIES TYPE

PEDRO ISAZA

Abstract. In this article we consider the problem of unique continuation for high-order equations of Korteweg-de Vries type which include the kdV hierarchy. It is proved that if the differencewof two solutions of an equation of this form has certain exponential decay forx > 0 at two different times, thenwis identically zero.

1. Introduction

This article concerns a unique continuation principle for the equation

∂tu+ (−1)^k+1∂_xⁿu+P(u, ∂xu, . . . , ∂_x^pu) = 0, u=u(x, t), x, t∈R, (1.1) where n = 2k+ 1, k = 1,2. . ., and P is a polynomial in u, ∂_xu, . . . , ∂_x^pu, with p≤n−1. In particular, we will focus our attention to the case in whichP has the form

P(u, ∂_xu, . . . , ∂_xⁿ⁻²u) =

k+1

X

d=2

X

|m|=2(k+1−d)+1

a_d,m∂^m_x¹u . . . ∂^m_x^du≡

k+1

X

d=2

A_d(z), z= (u, ∂xu, . . . , ∂_xⁿ⁻²u),

(1.2)

where, ford∈Nand for integersm₁, . . . , m_d,m:= (m₁, . . . , m_d) is a multi-index with 0≤m1 ≤ · · · ≤md, |m|:=m1+· · ·+md, andad,m is a constant. We will refer to equation (1.1) withP defined by (1.2) as equation (1.1)-(1.2). We will also consider equation (1.1) when the nonlinearityP has orderp≤k.

The type of relation expressed in (1.2), between the degree and the order of each monomial ofP, is present in the nonlinearities of the collection of equations known as the KdV (Korteweg-de Vries) hierarchy. This set of equations was introduced by Lax [15] in the process to determine the functions u = u(x, t) for which the eigenvalues of the operator L := _dx^d2₂ −u(·, t) remain constant as t evolves. This property had been already discovered by Gardner et al. in [3] for the solutions of the Korteweg-de Vries equation

∂_tu+∂_x³u+u∂_xu= 0.

2000Mathematics Subject Classification. 35Q53, 37K05.

Key words and phrases. Nonlinear dispersive equations; unique continuation;

estimates of Carleman type.

c

2013 Texas State University - San Marcos.

Submitted July 21, 2013. Published November 13, 2013.

Supported by DIME, Universidad Nacional de Colombia, Medell´ın, grant 15401.

1

Lax [15] showed that this property holds for the solutions of the equations

∂tu+ [Bk(u), L] = 0, (1.3)

where [B, L] :=BL−LBdenotes the commutator of B andL, andBk(u) is the skew-adjoint operator defined by

B_k(u) =b_k d^2k+1 dx^2k+1 +

k−1

X

j=0

b_k,j(u) d^2j+1

dx^2j+1 + d^2j+1

dx^2j+1b_k,j(u),

with the coefficientsb_k,j(u) chosen in such a way that the operator [B_k(u), L] has order zero. It was proved in [16] that the equations in the KdV hierarchy (1.3) can be written in the form∂tu+∂xGk+1(u) = 0, were the functionsGk(u) are the gradients of the functionals Fk(u) which define the conservation laws of the KdV equation. The gradientsGk satisfy the following recursion formula due to Lenard (see [4] and [20]):

∂xGk+1 =cJ Gk, whereJ =∂_x³+2

3u∂x+1 3∂xu .

This formula can be applied to obtain a derivation of the equations in the hierarchy.

Starting with G0(u) = 3, with k = 0 we get the transport equation, with k = 1 the KdV equation, and, with k = 2, k = 3, and k = 4, we respectively find the equations

∂_tu+∂⁵_xu−10u∂_x³u−20∂_xu∂_x²u+ 30u²∂_xu= 0,

∂_tu+∂_x⁷u+ 14u∂_x⁵u+ 42∂_xu∂_x⁴u+ 70∂_x²u∂_x³u+ 70u²∂_x³u + 280u∂_xu∂_x²u+ 70(∂_xu)³+ 140u³∂_xu= 0,

∂tu+∂_x⁹u+ X

m1+m2=7 0≤m1≤m2

a2,m∂_x^m1u∂_x^m2u

+ X

m₁+m₂+m₃=5 0≤m1≤m2≤m3

a3,m∂^m_x¹u∂_x^m2u∂^m_x³u+· · ·+a5,mu⁴∂xu= 0,

(1.4)

for certain constantsad,m, withd= 2, . . . ,5 and|m|= 2(5−d) + 1.

In spite of computational difficulties, it is possible to obtain exact expressions for all the equations in the hierarchy (see [1]). However, following a simple procedure, and without obtaining the explicit values for the coefficients, it can be proved (see [5]) that the equations in the KdV hierarchy (1.3) have the form of (1.1)-(1.2).

Whenk is even we have made the change of variablex7→ −xand thus the linear term∂_xⁿuhas been transformed into (−1)^k+1∂_xⁿuin (1.1).

The aspects of local and global well-posedness of the initial value problem (IVP) associated with the general equation (1.1) have been considered in [10] and [11], where Kenig, Ponce, and Vega proved that the (IVP) is locally well-posed in weighted spaces H^s(R)∩L²(|x|^mdx) if s ≥ s0(k), for some s0(k) and some in- tegerm=m(k).

For the (IVP) associated to (1.1)-(1.2), in [21], Saut proved the existence of global solutions for initial data in Sobolev spacesH^m(R) for m ≥k, integer. By using a variant of Bourgain spaces, in [5], Gr¨unrok proved the local well-posedness for the (IVP) of equation (1.1)-(1.2) in the context of the spaces

Hb_s^r(R) :={f | kfks,r:=k(1 +ξ²)^s/2fb(ξ)k_Lr0 ξ <∞},

with

r∈(1, 2k

2k−1], ands > k−3 2 − 1

2k+2k−1 2r⁰ .

