Lower Bound on the Radius of Analyticity of Solutions for Fourth-Order Nonlinear Schrödinger Equations on R 2

Abstract

We present lower bounds for the uniform radius of spatial analyticity of solutions to the fourth-order nonlinear Schrödinger equations. The main ingredients in the presentation of our results are a method of approximate conservation law in modified Gevrey spaces, Strichartz-type estimates, and Sobolev embedding.

Keywords

fourth-order nonlinear Schrödinger equation modified Gevrey spaces approximate conservation law Strichartz estimate radius of spatial analytic

1. Introduction

Consider the Cauchy problem for the fourth-order nonlinear Schrödinger equations (p-4NLS)

{\begin{cases} i \partial_{t} u + Δ^{2} u - Δ u + | u |^{p - 1} u = 0, (x, t) \in R^{2} \times R, \\ u (x, 0) = g (x), \end{cases}

(p-4NLS)

where

p = 2 k + 1

for

k = 1, 2, \dots

, and

u

is a complex-valued function. The fourth-order Schrödinger equation has been introduced by Karpman (1996) and Karpman and Shagalov (2000) to take into account the effect of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. The mass and energy

\begin{aligned} M (t) & = \frac{1}{2} \int_{R^{2}} | u |^{2} d x, \\ E (t) & = \frac{1}{2} \int_{R^{2}} (| u |^{2} + | Δ u |^{2} + | ▽ u |^{2} + \frac{2}{p + 1} | u |^{p + 1}) d x, \end{aligned}

are conserved by the flow of (p-4NLS), that is,

M (t) = M (0) and E (t) = E (0) for all t .

The well-posedness of the Cauchy problem (p-4NLS) with data in the Sobolev spaces

H^{s} (R^{2})

was studied extensively. In particular, Guo (2010) established a global well-posedness result in

H^{s} (R^{2})

for

s > 1 + \frac{[(p - 1) / 2] d - 9 + \sqrt{{2 p - [(p - 1) / 2] d + 5}^{2} + 16}}{2 p - 2},

such that

4 < [(p - 1) / 2] d < 2 p .

In a paper, Başakoğlu et al. (2023) on arxiv, the authors improved the result of Guo for initial data with Sobolev regularity

s > 2 - [3 / (2 p - 2)]

. In fact, the result in Başakoğlu et al. (2023) covers the cubic and quintic 4NLS equations. Our interest here is to study the asymptotic lower bound of spatial analyticity for solutions

u (x, t)

to the Cauchy problem (p-4NLS). For a given real analytic initial data

g (x)

that has an analytic extension to a complex strip of width

2 σ_{0}

, we need to check whether the solution is analytic or not in a complex strip of width

2 σ (t)

for all time

t \in R

. We also estimate the lower bound of the radius of spacial analyticity

σ (t)

at each time

t

. A class of analytic functions space suitable to the study of the spatial analyticity of the solution is analytic Gevrey space

G^{σ, s} := G^{σ, s} (R^{2})

, which was introduced by Foias and Temam (1989). This space is defined by the norm

{‖ f ‖}_{G^{σ, s}} = {‖ e^{σ | ξ |} ⟨ ξ ⟩^{s} \hat{f} (ξ) ‖}_{L_{ξ}^{2} (R^{2})} (σ \geq 0),

where

⟨ ξ ⟩ = \sqrt{1 + ξ^{2}}

and

\hat{f} (ξ)

denotes the spatial Fourier transform of

f (x)

given by

\hat{f} (ξ) = \int_{R^{2}} e^{- i x ξ} f (x) d x .

For

σ = 0

, the space

G^{σ, s} (R^{2})

coincides with the Sobolev space

H^{s} := H^{s} (R^{2})

endowed with the norm

‖ f ‖_{H^{s}} = ‖ ⟨ ξ ⟩^{s} \hat{f} (ξ) ‖_{L_{ξ}^{2} (R^{2})},

while for

σ > 0

, any function in

G^{σ, s} (R^{2})

has a radius of analyticity of at least

σ

at each point

x \in R^{2}

. Of course, this fact is contained in the Paley–Wiener Theorem, whose proof can be found in Katznelson (1976) for

s = 0

and

d = 1

, which applies also for

s \in R

and

d = 2

with some modification.

Paley--Wiener Theorem

Let $σ > 0, s \in R$ . Then, a function $f \in G^{σ, s}$ 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 + i y : x, y \in R^{2} and | y | < σ},

and satisfies the bound

sup_{| y | < σ} | | F (x + i y) | |_{H_{x}^{s}} < \infty .

Dufera et al. (2022) introduced the modified Gevrey space $H^{σ, s} := H^{σ, s} (R^{2})$ from the Gevrey space $G^{σ, s} (R^{2})$ by replacing the exponential weight $e^{σ | ξ |}$ with the hyperbolic weight $\cosh (σ | ξ |)$ , which is currently used by mathematicians to obtain a better lower bound on the radius of spatial analyticity for solutions to nonlinear dispersive partial differential equations (PDEs; see, e.g., Esfahani & Tesfahun, 2024; Getachew & Belayneh, 2024; Getachew et al., 2024; Mebrate et al., 2024; Zhang et al., 2024, and the references therein). For further information on the radius of spatial analyticity for different classes of nonlinear dispersive PDEs, we refer to the readers in Tesfahun (2017, 2019a, 2019b), Selberg and da Silva (2015), Selberg and Tesfahun (2015), Figueira and Panthee (2024), and the references therein. In fact, the modified Gevrey space is defined via the norm

{‖ f ‖}_{H^{σ, s}} = {‖ \cosh (σ | ξ |) ⟨ ξ ⟩^{s} \hat{f} (ξ) ‖}_{L_{ξ}^{2} (R^{2})} (σ \geq 0) .

