On the radius of spatial analyticity for defocusing nonlinear Schr\"odinger equations

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schr\"odinger equations $iu_t + \Delta u = |u|^{p-1}u$, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ cannot decay faster than $1/|t|$ as $|t|\rightarrow\infty$. This extends the previous work of Tesfahun \cite{Te} for the cubic case $p=3$ to the cases where $p$ is any odd integer greater than $3$.


1.2)
While the well-posedness in Sobolev spaces is well-understood, much less is known about spatial analyticity of the solutions to the above Cauchy problem. Our attention in this paper will be focused on the situation where we consider a real-analytic initial data with uniform radius of analyticity σ 0 > 0, so there is a holomorphic extension to a complex strip S σ0 = {x + iy : x, y ∈ R d , |y| < σ 0 }.
The question is then whether this property may be continued analytically to a complex strip S σ(t) for all later times t, but with a possibly smaller and shrinking radius of analyticity σ(t) > 0. This type of question was first introduced by Kato and Masuda [10], and Bona and Grujić [2] gave an explicit lower bound of the radius σ(t) for the Korteweg-de Vries equation. In fact, it is shown that σ(t) ≥ e −ct 2 for large t which shows that the radius can decay to zero at most at an exponential rate. Later, this exponential decay was improved to an algebraic lower bound, ct −1/2 , by Bona, Grujić and Kalisch [3]. See [15,19,9] for further refinements. We also refer the reader to [4,18,13,16,14] for other nonlinear dispersive equations like Schrödinger, Klein-Gordon and Dirac-Klein-Gordon equations.
In the present paper we shall work on the Cauchy problem (1.1), motivated by an earlier work on the defocusing cubic nonlinear Schrödinger equation, the case p = 3 in (1.1), by Tesfahun [18] who gave a lower bound of the radius, σ(t) ≥ ct −1 , for large t. It will turn out that it is still possible for the other cases p > 3 to have the same lower bound.
A nice choice of analytic function space suitable to study spatial analyticity of solution is the Gevrey space G σ,s (R d ), σ ≥ 0, s ∈ R, with the norm where ∇ = 1 + |∇| 2 . In fact, according to the Paley-Wiener theorem 1 (see e.g. [11], p. 209), a function f belongs to G σ,s with σ > 0 if and only if it is the restriction to the real line of a function F which is holomorphic in the strip S σ = {x + iy : x, y ∈ R d , |y| < σ} and satisfies sup |y|<σ F (x+ iy) H s x < ∞. In other words, every function in G σ,s with σ > 0 has an analytic extension to the strip S σ . This is one of the key properties of the Gevrey space and shows that the following result gives an algebraic lower bound on the radius of analyticity σ(t) of the solution to (1.1) as the time t tends to infinity. Theorem 1.1. Let d = 1, 2. Let u be the global C ∞ solution of (1.1) with any odd integer p > 3 and u 0 ∈ G σ0,s (R d ) for some σ 0 > 0 and s ∈ R. Then, for all t ∈ R with σ(t) ≥ c|t| −1 as |t| → ∞. Here, c > 0 is a constant depending on u 0 G σ 0 ,s (R d ) , σ 0 , s and p.
Only when d = 1, 2 does the existing well-posedness theory in H s (see (1.2)) guarantee the existence of the global C ∞ solution in the theorem, given initial data u 0 ∈ G σ0,s for all σ 0 > 0 and s ∈ R. Indeed, observe first that G 0,s coincides with the Sobolev space H s and the embeddings for all 0 ≤ σ ′ < σ and s, s ′ ∈ R. As a consequence of this embedding with σ ′ = 0 and the existing well-posedness theory in H s ′ , the Cauchy problem (1.1) has a unique smooth solution for all time, given initial data u 0 ∈ G σ0,s for all σ 0 > 0 and s ∈ R. The outline of this paper is as follows: In Section 2 we present some preliminaries which will be used for the proof of Theorem 1.1. In Section 3 we obtain some multilinear estimates in Gevrey-Bourgain spaces. By making use of a contraction argument involving these estimates, we prove that in a short time interval 0 ≤ t ≤ δ with δ > 0 depending on the norm of the initial data, the radius of analyticity remains strictly positive. Next, we prove an approximate conservation law, although the conservation of G σ0,1 -norm of the solution does not hold exactly, in order to control the growth of the solution in the time interval [0, δ], measured in the data norm G σ0,1 . Section 4 is devoted to the proofs of such a local result and an approximate conservation law. In the final section, Section 5, we finish the proof of Theorem 1.1 by iterating the local result based on the conservation law.
Throughout this paper, the letter C stands for a positive constant which may be different at each occurrence.