Herebdenotes the Fourier transform and 1/r+1/r⁰= 1. We also refer to the articles [9, 14, 17, 18, 19], which, among others, consider the problem of well-posedness for high order equations of KdV-type and especially for the equations of order five (k= 2).

Our main goal is to prove continuation principles for (1.1)-(1.2) with n ≥ 5, which include the KdV hierarchy, and for the equations (1.1) with n ≥ 5 and p≤k. Roughly speaking, we will prove that if the differencew:=u1−u2 of two sufficiently smooth solutions of equation (1.1)-(1.2) decays as exp(−x^4/3₊ ⁺) at two different times, then w≡0. (Here x+ := ¹₂(x+|x|), and 4/3⁺ means 4/3 + for arbitrarily small >0). For (1.1) with p≤k we have a similar result ifw decays as exp(−ax^n/(n−1)₊ ) for a > 0 sufficiently large at two different times. This last result is coherent with the decay exp(−cx^n/(n−1)₊ ) of the fundamental solution of the linear problem associated with (1.1) (see [23]). When the nonlineariy P has higher order as in (1.1)-(1.2), it is then necessary to impose a stronger decay onw.

The aspect of unique continuation has been studied for a variety of non-linear dispersive equations, and especially for the KdV and Schr¨odinger equations. Saut and Sheurer [22] considered a class of nonlinear dispersive equations, which includes the KdV equation, and proved that if a solution uof one of such equations van- ishes in an open set Ω of the space-time space, then uvanishes in all horizontal components of Ω, that is, in the set{(x, t) :∃y with (y, t)∈Ω}.

By using methods of complex analysis, Bourgain [2] proved that if a solutionuof the KdV equation is supported in a compact set{(x, t) :−B≤x≤B, t0≤t≤t1}, thenuvanishes identically.

Kenig, Ponce and Vega [12], considered a solution of the KdV equation which vanishes only in two half lines [B,+∞)× {t0}and [B,+∞)× {t1}, and proved that this solution must be identically zero. A similar result was proved in [13] for the differencew=u1−u2of two solutions of the KdV equation. Escauriaza et al. [8]

refined this result by only imposing the condition that w(·, t0) and w(·, t1) decay as exp(−ax^γ₊), forγ= 3/2 anda >0 sufficiently large, together with a additional hypothesis of polynomial decay foru₁ andu₂. This result is obtained by applying two types of estimates for the functionw: Carleman type estimates, which express a boundedness of the inverse of the linear operator∂_t+∂_x³ inL^p−L^q-spaces with exponential weight; and a so-called lower estimate which bounds the L²-norm of win a small rectangle at the origin with the H²-norm of w in a distant rectangle [R, R+ 1]×[0,1].

For the fifth-order equation (1.1) (k= 2), Dawson [6] proved a result similar to that in [8] withγ= 4/3⁺ for the general casep≤n−1 = 4, and withγ= 5/4 for the casep≤2.

In this article we consider equations (1.1) and (1.1)-(1.2) with arbitrary order nand prove the continuation principles stated in Theorem 1.1 and 1.2 below. For that, we follow the method traced in [8]. The greatest difficulty in this process is to manage the huge amount of terms arising in the computations of the operators involved in the lower estimate. We consider that the main contribution of our work is the presentation of a clear and organized procedure to obtain the lower estimate (see Lemma 4.1 and Theorem 4.3).

We now state our main results.

Theorem 1.1. For oddn≥5 ,k= (n−1)/2, andα > ⁿ⁺¹₃ , suppose thatu1, u2

are in C([0,1];Hⁿ⁺¹(R)∩L²((1 +x+)^2αdx))are two solutions of the equation

∂tu+ (−1)^k+1∂_xⁿu+P(u, ∂xu, . . . , ∂_xⁿ⁻²u) = 0 (1.5) withP as in (1.2), and letw:=u1−u2. If

w(0), w(1)∈L²(e^2x4/3++ dx) (1.6) for some >0, thenw≡0.

The proof of this theorem can be adapted to obtain a similar continuation prin- ciple for equation (1.1) whenp≤k. In this case we require a weaker decay forw(0) andw(1) and consider some minor modifications in the polynomial decay hypoth- esis for u₁ and u₂. For the sake of simplicity we state this result without making special emphasis in the optimal value ofα.

Theorem 1.2. For odd n ≥ 5, k = (n−1)/2, and α0 > 0 sufficiently large, suppose that u1, u2∈C([0,1];Hⁿ⁺¹(R)∩L²((1 +|x|)^2α0dx))are two solutions of the equation

∂_tu+ (−1)^k+1∂_xⁿu+P(u, ∂_xu, . . . , ∂^p_xu) = 0 (1.7) with p≤k. Definew:=u1−u2. Then, there is a >0, which depends only on n, such that if

w(0), w(1)∈L²(e^{2axn/(n−1)+} dx), (1.8) thenw≡0.

The article is organized as follows: In section 2 we prove that the exponential decay forw in the semi-axisx >0 is preserved in time. In section 3 we establish the Carleman type estimates and in section 4 we prove the lower estimates. Finally we give the proofs of Theorems 1.1 and 1.2 in section 5.

Throughout the paper the lettersC andcwill denote diverse positive 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 notationsC_a,C(a), andc_ato make emphasis in the fact that the constants depend upona. We frequently writef(·_s) to denote a functions7→f(s). For a setA,χ_A will denote the characteristic function of A. The symbolsband ˇ will denote the Fourier and the inverse Fourier transform, respectively. The notationsb

x and ˇ^ξ will emphasize the facts that the Fourier transform and its inverse are taken with respect to specific variables x and ξ, respectively. For 1 ≤ p, q < ∞, A, B ⊆ R, D=A×B, andf =f(x, t) we will denote

kfk^p_Lp

xL^q_t(D):=

Z

A

Z

B

|f(x, t)|^qdt^p/q dx .