The two weights are equivalent in the sense that

\frac{1}{2} e^{σ | ξ |} \leq \cosh (σ | ξ |) \leq e^{σ | ξ |},

As a consequence of this, the

‖ \cdot ‖_{G^{σ, s} (R^{2})}

and

‖ \cdot ‖_{H^{σ, s} (R^{2})}

-norms are equivalent. That is

‖ f ‖_{G^{σ, s} (R^{2})} \sim ‖ f ‖_{H^{σ, s} (R^{2})},

(1)

and hence the statement of Paley–Wiener Theorem still holds for functions in

H^{σ, s} (R^{2})

. The reason for considering the

H^{σ, s} (R^{2})

-spaces is that

\cosh (σ | ξ |)

satisfies the estimate

\cosh (σ | ξ |) - 1 \leq (σ | ξ |)^{2 β} \cosh (σ | ξ |), β \in [0, 1],

(2)

which follows from

\cosh x - 1 \leq \cosh x and \cosh x - 1 \leq x^{2} \cosh x, x \in R^{2} .

In view of (2), an application of approximate conservation law in the modified Gevrey spaces can be used to provide a decay rate of order

| t |^{- 1 / 2 β}

for some

β \in (0, 1]

The modified Gevrey spaces satisfy the following embedding property: For all $0 \leq σ^{'} < σ$ and $s, s^{'} \in R$ , we have

H^{σ, s} \subset H^{σ^{'}, s^{'}}, i.e., ‖ f ‖_{H^{σ^{'}, s^{'}}} \leq C ‖ f ‖_{H^{σ, s}} .

In particular,

H^{σ, s} \subset H^{s^{'}}, i.e., ‖ f ‖_{H^{s^{'}}} \leq C ‖ f ‖_{H^{σ, s}} .

(3)

As the consequence of the property (3) and the existing well-posedness theory in $H^{2} (R^{2})$ , the Cauchy problem (p-4NLS) with initial data $g \in H^{σ_{0}, 2} (R^{2})$ for all $σ_{0} > 0$ has a unique and global in time solution.

Theorem 1.1

Let $g \in H^{σ_{0}, 2} (R^{2})$ for $σ_{0} > 0$ . Then the global solution $u$ of (p-4NLS) satisfies

u (t) \in H^{σ (t), 2} (R^{2}), for all t \in R,

with the radius of spatial analyticity

σ (t)

satisfying an asymptotic lower bound

σ (t) \geq c | t |^{- \frac{1}{2}} as | t | \to \infty,

(4)

where

c > 0

is a constant depending on the initial data norm.

By time-reversal symmetry of (p-4NLS), we may from now on restrict ourselves to positive times $t \geq 0$ . The first step in the proof of Theorem 1.1 is to show that in a short time interval $0 \leq t \leq T$ , the radius of analyticity remains strictly positive, where $T > 0$ depends on the initial data norm. This is proved by a contraction mapping argument involving energy estimates, Hölder’s inequality, Sobolev embedding, and Strichartz-type estimates, which will be given in the next section. The next step is to improve the control on the growth of the solution in the time interval $[0, T]$ , measured in the data norm $H^{σ_{0}, 2}$ . To achieve this, we show that, although the conservation of $H^{σ_{0}, 2}$ -norm of solution does not hold exactly, it does hold in an approximate sense. This approximate conservation law will allow us to iterate the local result and obtain Theorem 1.1.

Notation: Throughout this paper, we use $C$ to denote various constants whose values do not affect the result of our work and may vary from one line to the next line. For any positive quantities $a$ and $b$ , we use $a ≲ b$ if $a \leq C b$ , and $a \sim b$ if $a ≲ b ≲ a$ . Moreover, we write $a +$ if there is a small enough $ϵ > 0$ such that $a + := a + ϵ$ .

2. Preliminaries

In this section, we discuss the function spaces, linear estimates, and bilinear estimates, which are utilized in the sequel.

2.1. Function Spaces

The Bourgain space $X^{s, b} := X^{s, b} (R^{1 + 2})$ associated with (p-4NLS) is defined to be the completion of the Schwartz space $S_{x, t} (R^{1 + 2})$ with respect to the norm

‖ u ‖_{X^{s, b}} = ‖ ⟨ ξ ⟩^{s} ⟨ τ - ϕ (ξ) ⟩^{b} \tilde{u} (τ, ξ) ‖_{L_{τ, ξ}^{2} (R^{1 + 2})},

and the restriction to time slab

(0, T) \times R^{2}

of the Bourgain space, denoted

X_{T}^{s, b}

, is a Banach space when equipped with the norm

‖ u ‖_{X_{T}^{s, b}} = inf {‖ v ‖_{X^{s, b}} : v = u on (0, T) \times R^{2}} .

Here $s, b \in R$ , $ϕ (ξ) := ξ^{4} + ξ^{2}$ , and $\tilde{u} (ξ, τ) := F_{x, t} [u] (ξ, τ)$ denote the space–time Fourier transform of $u (x, t)$ given by

\tilde{u} (ξ, τ) = \int_{R^{1 + 2}} e^{- i (t τ + x ξ)} u (x, t) d x d t .

For $s > 1$ and $2 \leq r \leq \infty$ , from the spatial Sobolev embedding, we have

X^{s, \frac{1}{2} +} \subseteq C_{t} H_{x}^{s} \subseteq L_{t}^{\infty} H_{x}^{s} \subseteq L_{t}^{\infty} L_{x}^{r} .