Preliminaries
In this section we introduce some function spaces and linear estimates which will be used for the proof of Theorem 1.1 in later sections.
For s, b ∈ R, we use X s,b = X s,b (R 1+d ) to denote the Bourgain space defined by the norm The restriction of the Bourgain space, denoted X s,b δ , to a time slab (0, δ) × R d is a Banach space when equipped with the norm We also need to introduce the Gevrey-Bourgain space X σ,s,b = X σ,s,b (R 1+d ) defined by the norm f X σ,s,b = e σ|D| f X s,b .
Its restriction X σ,s,b δ to a time slab (0, δ) × R d is defined in a similar way as above, and when σ = 0 it coincides with the Bourgain space X s,b .
The X σ,s,b -estimates in Lemmas 2.1, 2.2 and 2.3 follow easily by substitution f → e σ|D| f using the properties of X s,b -spaces and the restrictions thereof. When σ = 0, the proofs of Lemmas 2.1 and 2.2 can be found in Section 2.6 of [17], and Lemma 2.3 follows by the argument used for Lemma 3.1 of [6].
where C > 0 is a constant depending only on b.
where χ I (t) is the characteristic function of I, and the constant C > 0 depends only on b.
for Schrödinger-admissible pair (q, r), i.e., Next, consider the linear Cauchy problem for the Schrödinger equation By Duhamel's principle the solution can be then written as where the Fourier multiplier e it∆ with symbol e −it|ξ| 2 is given by Then the following is the standard energy estimate in X s,b δ -spaces (see e.g. [12,18]). Lemma 2.5. Let σ ≥ 0, s ∈ R, 1/2 < b ≤ 1 and 0 < δ ≤ 1. Then we have Here the constant C > 0 depends only on b.

Multilinear estimates in Gevrey-Bourgain spaces
In this section we obtain a couple of multilinear estimates in Gevrey-Bourgain spaces, Proposition 3.1, which will play a key role in obtaining the local well-posedness (Theorem 4.1) and almost conservation law (Theorem 4.2) in the next section. With the aid of Proposition 3.1, we also deduce two more estimates which are also important in obtaining the almost conservation law.
From now on, p will always denote an odd integer greater than 3. For the sake of brevity, we also let (3.1) where U j denotes u j or u j .
Proof. First we prove (3.2). By duality, it is enough to prove By Hölder's inequailty, the Sobolev embedding and then (2.1), we get as desired.
Next, we prove the second estimate (3.3). By duality we first observe that Again by Hölder's inequailty, the Sobolev embedding and (2.1), we get as desired.
Finally, we prove (3.4). First we consider the case σ = 0. We note that Hence it is enough to show that and by symmetry which are direct consequences of (3.2). For instance, Note here that s 0 < 1. Now we consider the case σ > 0. Without loss of generality, we may assume U k = u k for each k ∈ {1, 2, · · · , p}. Let v j = e σ|D| u j . Then (3.4) reduces to We first write the space-time Fourier transform of where we used * to denote the conditions τ = p j=1 τ j and ξ = p j=1 ξ j . In fact, observe that A similar argument can be made for multiplications of four or more terms.
Let w j (τ, ξ) = | v j (τ, ξ)|. Then using the fact that e σ(|ξ|− p j=1 |ξj |) ≤ 1, which follows from the triangle inequality, we obtain Applying the above case σ = 0 to the last expression, we get p j=1 w j Now we have the desired result.
With the aid of Proposition 3.1, we deduce two more estimates, which, along with the function f defined here, will play a crucial role in obtaining the almost conservation law, Theorem 4.2, in the next section. For all b > 1/2 we then have

Local well-posedness and almost conservation law
In this section we shall establish the local well-posedness in Subsection 4.1 and the almost conservation law in Subsection 4.2 by making use of the multilinear estimates obtained in the previous section.