We will use similar definitions whenp=∞orq=∞and also forkfk_L^q

tL^p_x(D). 2. Exponential decay

In this section we prove that if the differencew of two solutions of (1.5) decays exponentially at t = 0, then this decay is preserved at all positive times. This property will be crucial for the application of the Carleman estimates in the proofs of Theorem 1.1 and 1.2.

Theorem 2.1. For oddn≥5,k= (n−1)/2, andα >(n+ 1)/4, letu1, u2 are in C([0,1];Hⁿ⁺¹(R)∩L²((1 +x+)^2αdx))) be two solutions of (1.1)-(1.2), and define w:=u1−u2. Letβ >0 and suppose thatw(0)∈L²(e^βxdx). Then

sup

t∈[0,1]

kw(t)k_L2(e^βxdx)<∞. (2.1) Proof. Let us denotezi = (ui, ∂xui, . . . ∂_xⁿ⁻²ui), i= 1,2. Then w is a solution of the differential equation

∂tw+ (−1)^k+1∂_xⁿw+P(z1)−P(z2) = 0. (2.2) We will first prove that the theorem is valid provided we can construct a se- quence{φN}N∈N of nondecreasing functions inC^∞(R) satisfying for allx∈Rthe conditions

ϕ_N(x)→e^βx as N → ∞ and 0≤ϕ_N(x)≤C e^βx, (2.3) ϕ_N(x)≤C_N(1 +x₊)^(k+2)/4, (2.4)

|ϕ^(j)_N (x)| ≤Cjϕ⁰_N(x) forj = 2,3. . . , n= 2k+ 1 andϕ⁰_N(x)≤CϕN(x), (2.5) ϕ_N(x)≤C(1 +x₊)ϕ⁰_N(x), (2.6) where the constantsC andCj are independent ofN.

We multiply (2.2) by ϕNu and, for t fixed, integrate in R. Thus, by applying integration by parts we obtain

1 2

d dt

Z

ϕNw²=−2k+ 1 2

Z

ϕ⁰_N(∂_x^kw)²+ck−1

Z

ϕ⁽³⁾_N (∂_x^k−1w)²+. . . +c1

Z

ϕ^(2k−1)_N (∂xw)²+1 2

Z

ϕ^(2k+1)_N w²

− Z

(P(z1)−P(z2))ϕNw .

(2.7)

The integration by parts is justified as follows: since there is a constantC >0 such thatk(1 +x₊)^αu_i(t)kL² ≤C, andkui(t)kHⁿ⁺¹(R)≤Cfor allt∈[0,1] andi= 1,2, by using integration by parts and truncation functions, it can be proved that the following interpolation property holds:

k(1 +x+)^{α(1−(n+1)j )}∂_x^jui(t)kL²(R≤C , for allt∈[0,1];i= 1,2; j= 0, . . . , n+ 1.

(2.8) Sinceα >(n+ 1)/4, it follows that, for 0≤j≤k, (1 +x+)^(k+2)/4∂_x^jw(t)∈L²(R, and thus, from (2.4) and (2.5) ϕ^(l)∂_x^jw(t)∈ L²(R for all positive integersl. This implies that all the terms which appear in the procedure to obtain (2.7) are in a right setting for the application of the integration by parts.

Let us estimate the last term on the rand-hand side of (2.7). From (1.2) we have that

P(z) =

k+1

X

d=2

Ad(z) where Ad(z) = X

|m|=n−2(d−1) 0≤m₁≤···≤m_d

ad,m∂_x^m1u . . . ∂_x^mdu , (2.9)

and thus

| Z

P(z1)−P(z2)

ϕNw dx|=|

k+1

X

d=2

Z

Ad(z1)−Ad(z2)

ϕNw dx| ≡ |

k+1

X

d=2

Id|. (2.10) It is easily seen that

Ad(z1)−Ad(z2) = X

|m|=n−2(d−1) 0≤m₁≤···≤m_d

ad,m

∂_x^m1w∂_x^m2u1. . . ∂^m_x^du1

+∂_x^m1u2∂_x^m2w . . . ∂_x^mdu1+∂_x^m1u2∂_x^m2u2. . . ∂_x^mdw .

(2.11)

We estimate I₂, which, having the derivatives of the highest order, is the most critical term in (2.10). From (2.11),

I₂= Z

A₂(z₁)−A₂(z₂)

ϕ_Nw dx

= X

m₁+m₂=n−2 0≤m1≤m2

a2,m

Z

(∂_x^m1w∂_x^m2u1+∂_x^m1u2∂_x^m2w)ϕNw dx. (2.12)

We estimate only the second term on the right-hand side of (2.12), the other term being similar. We apply integration by parts to reduce the order of ∂_x^m2w and obtain that

Z

(∂^m_x¹u2) (∂_x^m2w)ϕNw= X

r₁+r₂+2r₃=m₁+m₂=n−2 r1≥m1

cr₁,r₂

Z

(∂^r_x¹u2)ϕ^(r_N²⁾(∂_x^r3w)² (2.13) To analyze the terms in this sum we consider the casesr2= 0 and r2≥1:

(i) Ifr₂= 0 andr₃= 0, thenr₁=n−2 and we bound the corresponding integral in (2.13) by

Ck∂_xⁿ⁻²u2(t)kL^∞

Z

ϕNw²≤C Z

ϕNw²,

whereCis independent oft∈[0,1] by the Sobolev embedding ofH¹(R) inL^∞(R).