Here, we used the notations

L_{t}^{\infty} H_{x}^{s} := L_{t}^{\infty} (R; H_{x}^{s} (R^{2}))

and

C_{t} H_{x}^{s} := C (R; H_{x}^{s} (R^{2}))

. In particular, for

T > 0

, we have

| | u | |_{L_{T}^{\infty} L_{x}^{\infty}} ≲ | | u | |_{X_{T}^{1 +, \frac{1}{2} +}},

(5)

where

L_{T}^{\infty} L_{x}^{\infty} := L_{t}^{\infty} ([0, T]; L_{x}^{\infty} (R^{2}))

We also need the Grevey–Bourgain space denoted $X^{σ, s, b} \dot{=} X^{σ, s, b} (R^{1 + 2})$ defined by the norm

‖ u ‖_{X^{σ, s, b}} = {‖ e^{σ | D |} u ‖}_{X^{s, b}},

(6)

where

D = - i ▽

with Fourier symbol

ξ

. In the case

σ = 0

, the Gevrey–Bourgain space coincides with the Bourgain space. Moreover, the restriction to time slab

(0, T) \times R^{2}

of the Gevrey–Bourgain spaces is denoted by

X_{T}^{σ, s, b}

, which is a Banach space equipped with the norm

‖ u ‖_{X_{T}^{σ, s, b}} = {‖ e^{σ | D |} u ‖}_{X_{T}^{s, b}} .

2.2. Space–Time Estimates

Consider the linear Cauchy problem, for given $F (x, t)$ and $g (x)$

{\begin{cases} i \partial_{t} u + Δ^{2} u - Δ u = F, \\ u (x, 0) = g (x), \end{cases}

whose solution is given by Duhamel’s integral formula

u (t) = W (t) g + i \int_{0}^{t} W (t - t^{'}) F (t^{'}) d t^{'},

(7)

where

W (t) g

is the free wave defined via its Fourier transform as

\hat{W (t) g} = e^{i t (- ξ^{4} + ξ^{2})} \hat{g}

. Then, the free wave

W (t) g

satisfies the following space–time estimates.

Lemma 2.1

(Cui & Guo 2007, Theorem 2.2). Let $0 \leq α \leq 1$ , $[2 / (1 - α)] \leq r \leq \infty$ , and $2 \leq q \leq \infty$ such that

\frac{4}{q} = 2 (\frac{1}{2} - \frac{1}{r}) + α .

Then, for any

T > 0

, and

g \in L_{x}^{2}

, we have

\begin{aligned} {‖ | D |^{α} W (t) g ‖}_{L_{t}^{q} L_{x}^{r}} \leq C ‖ g ‖_{L_{x}^{2}}, \end{aligned}

(8)

\begin{aligned} {‖ ⟨ D ⟩^{α} W (t) g ‖}_{L_{t}^{q} L_{x}^{r}} \leq C ‖ g ‖_{L_{x}^{2}} . \end{aligned}

(9)

Here, we used the notation

L_{t}^{q} L_{x}^{r} := L_{t}^{q} L_{x}^{r} ([0, T]; R^{2})

L_{x}^{2} = L_{x}^{2} (R^{2})

, and the constant

C > 0

depends on

q, r, T,

and

α

From Lemma 2.1 and a standard transference principle, it is standard to deduce the following space–time estimates:

\begin{aligned} {‖ | D |^{α} u ‖}_{L_{t}^{q} L_{x}^{r}} \leq C ‖ u ‖_{X^{0, b}}, \end{aligned}

(10)

\begin{aligned} {‖ ⟨ D ⟩^{α} u ‖}_{L_{t}^{q} L_{x}^{r}} \leq C ‖ u ‖_{X^{0, b}}, \end{aligned}

(11)

for

b > 1 / 2

and

T, q, r, C

as in Lemma 2.1. In particular, for the biharmonic admissible pair

(q, r)

satisfying

4 / q = 2 [(1 / 2) - (1 / r)]

, we have

{‖ u ‖}_{L_{t}^{q} L_{x}^{r}} \leq C ‖ u ‖_{X^{0, b}},

(12)

The next four lemmas are key properties of $X^{σ, s, b}$ -spaces. Proofs of these lemmas can be found in Tao (2006) for $σ = 0$ , while for $σ > 0$ , the properties of $X^{s, b}$ and its restrictions carry over to $X^{σ, s, b}$ because of the substitution $u \to e^{σ | D |} u$ .

Lemma 2.2

Let $σ \geq 0, s \in R$ , and $b > 1 / 2$ . We have

sup_{t \in R} ‖ u ‖_{G^{σ, s} (R^{2})} \leq C ‖ u ‖_{X^{σ, s, b}},

where

C > 0

is a constant depending only on

b

Lemma 2.3

Let $σ \geq 0, s \in R, - 1 / 2 < b < 1 / 2$ , and $0 < T \leq 1$ . Then

{‖ χ_{(0, T)} u ‖}_{X^{σ, s, b}} \leq C ‖ u ‖_{X_{T}^{σ, s, b}},

where

χ_{(0, T)}

is the characteristic function of

[0, T]

and

C > 0

is a constant depends only on

b

Lemma 2.4

Let $σ \geq 0$ , $s \in R$ , $1 / 2 < b \leq 1$ , and $0 < T \leq 1$ . Then, for all $g \in G^{σ, s}$ and $F \in X_{T}^{σ, s, b - 1}$ , we have the estimates

\begin{aligned} {‖ W (t) g ‖}_{X_{T}^{σ, s, b}} & ≲ {‖ g ‖}_{H^{σ, s}} \\ {‖ \int_{0}^{t} W (t - t^{'}) F (t^{'}) d t^{'} ‖}_{X_{T}^{σ, s, b}} & ≲ {‖ F ‖}_{X_{T}^{σ, s, b - 1}} . \end{aligned}

3. Local Well-Posedness Result

In this section, we use the contraction mapping principle, and one-dimensional Sobolev embedding to prove local well-posedness of the Cauchy problem (p-4NLS) in $H^{σ, 2}$ for $σ > 0$ .