4.1.
Local well-posedness. Based on Picard's iteration in the X σ,1,b δ -space and Lemma 2.1, we establish the following local well-posedness in G σ,1 , with a lifespan δ > 0. Equally the radius of analyticity remains strictly positive in a short time interval 0 ≤ t ≤ δ, where δ > 0 depends on the norm of the initial data. We use the notation A± = A ± ǫ for sufficiently small ǫ > 0.
for some constant c 0 > 0 depending only on p, and the solution u satisfies Proof. Fix σ > 0 and u 0 ∈ G σ,1 . By Lemma 2.1 we shall employ an iteration argument in the space X σ,1,b δ . Let u (n) ∞ n=0 be the sequence defined by and iu (n) t + ∆u (n) = |u (n−1) | p−1 u (n−1) , for n ∈ Z + . Applying (2.2), we first write By Lemma 2.5 we have and Lemmas 2.5 and 2.2 combined imply Applying (3.4) to the second term in the right-hand side of (4.3), we obtain By induction together with (4.2) and (4.4), it follows that for all n ≥ 0 with a choice of Furthermore, Lemma 2.5 and Lemma 2.2 with the same choice of δ yield

applying (3.4), we now get
By (4.5) and the choice of δ, this implies which guarantees the convergence of the sequence u (n) ∞ n=0 to a solution u with the bound (4.5). Now assume that u and v are solutions to the Cauchy problem (1.1) for initial data u 0 and v 0 , respectively. Then similarly as above, again with the same choice of δ and for any δ ′ such that 0 < δ ′ < δ, we have is sufficiently small, which proves the continuous dependence of the solution on the initial data.
Lastly it remains to show the uniqueness of solutions. Assume u, v ∈ C t G σ,1 are solutions to (1.1) for the same initial data u 0 and let w = u − v. Then w satisfies iw t + ∆w = |u| p−1 u − |v| p−1 v. Multiplying both sides byw and taking imaginary parts thereon, we have Re(ww t ) + Im(w∆w) = Im(|u| p−1 uw − |v| p−1 vw).
By Grönwall's inequality, we now conclude that w = 0.
which are the conservation of mass and energy, respectively. Define a quantity: Then one can easily see This quantity is approximately conservative in the sense that, although it fails to hold for σ > 0, the discrepancy between both sides is bounded as well as the quantity reduces to the conservation of mass and energy in the limit σ → 0. This approximate conservation will allow us (see Section 5) to repeat the local result on successive shorttime intervals to reach any target time T > 0, by adjusting the strip width parameter σ according to the size of T .
Note here that ∂ t |∇v| 2 = 2Re(∇v · ∇v t ), ∇v · ∆∇v = ∇ · (∇v∆v) − |∆v| 2 and using (4.8). It then follows that Here we also used the fact that Similarly as in Step 1, integrating (4.10) in space yields and then integrating in time over the interval [0, δ ′ ] we have Now we estimate the second integral on the right-hand side in the above. By Hölder's inequality, we obtain R 1+d Therefore, by (4.11), (4.14) and (4.15), we get (4.16) Step 3. Adding (4.9) and (4.16) yields Here we used the fact that s 1 ≤ s 0 ≤ 1. This shows the desired estimate (4.7).

Proof of Theorem 1.1
Following the argument in [18], we first consider the case s = 1. By the embedding (1.3), the general case s ∈ R will reduce to s = 1 as shown in the end of this section.