Ifr2= 0 andr3≥1, then the maximum value ofr1in (2.13) isn−4. Therefore, using the fact that ϕN ≤ C(1 +x+)ϕ⁰_N we bound the corresponding integral in (2.13) by

C max

0≤r1≤n−4k(1 +x₊)∂_x^r1u₂(t)kL^∞

Z

ϕ⁰_N(∂_x^r3w)². (2.14) From (2.8) it can be seen that if Ψ∈C^∞(R) is a truncation function with Ψ≡0 in (−∞,1], and Ψ ≡ 1 in [2,+∞), then Ψ(·)(1 +x+)^{α(1−n+1j+1)}∂_x^jui(t) ∈ H¹(R), i= 1,2,j= 0,1, . . . , n, and

kΨ(·)(1 +x+)^{α(1−j+1n+1)}∂_x^jui(t)k_H1(R)

≤C(k(1 +x₊)^αu_i(t)k_L2(R,ku_i(t)k_Hn+1(R))≤C for allt∈[0,1]. (2.15) In particular, forj= 0, . . . , n−4,α(1−_n+1^j+1)>ⁿ⁺¹₄ (1−ⁿ⁻³_n+1) = 1, and thus from the Sobolev embedding ofH¹in L^∞ we conclude that

0≤j≤n−4max k(1 +x₊)∂_x^ju_i(t)k_L∞(R)≤C, (2.16) withC independent oft∈[0,1]. Thus (2.14) is bounded by CR

ϕ⁰_N(∂_x^r3w)².

(ii) Ifr2≥1, thenr1≤n−3. From (2.5),ϕ^(r2)≤Cr₂ϕ⁰ ≤Cϕ⁰for 1≤r2≤n−2.

Thus we bound the corresponding term in (2.13) by C max

0≤r1≤n−3k∂^r_x¹u2(t)kL^∞

Z

ϕ⁰_N(∂_x^r3w)²≤C Z

ϕ⁰_N(∂_x^r3w)². (2.17) Now, let us determine the maximum value of r₃ appearing in (2.13). For that, we observe thatr1+r2+ 2r3=n−2 is odd, and thus the maximum value of r3

occurs when (r1, r2) = (0,1) or (1,0), which then givesr3≤(n−3)/2 =k−1.

Gathering the estimates of the cases (i) and (ii) above, and taking into account thatr3≤k−1, we conclude that

|I2| ≤C

k−1

X

j=1

Z

ϕ⁰_N(∂_x^jw)²dx+C Z

ϕ_Nw². (2.18)

Proceeding in a similar way, we obtain the same bound for|I3|, . . .|Ik+1|, and thus for the left hand side of (2.10). Therefore, returning to (2.7) and using the fact that, from condition (2.5),|φ^(j)| ≤C φ⁰, j= 1, . . .2k+ 1, we obtain

1 2

d dt

Z

ϕNw²≤ −2k+ 1 2

Z

ϕ⁰_N(∂^k_xw)²+C

k−1

X

j=1

Z

ϕ⁰_N(∂_x^jw)²dx+C Z

ϕNw² (2.19) To handle the terms in (2.19) having derivatives ∂^j_xw withj = 1, . . . , k−1, we will show that givenε >0 there is a constantC_ε>0 such that forj= 1, . . . , k−1

Z

ϕ⁰_N(∂_x^jw)²≤ε Z

ϕ⁰_N(∂_x^kw)²+Cε

Z

ϕNw². (2.20) In fact, we first prove that

Z

ϕ⁰_N(∂^j_xw)²≤ε Z

ϕ⁰_N(∂_x^j+1w)²+C_ε Z

ϕ_Nw². (2.21) This can be seen by induction: by applying integration by parts, Young’s inequality

|xy| ≤_2ε¹x²+^ε₂y², and the properties ofϕN we can see that (2.21) is valid forj = 1.

If we assume that (2.21) is valid forj−1, then, again integrating by parts and using Young’s inequality,

Z

ϕ⁰_N(∂_x^jw)²= 1 2 Z

ϕ⁽³⁾_N (∂_x^j−1w)²− Z

ϕ⁰_N∂_x^j−1w ∂^j+1_x w

≤C Z

ϕ⁰_N(∂_x^j−1w)²+ 1 2ε

Z

ϕ⁰_N(∂_x^j−1w)²+ε 2 Z

ϕ⁰_N(∂^j+1_x w)². If we apply the induction hypothesis at levelj−1, with _C+1/(2ε)^1/2 instead ofε, then we have

Z

ϕ⁰_N(∂_x^jw)²≤ 1 2 Z

ϕ⁰_N(∂_x^jw)²+C_ε Z

ϕ_Nw²+ε 2

Z

ϕ⁰_N(∂_x^j+1w)², (2.22) which gives (2.21). From a repeated application of (2.21) we obtain (2.20).

Therefore, taking into account that the first term on the right-hand side of (2.19) is negative, we can apply (2.20) withεsufficiently small, to absorb with this negative term the integrals containing (∂_x^jw)²in (2.19). Thus we conclude that

d dt

Z

ϕNw²≤Cε

Z

ϕNw², (2.23)

which, from Gronwall’s inequality and (2.3) implies that Z

ϕNw(t)²dx≤C Z

ϕNw(0)²dx≤C Z

e^βxw(0)²dx , for allt∈[0,1], where C is independent of t∈ [0,1] and N. Since ϕ_N(x) →e^βx as N → ∞, the conclusion of the theorem will follow by applying Fatou’s Lemma on the left-hand side of the former inequality.