Theorem 3.1
Let $P \geq 3$ be an odd integer and $σ > 0$ . Given $g \in H^{σ, 2} (R^{2})$ , then, there exists a time $T > 0$ and a unique solution $u (t) \in C ([0, T]; H^{σ, 2} (R^{2}))$ of the Cauchy problem (p-4NLS), where the existence time is given by
$T \sim ‖ u_{0} ‖_{H^{σ, 2} (R^{2})}^{- (p - 1)} .$
(13)
Moreover, the map from data to solution is continuous, and for $1 / 2 < b < 1$ the solution $u$ satisfies the bound
$‖ u ‖_{X_{T}^{σ, 2, b}} \leq C ‖ u_{0} ‖_{H^{σ, 2} (R^{2})},$
(14)
where $C > 0$ is a constant depending only on $b$ .

The following lemma helps us to prove Theorem 3.1.

By applying $\cosh (σ | D |)$ to (7) and taking the $H^{2}$ norm on both sides for $F (t^{'}) = u (t^{'}) |^{p - 1} u (t^{'})$ , we obtain the energy inequality
$sup_{t \in [0, T]} | | u | |_{H^{σ, 2}} ≲ | | g | |_{H^{σ, 2}} + \int_{0}^{t} {‖ | u (t^{'}) |^{p - 1} u (t^{'}) ‖}_{H^{σ, 2}} d s .$
(15)

Now, we use the following lemma to estimate the nonlinear term in (15).
Lemma 3.2
Let $σ > 0$ and $p = 2 k + 1$ for $k = 1, 2, 3, \dots$ . Then, for any $u \in H^{σ, 2}$ , we have the nonlinear estimate
${‖ | u |^{p - 1} u ‖}_{H^{σ, 2}} ≲ ‖ u ‖_{H^{σ, 2}}^{p} .$
(16)
Proof.
Setting $v_{σ} := \cosh (σ | D |) u$ and applying Plancherel’s theorem, (16) reduced to
${‖ ⟨ ξ ⟩^{2} \cosh (σ | ξ |) F_{x} (| sech (σ | D |) v_{σ} |^{p - 1} sech (σ | D |) v_{σ}) (ξ) ‖}_{L_{ξ}^{2}} ≲ | | v_{σ} | |_{H^{2}}^{p} .$
(17)

By applying Plancherel’s theorem, Hölder’s inequality, and two-dimensional Sobolev embedding, the left-hand side (LHS) of (17) is estimated as
$\begin{aligned} LHS of (17) & = {‖ ⟨ ξ ⟩^{2} \cosh (σ | ξ |) F_{x} (| sech (σ | D |) v_{σ} |^{p - 1} sech (σ | D |) v_{σ}) (ξ) ‖}_{L_{ξ}^{2}} \\ \leq {‖ \int_{ζ} ⟨ ξ ⟩^{2} | \cosh (σ | ξ |) \prod_{j = 1}^{p} sech (σ | ξ_{j} |) | \prod_{j = 1}^{p} | \hat{v_{σ}} (ξ_{j}) | d ξ ‖}_{L_{ξ}^{2}} \\ ≲ {‖ \int_{ζ} \prod_{j = 1}^{p - 1} | \hat{v_{σ}} (ξ_{j}) | \cdot ⟨ ξ_{p + 1} ⟩^{2} | \hat{v_{σ}} (ξ_{p + 1}) | d ξ ‖}_{L_{ξ}^{2}} \\ \sim {‖ v_{σ}^{p - 1} \cdot ⟨ D ⟩^{2} v_{σ} ‖}_{L_{x}^{2}} \\ ≲ {‖ v_{σ}^{p - 1} ‖}_{L_{x}^{\infty}} {‖ ⟨ D ⟩^{2} v_{σ} ‖}_{L_{x}^{2}} \\ ≲ {‖ v_{σ} ‖}_{L_{x}^{\infty}}^{p - 1} {‖ v_{σ} ‖}_{H^{2}} \\ \sim {‖ v_{σ} ‖}_{H^{2}}^{p} \\ \sim ‖ u ‖_{H^{σ, 2}}^{p}, \end{aligned}$
where $ζ$ is the hyperplane defined by $ξ = \sum_{j = 1}^{p} ξ_{j}$ and $d ξ = \prod_{j = 1}^{p} d ξ_{j}$ . In fact, to obtain the third line, we used the following estimate from Dufera et al. (2022):
$| \cosh (σ | ξ |) \prod_{j = 1}^{p} sech (σ | ξ_{j} |) | \leq 2^{p} for any integer p \geq 3.$

Proof of Theorem 3.1. Proof of Theorem 3.1.

From (7), we define a mapping

Γ u (t) = W (t) g + i \int_{0}^{t} W (t - t^{'}) u (t^{'}) |^{p - 1} u (t^{'}) d t^{'} .

(18)

Let

σ > 0

, and

T > 0

to be determined later. Consider the closed ball

B

X

B = {u \in X : sup_{t \in [0, T]} ‖ u ‖_{H^{σ, 2}} \leq 2 ‖ g ‖_{H^{σ, 2}}},

where

X := C ((0, T); H^{σ, 2}) .