In this way, the proof of theorem 2.1 will be complete if we construct a sequence of functions ϕN, satisfying the conditions (2.3) to (2.6). For that we proceed as follows:

Let ˜φ∈C^∞(R) be a nonincreasing function such that ˜φ(x) = 1 forx∈(−∞,0], and ˜φ(x) = 0 forx∈[1,∞). For each N∈NletφN(x)≡φ(x) := ˜φ(x−N). Thus φis supported in (−∞, N+ 1], and (1−φ) in [N,+∞). We define

θ_N(x)≡θ(x) :=φβe^βx+ (1−φ)βe^βN, (2.24) ϕ_N(x)≡ϕ(x) :=

Z x

−∞

θ(x⁰)dx⁰. (2.25)

Let us show thatϕN satisfies the conditions (2.3)–(2.6).

Taking into account the support of (1−φ), we see that 0≤θ(x)≤φβe^βx+ (1− φ)βe^βx = βe^βx. Thus, by integratingϕ⁰ we have that 0 ≤ϕ(x) ≤e^βx. Besides, from the definition of ϕit is clear thatϕ_N(x)→e^βx as N → ∞. Thus ϕsatisfies (2.3).

To prove (2.4) it suffices to observe that for x ≤ N, ϕ(x) ≤ e^βN ≤ C_N(1 + x+)^(k+2)/4, while forx > N,

ϕ(x)≤ Z N+1

−∞

βe^βx0dx⁰+ Z x

N

βe^βNdx⁰≤e^β(N+1)+xβe^βN ≤C_N(1 +x₊)^(k+2)/4, (2.26) sincek≥2. Thus we have (2.4).

We proceed now to prove (2.6). For x ≤ N, ϕ(x) = e^βx = _β¹ϕ⁰(x) ≤ C(1 + x+)ϕ⁰(x). Ifx > N, then, from (2.26), and using the fact thatx≥1, we see that

ϕ(x)≤e^β(N+1)+xβe^βN≤(1

β + 1)e^βxβe^βN. (2.27) On the other hand, forx > N,

xϕ⁰(x) =xθ(x)≥N φβe^βN+x(1−φ)βe^βN. (2.28) Therefore, from (2.27) and (2.28), taking into account the supports ofφand (1−φ) we observe that for x > N + 1, xϕ⁰(x) ≥Cϕ(x), while for N < x < N+ 1, we conclude that

xϕ⁰(x)≥N φβe^βN+N(1−φ)βe^βN =N βe^βN ≥ 1

2(N+1)βe^βN≥ 1

2xβe^βN ≥Cϕ(x), from which (2.6) follows.

Finally, we verify (2.5). We observe that that forj= 1,2. . ., and fixedβ >0,

|ϕ^(j+1)|=|θ^(j)|=|φβ^1+je^βx+

j

X

l=1

c_j,lφ^(l)β^j−lβe^βx−φ^(j)βe^βN|

≤β^jφβe^βx+Cj(1 +β)^j−1(βe^β(N+1)+βe^βN)χ_[N,N+1]

≤β^jθ+C_jβe^βNχ_[N,N+1]

=β^jθ+Cj(βφe^βN+ (1−φ)βe^βN)χ[N,N+1]

≤β^jθ+Cjθ=Cjϕ⁰,

where Cj depends onβ and j but is independent of N. Thus, the first inequality in (2.5) is proved. For the inequality φ⁰ ≤Cφin (2.5) we proceed by integrating the inequalityφ⁰⁰≤Cφ⁰ already established. This completes the proof of Theorem

2.1.

Remark 2.2. For the case of equation (1.1), withp≤k, we can establish a result similar to Theorem 2.1, by making minor modifications and some simplifications in the former proof. In the simple casep≤1, for example for the equation

∂tu+ (−1)^k+1∂_xⁿu=−u∂xu,

it is possible to follow the procedure of the proof of Theorem 2.1 to establish, without the hypothesis of polynomial decay, that the exponential decay att= 0 is preserved fort∈[0,1]. This can be done by takingϕN(x) :=Rx

−∞θN(x⁰)dx⁰ as in (2.25), withθN(x)≡θ(x) :=φβe^βx+ (1−φ)βe^{−β(x−2N)}, instead of the functions θN defined in (2.24). This functions ϕN are bounded and satisfy (2.3) and (2.5) which is enough for this case.

3. Estimates of Carleman type

In this section we obtain boundedness properties of the linear operator (∂t+ (−1)^k+1∂_xⁿ)⁻¹, and its spatial derivatives up to ordern−1, in spaces of the type L^p−L^q with exponential weighte^λx. We keep our exposition simple since we only use values ofpandqin the set{1,2,+∞}.

LetD:=R×[0,1] and, forR∈R, letD_R:={(x, t)|x≥R , t∈[0,1]}. We will denote

k · k_L^p_xL^q

T :=k · k_L^p_xL^q

t(D), k · k_L^p

x≥RL^q_T :=k · k_L^p_xL^q

t(DR), k · k_L^q

TL^p_x:=k · k_L^q

tL^p_x(D), k · k_L^q

TL^p_x≥R :=k · k_L^q

tL^p_x(D_R).

Theorem 3.1. Forn= 2k+ 1withk∈N, letvbe a function inC([0,1];Hⁿ(R))∩

C¹([0,1];L²(R) such thatsuppv(t)⊂[−M, M] for all t ∈[0,1], for someM >0.

Then, forλ >2 we have

ke^λxvkL^∞_TL²_x ≤Cke^λx(|v(0)|+|v(1)|)k²_L(R+Cke^λx(∂t+ (−1)^k+1∂_xⁿ)vk_L1 TL²_x.

(3.1)

n−1

X

j=1

ke^λx∂_x^jvk_L∞

xL²_T ≤Cλⁿ⁻¹k(|Jⁿ(e^λxv(1))|+|Jⁿ(e^λxv(0))|k²_L(R +ke^λx(∂t+ (−1)^k+1∂_xⁿ)vk_L1

xL²_T,

(3.2)

whereC is independent of λ >2 andM, and(J f)b:= (1 +|ξ|²)^1/2fb.

Reasoning formally, suppose thate^λx(∂_t+(−1)^k+1∂_xⁿ)g=h, and denotef =e^λxg and T₀ = [e^λx(∂_t+ (−1)^k+1∂_xⁿ)e^−λx]⁻¹. Then, f = T₀h. Since e^λx∂_xe^−λxf = (∂x−λ)f, we have thate^λx∂_xⁿe^−λxf = (∂x−λ)ⁿf, and thus the multipier operator representingT0 via the Fourier transform is given by

(T₀h)b(ξ, τ) = bh

iτ+ (−1)^k+1(iξ−λ)ⁿ ≡m₀bh. (3.3)

We will writem0 as

m0= −i

τ−(ξ+iλ)ⁿ. (3.4)

Since for a positive integerj,e^λx∂_x^jg= (∂x−λ)^jf, we have that e^λx∂_x^jg= (∂x−λ)^jT0h≡Tjh= [(iξ−λ)^jm0bh]ˇ

=h−i^j+1(ξ+iλ)^j τ−(ξ+iλ)ⁿ bhi

ˇ ≡[m_jbh]ˇ.

(3.5)

Lemma 3.2. Let h∈L¹(R²). Then there is C >0 independent of h andλ > 0 such that

kT0hkL^∞_t L²_x≤Ckhk_L¹

tL²_x. (3.6)

Proof. Leta(ξ) andb(ξ) be the real and imaginary parts of−(ξ+iλ)ⁿ, respectively.

Then

m0= −i

τ+a(ξ) +ib(ξ). We recall that fora∈Randb6= 0

1 τ+a+ib

ˇ^τ(t)

=ce^−iat[e^−btχ_[0,∞](t)χ_(0,∞)(b) +e^btχ_[−∞,0)(t)χ_(−∞,0)(b)] =:G_a,b(t).

(3.7) Since |Ga,b| ≤c, by taking inverse Fourier transform in the variable τ and using convolutions, it follows that fort∈R

|[T₀h(t)]b(ξ)|=| Z ∞

−∞

G_a(ξ),b(ξ)(t−s)h(s)(ξ)d ds| ≤C Z ∞

−∞

|h(s)(ξ)|d ds, for those values of ξ such that b(ξ) 6= 0 (a finite set). In this way, applying Plancherel’s identity and Minkowski’s integral inequality we obtain (3.6).

Lemma 3.3. Let h∈L¹(R²). Then there isC >0, independent of handλ >2, such that

kTjhk_L∞

xL²_t ≤Ckhk_L1

xL²_t forj= 1, . . . , n−1. (3.8) Proof. From (3.4) and (3.5), it follows that

(Tjh)b= (iξ−λ)^jm0bh=−i^j+1(ξ+iλ)^jbh

τ−(ξ+iλ)ⁿ =mjbh. (3.9) Let us denoteθ:= (ξ+iλ)/τ^1/n. Then

m_j = C τ^1−j/n

θ^j

1−θⁿ = C τ^1−j/n

n

X

l=1

cl

θ−r_l, (3.10)

whererl:=al+ibl,l= 1, . . . , n, are then^th-roots of 1, and theclcan be computed by L’Hopital’s rule to obtain that

c_l= lim

θ→rl

(θ−r_l)θ^j

1−θⁿ =− 1 nr^n−j−1_l . Therefore,

mj = C

τ^1−(j+1)/n

n

X

l=1

cl

ξ+iλ−τ^1/nr_l = C τ^1−(j+1)/n

n

X

l=1

cl

ξ−τ^1/na_l+i(λ−τ^1/nb_l).

Taking the inverse Fourier transform with respect to the variableξ, observing that for fixed λ, λ−τ^1/mbl 6= 0 for alll, except for a finite number of values ofτ, and applying (3.7) (withξandxinstead ofτ andt, respectively) we obtain a collection of bounded functionsG₁, . . . , G_lofxandτ such that

[mj(·ξ, τ)]ˇ^ξ(x) = C τ^1−(j+1)/n

n

X

l=1

clGl(x, τ).

If|τ|>1, then it is clear that

|[mj(·, τ)]ˇ^ξ(x)| ≤C, (3.11) withCindependent ofλ,xand|τ|>1. We can use (3.9) to prove that this function is bounded also for|τ| ≤1. To do this we will consider only the casej=n−1 = 2k, the other cases being similar. Let us observe from (3.5) that

m2k− iⁿ ξ+iλ

=

(ξ+iλ)ⁿ⁻¹

τ−(ξ+iλ)ⁿ + 1 ξ+iλ

=

τ

(τ−(ξ+iλ)ⁿ)(ξ+iλ)

≤ 2

|ξ+iλ|ⁿ⁺¹

which belongs to L¹_ξ, since λ >2 and|τ| ≤1. From the Fourier inversion formula it can be seen that|[|ξ+iλ|⁻ⁿ⁻¹]ˇ(x)| ≤C, withC independent of λ >2. Thus, by taking inverse Fourier transform with respect to the variable ξand taking into account that from (3.7) [(ξ+iλ)⁻¹]ˇ^ξ(x) is a bounded function ofx, with bound independent ofλ, we see, together with the estimate already obtained for|τ| ≤1, that (3.11) is valid for allxand all but a finite number of values ofτ.