Applying the $H^{σ, 2}$ -norm on both sides of (18), and then using Lemmas 2.4 and 3.2 gives

\begin{aligned} sup_{t \in [0, T]} ‖ Γ u ‖_{H^{σ, 2}} & \leq ‖ g ‖_{H^{σ, 2}} + \int_{0}^{T} {‖ | u (t^{'}) |^{p - 1} u (t^{'}) ‖}_{H^{σ, 2}} d t^{'} \\ \leq ‖ g ‖_{H^{σ, 2}} + C T sup_{t \in [0, T]} ‖ u ‖_{H^{σ, 2}}^{p} . \end{aligned}

(19)

Now, for any $u \in B$ , it follows from (19) that

sup_{t \in [0, T]} ‖ Γ u ‖_{H^{σ, 2}} \leq 2 C ‖ g ‖_{H^{σ, 2}},

(20)

for

T

small enough so that

T = \frac{1}{2 C {(2 ‖ g ‖_{H^{σ, 2}})}^{p - 1}} .

(21)

It follows from (20) that

Γ

maps

B

into itself.

For any $u, v \in B$ , it follows from (15), Lemma 3.2 and (21) that

\begin{aligned} sup_{t \in [0, T]} ‖ Γ u - Γ v ‖_{H^{σ, 2}} & \leq \int_{0}^{T} {‖ | u (t^{'}) |^{p - 1} u (t^{'}) - | v (t^{'}) |^{p - 1} v (t^{'}) ‖}_{H^{σ, 2}} d t^{'} \\ \leq C T sup_{t \in [0, T]} ‖ u - v ‖_{H^{σ, 2}} {‖ {(| u | + | v |)}^{p - 1} ‖}_{H^{σ, 2}} \\ \leq C T {(2 ‖ g ‖_{H^{σ, 2}})}^{p - 1} sup_{t \in [0, T]} ‖ u - v ‖_{H^{σ, 2}} \\ \leq \frac{1}{2} sup_{t \in [0, T]} ‖ u - v ‖_{H^{σ, 2}}, \end{aligned}

which implies that

Γ

is a contraction map on

B

. So by the contraction mapping principle, the Cauchy problem (p-4NLS) has a unique solution in

X

with existence time as in (21). Continuous dependence on the initial data can be shown in a similar way, using the difference estimate. This concludes the proof of Theorem 3.1.

4. Approximate Conservation Law

For the existence of a global solution, we will apply the local well-posedness result repeatedly to cover time intervals of arbitrary length. The following approximate conservation law will allow us to repeat the local result on successive short time intervals to reach any target time $T_{*} > 0$ , by adjusting the strip width $σ$ according to the size of $T_{*} > 0$ . To do this, define the function $v_{σ}$ by

v_{σ} (x, t) := \cosh (σ | D |) u (x, t),

and then applying

\cosh (σ | D |)

to both sides of (p-4NLS) to obtain

i \partial_{t} v_{σ} + Δ^{2} v_{σ} - Δ v_{σ} + | v_{σ} |^{p - 1} v_{σ} = N (v_{σ}),

(22)

where

N (v_{σ}) := {| v_{σ} |^{p - 1} v_{σ} - \cosh (σ | D |) (sech (σ | D |) v_{σ} [sech (σ | D |) v_{σ}]^{p - 1})} .

(23)

Now, define a modified mass $+$ energy functional associated with $v_{σ}$ by

E_{σ} (t) := \frac{1}{2} \int_{R^{2}} (| v_{σ} |^{2} + | Δ v_{σ} |^{2} + | ▽ v_{σ} |^{2} + \frac{2}{p + 1} | v_{σ} |^{p + 1}) d x .

(24)

Since

v_{0} = u

and

E (t) + M (t) = E (0) + M (0)

, we have the conservation

E_{0} (t) = E_{0} (0) \forall t .

However, for

σ > 0

, the modified mass

+

energy functional fail to be conserved in time and we will prove an approximate conservation law by establishing a growth estimate. To do this, we first differentiating

E_{σ} (t)

, and then using (22) and (23) and integration by parts, we obtain

\begin{aligned} \frac{d}{d t} E_{σ} (t) & = Re \int_{R^{2}} \bar{v_{σ}} \partial_{t} v_{σ} + \bar{Δ v_{σ}} Δ \partial_{t} v_{σ} + ▽ v_{σ} \partial_{t} ▽ v_{σ} + | v_{σ} |^{p - 1} \bar{v_{σ}} \partial_{t} v_{σ} d x \\ = Re \int_{R^{2}} (\bar{v_{σ} + Δ^{2} v_{σ} - Δ v_{σ} + v_{σ} |^{p - 1} v_{σ}}) \partial_{t} v_{σ} d x \\ = - Re i \int_{R^{2}} (\bar{v_{σ} + Δ^{2} v_{σ} - Δ v_{σ} + | v_{σ} |^{p - 1} v_{σ}}) (- Δ^{2} v_{σ} + Δ v_{σ} - | v_{σ} |^{p - 1} v_{σ} + N (v_{σ})) d x \\ = Im \int_{R^{2}} (| Δ v_{σ} |^{2} - | ▽ v_{σ} |^{2} - | v_{σ} |^{p - 1} v_{σ} |^{2} + | Δ^{2} v_{σ} - Δ v_{σ} + | v_{σ} |^{p - 1} v_{σ} |^{2}) d x \\ + Im \int_{R^{2}} (\bar{v_{σ} + Δ^{2} v_{σ} - Δ v_{σ} + | v_{σ} |^{2} v_{σ}}) N (v_{σ}) d x \\ = Im \int_{R^{2}} [\bar{v_{σ} + Δ^{2} v_{σ} - Δ v_{σ} + | v_{σ} |^{2} v_{σ}}] N (v_{σ}) d x . \end{aligned}

Consequently, integrating over

[0, s]

with

s \leq T

gives

E_{σ} (s) = E_{σ} (0) + I_{σ}^{(1)} (s) + I_{σ}^{(2)} (s) + J_{σ} (s),

(25)

where

\begin{aligned} I_{σ}^{(1)} (s) & = Im \int_{0}^{s} \int_{R^{2}} (\bar{v_{σ} + Δ^{2} v_{σ}}) N (v_{σ}) d x d t, \end{aligned}

(26)

\begin{aligned} I_{σ}^{(2)} (s) & = - Im \int_{0}^{s} \int_{R^{2}} Δ v_{σ} N (v_{σ}) d x d t, \end{aligned}

(27)

\begin{aligned} J_{σ} (s) & = Im \int_{0}^{s} \int_{R^{2}} v_{σ} | v_{σ} |^{2} v_{σ} N (v_{σ}) d x d t, \end{aligned}

(28)

for

N (v_{σ})

as in (23).

The integrals given in (26)–(28) satisfy the following nonlinear estimates:

\begin{aligned} sup_{s \in [0, T]} | I_{σ}^{(m)} (s) | & \leq C σ^{2} | | v_{σ} | |_{X_{δ}^{2, b}}^{p + 1}, for m = 1, 2, \end{aligned}

(29)

\begin{aligned} sup_{s \in [0, T]} | J_{σ} (s) | & \leq C σ^{2} | | v_{σ} | |_{X_{δ}^{2, b}}^{2 p} . \end{aligned}

(30)

To prove the estimates (26)–(28), we need the following lemma from Dufera et al. (2022, Lemma 3).

Lemma 4.1

Let $p > 1$ be any integer and $ξ_{j} \in R^{2}$ such that $| ξ_{1} | \leq | ξ_{2} | \leq \dots \leq | ξ_{p} |$ and $ξ = \sum_{j = 1}^{p} ξ_{j}$ . Let $K (σ | ξ |) = 1 - \cosh (σ | ξ |) \prod_{j = 1}^{p} sech (σ | ξ_{j} |)$ . Then

| K (σ | ξ |) | \leq 2^{p} σ^{2} \sum_{j \neq k = 1}^{p} | ξ_{j} | | ξ_{k} | \leq 2^{p} σ^{2} | ξ_{p - 1} | | ξ_{p} | .

Proof Proof of estimate (29) for

m = 1

By symmetry of our argument we may assume that $| ξ_{1} | \leq | ξ_{2} | \leq \dots \leq | ξ_{p} |$ , and hence $| ξ | \leq p | ξ_{p} |$ . Setting

w_{σ} = F_{x}^{- 1} (| \hat{v_{σ}} |) .

(31)

Then by using Plancherel’s theorem, Lemma 4.1, and (31), we obtain

\begin{aligned} | I_{σ}^{(1)} (s) | & = | \int_{0}^{s} \int_{R^{2}} (1 + ξ^{4}) \bar{\hat{v_{σ}} (ξ, t)} \cdot \hat{N (v_{σ})} (ξ, t) d ξ d t | \\ \leq \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{4} | \hat{v_{σ}} (ξ, t) | | 1 - \cosh (σ | ξ |) \prod_{j = 1}^{p} sech (σ | ξ_{j} |) | \prod_{j = 1}^{p} | \hat{v_{σ}} (ξ_{j}, t) | d μ (ξ) d t, \\ \leq 2^{p} σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{4} | \hat{v_{σ}} (ξ, t) | (\sum_{j \neq k = 1}^{p} | ξ_{j} | | ξ_{k} |) \prod_{j = 1}^{p} | \hat{v_{σ}} (ξ_{j}, t) | d μ (ξ) d t \\ \leq C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{4} \hat{w_{σ}} (ξ, t) | ξ_{p - 1} | | ξ_{p} | \prod_{j = 1}^{p} \hat{w_{σ}} (ξ_{j}, t) d μ (ξ) d t, \end{aligned}

where

d μ (ξ)

is a measure

d μ (ξ) = δ (ξ - \sum_{j = 1}^{p} ξ_{j}) \prod_{j = 1}^{p} d ξ_{j} .

(32)

This measure imposes the condition

ξ = \sum_{j = 1}^{p} ξ_{j}

By apply Plancherel’s theorem, Hölder’s inequality, (12), (11), Sobolev embedding, and (31), we get

\begin{aligned} | I_{σ}^{(1)} (s) | & \leq C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{\frac{5}{2}} \hat{w_{σ}} (ξ, t) \prod_{j = 1}^{p - 2} \hat{w_{σ}} (ξ_{j}, t) | ξ_{p - 1} | \hat{w_{σ}} (ξ_{p - 1}, t) ⟨ ξ_{p} ⟩^{\frac{5}{2}} \hat{w_{σ}} (ξ_{p}, t) d μ (ξ) d t \\ \sim C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} F_{x} [⟨ D ⟩^{\frac{5}{2}} w_{σ}] (ξ, t) \cdot F_{x} [w_{σ}^{p - 2} \cdot | D | w_{σ} \cdot ⟨ D ⟩^{\frac{5}{2}} w_{σ}] (ξ, t) d ξ d t \\ \sim C σ^{2} \int_{0}^{T} \int_{R^{2}} ⟨ D ⟩^{\frac{5}{2}} w_{σ} \cdot w_{σ}^{p - 2} \cdot | D | w_{σ} \cdot ⟨ D ⟩^{\frac{5}{2}} w_{σ} d x d t \\ \leq C σ^{2} T^{\frac{1}{2}} ‖ w_{σ}^{p - 2} ‖_{L_{T}^{\infty} L_{x}^{\infty}} ‖ | D | w_{σ} ‖_{L_{T}^{\infty} L_{x}^{2}} ‖ ⟨ D ⟩^{\frac{5}{2}} w_{σ} ‖_{L_{T}^{4} L_{x}^{4}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} L_{x}^{\infty}}^{p - 2} ‖ | D | w_{σ} ‖_{X_{T}^{0, b}} ‖ ⟨ D ⟩^{2} w_{σ} ‖_{X_{T}^{0, b}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} H^{2}}^{p - 2} ‖ | D | w_{σ} ‖_{X_{T}^{0, b}} ‖ ⟨ D ⟩^{2} w_{σ} ‖_{X_{T}^{0, b}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{X_{T}^{2, b}}^{p + 1} \sim C σ^{2} ‖ v_{σ} ‖_{X_{T}^{2, b}}^{p + 1}, \end{aligned}

which is (29).