Hence, we can apply basic properties of convolution and Plancherel’s identity to conclude that

kT2khk_L∞

xL²_t ≤Ckhk_L1 xL²_t,

which concludes the proof of Lemma 3.3.

Proof of Theorem 3.1. We extendv to all t∈Rwith value zero in R−[0,1], and call this extension again v. Forε > 0, we consider a function η :=η_ε ∈ C^∞(Rt) such thatη_ε= 0 inR−[0,1],η_ε= 1 in [ε,1−ε],η⁰ ≥0 in [0, ε],η⁰≤0 in [1−ε,1], andη⁰ ≤η_εifε⁰< ε. Defineg:=η_ε(·_t)v. Then

e^λx(∂_t+ (−1)^k+1∂ⁿ_x)g=e^λxη⁰_εv+η_εe^λx(∂_t+ (−1)^k+1∂_xⁿ)v≡h_1,ε+h_2,ε≡h₁+h₂. Then, from (3.5),

e^λx∂_x^jg=Tjh1+Tjh2 forj= 0, . . . , n−1. (3.12) From Lemma 3.3,

kTjh2k_L∞

xL²_t ≤Ckh2k_L1

xL²_t =Ckηεe^λx(∂t+ (−1)^k+1∂_xⁿ)vk_L1

xL²_t. (3.13) ForT_jh₁, we see from (3.5) that (T_jh₁)b=C(ξ+iλ)^j(T₀h₁)b, and apply the Sobolev embedding fromH¹(R) toL^∞(R), Plancherel’s identity, and Lemma 3.2 to conclude

that

kTjh₁k_L∞

xL²_T ≤CkJ Tjh₁k_L²

TL²_x

≤Ck(1 +|ξ|)(|ξ|^j+λ^j)(T₀h₁)b

xk_L²

TL²_ξ

≤Ck(1 +λ)^j(1 +|ξ|)^j+1(T0h1)b

xk_L2 TL²_ξ

≤C(1 +λ)^jkT₀J^j+1h₁k_L^∞

t L²_x

≤Cλ^jkJ^j+1h₁k_L¹

tL²_x

=Cλ^jkη_ε⁰J^j+1(e^λxv)k_L1 tL²_x

≤Cλⁿ⁻¹kη_ε⁰Jⁿ(e^λxv)k_L1 tL²_x.

(3.14)

Hence, from (3.12), (3.13), and (3.14) we have that kη_εe^λx∂_x^jvk_L∞

xL²_T ≤Cλⁿ⁻¹kη⁰_εJⁿ(e^λxv)k_L1

tL²_x+Ckη_εe^λx(∂_t+ (−1)^k+1∂_xⁿ)vk_L1 xL²_t. We now makeε→0 in this inequality and apply Fatou’s Lemma on the left-hand side and the monotone convergence theorem in the second term of the right-hand side. For the first term on the right-hand side we use the fact that|η⁰_ε|acts as an approximation of the identity on each one of the time intervals (0, ε) and (1−ε,1).

Thus, we obtain (3.2) after adding up inj.

The proof of (3.1) is similar but we use Lemma 3.2 instead of Lemma 3.3. This

completes the proof.

4. Lower estimates

In this section we prove that theL²-norm ofwin a small rectangleQ= [0,1]× [r,1−r], withr∈(0,1) can be bounded by a multiple of theHⁿ⁻¹norm ofwin a distant rectangle [R, R+ 1]×[0,1].

Lemma 4.1. Let φ ∈ C^∞([0,1]) be a function with φ(0) = φ(1) = 0, and for R > 1 and α > 0 define ψ_α(x, t) := ψ(x, t) := α(_R^x +φ(t))², x ∈ R, t ∈ [0,1].

For n= 2k+ 1, suppose that g ∈C([0,1];Hⁿ(R))∩C¹([0,1];L²(R))is such that g(0) =g(1) = 0 and the support of g is contained in the set

A_1,5:={(x, t) :t∈[0,1], 1≤ x

R +φ(t)≤5}.

Then, there are constantsC=C(n)>0 andC=C(n,kφ⁰kL^∞,kφ⁰⁰kL^∞)>1 such that

n−1

X

j=0

α^n−j−12

R^n−j ke^ψα∂_x^jgk ≤Cke^ψα(∂_t+(−1)^k+1∂_xⁿ)gk for all α > CR^n/(n−1), (4.1) wherek · k:=k · kL²(D).

Proof. Let f := e^ψg and observe that e^ψ∂_x^jg = e^ψ∂_x^je^−ψf = (∂_x−ψ_x)^jf, and e^ψ∂_tg= (∂_t−ψ_t)f. Therefore, if we denote

T :=e^ψ(∂_t+ (−1)^k+1∂_xⁿ)e^−ψ = (∂_t−ψ_t) + (−1)^k+1(∂_x−ψ_x)ⁿ, then, to prove the inequality in (4.1) we must prove that

n−1

X

j=0

α^n−j−12

R^n−j k(∂_x−ψ_x)^jfk ≤CkT fk. (4.2)

Let B := −ψx = −^2α_Rϕ, where ϕ(x, t) := _R^x +φ(t). We will study the operator (∂x−ψx)ⁿ= (∂x+B)ⁿ in the following manner.

Each one of the terms in the expansion of this operator is of the formT1. . . Tn, where eachT_iis either∂_xorB. Formandlwithm+l=nwe will denote by [m, l]

the sum of the termsT₁. . . T_nin this expansion withT_i=∂_xformvalues ofi, and T_i=Bforlvalues ofi. The number of terms of [m, l] in the binomial expansion of (∂_x+B)ⁿ is then given by ⁿ_l

= _mⁿ

:=n!/m!l!. Applying integration by parts in D, for the class of functions satisfying the hypotheses given forg, we observe that [m, l] is a symmetric operator ifmis even and is an antisymmetric operator ifmis odd.