Proof Proof of estimate (29) for

m = 2

By symmetry of our argument we may assume that $| ξ_{1} | \leq | ξ_{2} | \leq \dots \leq | ξ_{p} |$ , and hence $| ξ | \leq p | ξ_{p} |$ . Setting

w_{σ} = F_{x}^{- 1} (| \hat{v_{σ}} |) .

(33)

Then, by using Plancherel’s theorem, Lemma 4.1, and (33), we obtain

\begin{aligned} | I_{σ}^{(2)} (s) | & = | \int_{0}^{s} \int_{R^{2}} ξ^{2} \bar{\hat{v_{σ}} (ξ, t)} \cdot \hat{N (v_{σ})} (ξ, t) d ξ d t | \\ \leq \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{2} | \hat{v_{σ}} (ξ, t) | | 1 - \cosh (σ | ξ |) \prod_{j = 1}^{p} sech (σ | ξ_{j} |) | \prod_{j = 1}^{p} | \hat{v_{σ}} (ξ_{j}, t) | d μ (ξ) d t, \\ \leq 2^{p} σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{2} | \hat{v_{σ}} (ξ, t) | (\sum_{j \neq k = 1}^{p} | ξ_{j} | | ξ_{k} |) \prod_{j = 1}^{p} | \hat{v_{σ}} (ξ_{j}, t) | d μ (ξ) d t \\ \leq C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{2} \hat{w_{σ}} (ξ, t) | ξ_{p - 1} | | ξ_{p} | \prod_{j = 1}^{p} \hat{w_{σ}} (ξ_{j}, t) d μ (ξ) d t, \end{aligned}

where

d μ (ξ)

is as given in (32).

By apply Plancherel’s theorem, Hölder’s inequality, (12), Sobolev embedding, and (33), we get

\begin{aligned} | I_{σ}^{(2)} (s) | & \leq C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} ⟨ ξ ⟩^{2} \hat{w_{σ}} (ξ, t) \prod_{j = 1}^{p - 1} \hat{w_{σ}} (ξ_{j}, t) ⟨ ξ_{p} ⟩^{2} \hat{w_{σ}} (ξ_{p}, t) d μ (ξ) d t \\ \sim C σ^{2} \int_{0}^{T} \int_{R^{2 p + 2}} F_{x} [⟨ D ⟩^{2} w_{σ}] (ξ, t) \cdot F_{x} [w_{σ}^{p - 1} \cdot ⟨ D ⟩^{2} w_{σ}] (ξ, t) d ξ d t \\ \sim C σ^{2} \int_{0}^{T} \int_{R^{2}} ⟨ D ⟩^{2} w_{σ} \cdot w_{σ}^{p - 1} \cdot ⟨ D ⟩^{2} w_{σ} d x d t \\ \leq C σ^{2} T ‖ w_{σ}^{p - 1} ‖_{L_{T}^{\infty} L_{x}^{\infty}} ‖ | D | w_{σ} ‖_{L_{T}^{\infty} L_{x}^{2}} ‖ ⟨ D ⟩^{2} w_{σ} ‖_{L_{T}^{\infty} L_{x}^{2}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} L_{x}^{\infty}}^{p - 1} ‖ ⟨ D ⟩^{2} w_{σ} ‖_{X_{T}^{0, b}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} H^{2}}^{p - 1} ‖ ⟨ D ⟩^{2} w_{σ} ‖_{X_{T}^{0, b}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{X_{T}^{2, b}}^{p + 1} \sim C σ^{2} ‖ v_{σ} ‖_{X_{T}^{2, b}}^{p + 1}, \end{aligned}

which is (29).

Proof Proof of (30)

Proceeding as above, we get

\begin{aligned} | J_{σ} (s) | & = | \int_{0}^{s} \int_{R^{2}} \bar{F_{x} [| v_{σ} |^{p - 1} v_{σ}] (ξ, t)} \cdot F_{x} [N (v_{σ})] (ξ, t) d ξ d t | \\ \leq \int_{0}^{T} \int_{R^{4 p}} | 1 - \cosh (σ | ξ |) \prod_{j = p + 1}^{2 p} sech (σ | ξ_{j} |) | \prod_{j = 1}^{2 p} | \hat{v_{σ}} (ξ_{j}, t) | d ν (ξ) d t \\ \leq 8 σ^{2} \int_{0}^{T} \int_{R^{4 p}} (\sum_{j \neq k = p + 1}^{2 p} | ξ_{j} | | ξ_{k} |) \prod_{j = 1}^{2 p} | \hat{v_{σ}} (ξ_{j}, t) | d ν (ξ) d t \\ \leq C σ^{2} \int_{0}^{T} \int_{R^{4 p}} | ξ_{2 p - 1} | | ξ_{2 p} | \prod_{j = 1}^{2 p} \hat{w_{σ}} (ξ_{j}, t) d ν (ξ) d t, \end{aligned}

where

d ν (ξ)

is the measure

d ν (ξ) = δ (\begin{matrix} ξ = \sum_{j = 1}^{p} ξ_{j} \\ ξ = \sum_{j = p + 1}^{2 p} ξ_{j} \end{matrix}) \prod_{j = 1}^{2 p} d ξ_{j},

which measure imposes the conditions

ξ = \sum_{j = 1}^{p} ξ_{j} = \sum_{j = p + 1}^{2 p} ξ_{j}

Then, using Plancherel’s theorem, Hölder’s inequality, (12), and Sobolev embedding, we obtain

\begin{aligned} | J_{σ} (s) | & \sim C σ^{2} \int_{0}^{T} \int_{R^{2}} F_{x} [w_{σ}^{p}] (ξ, t) \cdot F_{x} [w_{σ}^{p - 2} \cdot | D | w_{σ} \cdot | D | w_{σ}] (ξ, t) d ξ d t \\ \sim C σ^{2} \int_{0}^{T} \int_{R^{2}} w_{σ}^{2 p - 2} \cdot | D | w_{σ} \cdot | D | w_{σ} d x d t \\ \leq C σ^{2} T ‖ w_{σ}^{2 p - 2} ‖_{L_{T}^{\infty} L_{x}^{\infty}} {‖ (| D | w_{σ})^{2} ‖}_{L_{T}^{\infty} L_{x}^{1}} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} L_{x}^{\infty}}^{2 p - 2} {‖ (| D | w_{σ}) ‖}_{L_{T}^{\infty} L_{x}^{2}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{L_{T}^{\infty} H^{2}}^{2 p - 2} {‖ (| D | w_{σ}) ‖}_{X_{T}^{0, b}}^{2} \\ \leq C σ^{2} ‖ w_{σ} ‖_{X_{T}^{2, b}}^{2 p} \sim C σ^{2} ‖ v_{σ} ‖_{X_{T}^{2, b}}^{2 p} . \end{aligned}

This proves (30).

Theorem 4.2 (Almost conservation law)

Let $σ > 0$ , $g \in H^{σ, 2}$ and $u$ be the local solution of (p-4NLS) on $[0, T] \times R^{2}$ that is obtained in Theorem 3.1. Then

sup_{t \in [0, δ]} E_{σ} (t) \leq E_{σ} (0) + σ^{2} [1 + E_{σ}^{\frac{(p - 1)}{2}} (0)] E_{σ}^{\frac{(p + 1)}{2}} (0) .

(34)

In the limit as

σ \to 0

, we recover the conservation

E_{0} (t) = E_{0} (0)

Proof.

By (14), we have

‖ v_{σ} ‖_{X_{T}^{2, b}} = ‖ u ‖_{X_{T}^{σ, 2, b}} \leq C {‖ g ‖}_{H^{σ, 2}} = C {‖ v_{σ} (0) ‖}_{H^{2}} .

(35)

On the other hand,

\begin{aligned} ‖ v_{σ} (0) ‖_{H^{2}}^{2} = \int_{R^{2}} ⟨ ξ ⟩^{4} | \hat{v_{σ}} (ξ, 0) |^{2} d ξ & \leq C \int_{R^{2}} (1 + | ξ |^{4}) | \hat{v_{σ}} (ξ, 0) |^{2} d ξ \\ \sim C \int_{R^{2}} (| \hat{v_{σ}} (ξ, 0) |^{2} + | ξ |^{4} | \hat{v_{σ}} (ξ, 0) |^{2}) d ξ \\ \sim C \int_{R^{2}} (| v_{σ} (x, 0) |^{2} + | Δ v_{σ} (x, 0) |^{2}) d x \\ \leq C E_{σ} (0), \end{aligned}

which in turn implies

‖ v_{σ} ‖_{X_{T}^{2, b}} \leq C E_{σ}^{\frac{1}{2}} (0) .

(36)

Finally, use (36) in (29) and (30). Then, plugging the result into (25) gives the desired estimate in Theorem 4.2.

5. Proof of Theorem 1.1

Suppose that $u (\cdot, 0) = g (\cdot) \in H^{σ_{0}, 2} (R^{2})$ for some $σ_{0} > 0$ . Then $v_{σ_{0}} (\cdot, 0) = \cosh (σ_{0} | D |) g \in H^{2} (R^{2})$ . From two-dimensional Sobolev embedding, we have $H^{2} (R^{2}) \subset H^{(p - 1) / (p + 1)} (R^{2}) \subset L^{p + 1} (R^{2})$ , and hence we get

\begin{aligned} E_{σ_{0}} (0) & \leq C {‖ v_{σ_{0}} (\cdot, 0) ‖}_{H^{2} (R^{2})}^{2} + {‖ v_{σ_{0}} (\cdot, 0) ‖}_{L^{p + 1} (R^{2})}^{p + 1} \\ \leq C {‖ v_{σ_{0}} (\cdot, 0) ‖}_{H^{2} (R^{2})}^{2} + {‖ v_{σ_{0}} (\cdot, 0) ‖}_{H^{2} (R^{2})}^{p + 1} \\ < \infty . \end{aligned}

Following the argument used in Tesfahun (2017) (see also Selberg & da Silva, 2015; Selberg & Tesfahun, 2015), we can construct a solution on $[0, T_{*}]$ for arbitrarily large time $T_{*}$ , by applying the approximate conservation law in Theorem 4.2 so as to repeat the local result on successive short time intervals of size $T$ to reach $T_{*}$ , by adjusting the strip width $σ$ according to the size of $T_{*}$ . In doing so, we establish the bound

sup_{t \in [0, T_{*}]} E_{σ} (t) \leq 2 E_{σ_{0}} (0),

(37)

for

σ

satisfying

σ (t) \geq C t^{- 1 / 2} for all t \in [0, T_{*}] .

Hence, we have

E_{σ} (t) < \infty

for all

t \in [0, T_{*}]

, which implies

u (t) \in H^{σ (t), 2}

, and this concludes the proof of Theorem 1.1.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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