In this way, we write

(−1)^k+1T = (−1)^k+1(∂t−ψt) + X

0≤m≤n l+m=n

[m, l] :=S+A, (4.3)

where

S= [n−1,1] + [n−3,3] +· · ·+ [2, n−2] + [0, n] + (−1)^kψt≡S1+ (−1)^kψt, A= [n,0] + [n−2,2] +· · ·+ [3, n−3] + [1, n−1]−(−1)^k∂_t≡A₁−(−1)^k∂_t.

(4.4) Let us denote byh·,·ithe inner product in the (real) spaceL²(D). Then

kT fk²=h(S+A)f,(S+A)fi=kSfk²+kAfk²+ 2hSf, Afi ≥2hSf, Afi. (4.5) Now,

hSf, Afi=hS1f, A₁fi −(−1)^khS1f, ∂_tfi+ (−1)^khψtf, A₁fi − hψtf, ∂_tfi. (4.6) We will now estimate each one of the four terms on the right hand side of (4.6).

To estimate hS1f, A1fi we observe that this product is a sum of terms of the form h[m, n−m]f,[r, n−r]fi, with meven and rodd, saym= 2k1,r= 2k2+ 1, k1, k2∈ {0, . . . , k}. Using the fact thatBxx=ψxxx = 0, we can apply integration by parts to obtain

h[m, n−m]f,[r, n−r]fi=

k₁+k₂

X

j=0

Z

D

Pk₁,k₂,j(B, Bx)(∂_x^jf)² (4.7) where Pk1,k2,j(B, Bx) = ck1,k2,jB(n−m)+(n−r)−νB^ν_x, with ν+ 2j = m+r. Since B=−ψx=−^2α_Rϕ,Bx=−ψxx=−_R^2α2, we have that

P_k1,k2,j(B, B_x) =−c_k1,k2,j(2α)^2n−m−rϕ^{2n−m−r−ν} R^{2n−m−r+ν}

=−(2α)^{2n−2k1−2k2−1}c_k1,k2,jϕ^{2n−4k1−4k2+2j−2} R^2n−2j

≡α^{2n−2k1−2k2−1}Qk₁,k₂,j

(4.8)

Therefore,

hS1f, A1fi=

k

X

k1,k2=0

h[2k1, n−2k1]f,[2k2+ 1, n−2k2−1]fi

=

k

X

k₁,k₂=0

α^{2n−2k1−2k2−1}

k₁+k₂

X

j=0

Z

Qk₁,k₂,j(∂_x^jf)²

=

2k

X

j=0

X

k₁+k₂≥j 0≤k₁,k₂≤k

α^{2n−2k1−2k2−1} Z

Qk₁,k₂,j(∂_x^jf)². (4.9)

For each j in the former expression we will separate the term having the highest power ofα. Thus we write

hS1f, A1fi=

2k

X

j=0

α^2n−2j−1 X

k₁+k₂=j 0≤k1,k2≤k

Z

Qk₁,k₂,j(∂_x^jf)²

+

2k

X

j=0

X

k1+k2>j 0≤k1,k2≤k

α^{2n−2k1−2k2−1} Z

Qk₁,k₂,j(∂_x^jf)²

≡

2k

X

j=0

I_j+

2k

X

j=0

II_j.

(4.10)

We will concentrate upon the termsIjand will refer to the termsIIjas lower order terms (l.o.t.).

For eachjwe will now compute the termIj. Ifm≤nandl=n−m, then [m, l]

is a sum of operators of the formT =T1T2. . . Tn where Ti =∂x for m indices i, andT_i=B for the remaininglindicesi. We apply the product rule for derivatives to expandT and consider the terms in its expansion containing the derivatives of highest order: ∂_x^m and∂^m−1_x . This leads to (see the illustration below)

T =T₁. . . T_n=B^l∂_x^m+ [B^r1(∂_xB^l−r1) +B^r2(∂_xB^l−r2) +. . .

+B^rm(∂xB^l−rm)]∂_x^m−1+ l.d.t., (4.11) wherer₁, r₂, . . . r_ndepend upon the position of themoperators∂_xin the expression ofT, and the notation l.d.t. stands for “lower derivative terms”.

To illustrate this, let us take for example the case withn= 9, m= 3,l= 6, and consider the operatorT =B∂xB∂xBB∂xBB. Then,

T =B⁶∂_x³+B⁴(∂_xB²)∂²_x+B²(∂_xB⁴)∂_x²+B(∂_xB⁵)∂_x²+ l.d.t.

But, the operatorT^(∗):=T_n. . . T₂T₁, is also present in the expansion of [m, l], and T^(∗)=Tn. . . T1

=B^l∂_x^m+ [B^l−r1(∂_xB^r1) +B^l−r2(∂_xB^r2) +. . . +B^l−rm(∂xB^rm)]∂_x^m−1+ l.d.t.

(4.12)

Therefore, ifT 6=T^∗ we have from (4.11) and (4.12) that

T+T^(∗)= 2B^l∂_x^m+m(∂xB^l)∂_x^m−1+ l.d.t. (4.13) It can be seen that ifT =T^(∗), then the same expression is valid. Since there are

n m

terms in the expansion of [m, l] we conclude from (4.13) that [m, l] =

n m

[B^l∂_x^m+ml

2 B^l−1B_x∂_x^m−1] + l.d.t. (4.14) In this way, for m = 2k1, r = 2k2+ 1, j =k1+k2 = ¹₂(m+r−1), l =n−m, s=n−r, we apply integration by parts to observe that

h[m, l]f,[r, s]fi