The defocusing energy-supercritical inhomogeneous NLS in four space dimension

Xuan Liu School of Mathematics, Hangzhou Normal University, Hangzhou , 311121, China liuxuan95@hznu.edu.cn and Chengbin Xu School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai, 810008, China. xcbsph@163.com

Abstract.

In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schrödinger equation $iu_{t}+\Delta u=|x|^{-b}|u|^{\alpha}u$ in four space dimension, where $s_{c}:=2-\frac{2-b}{\alpha}\in(1,2)$ and $0<b<\min\{(s_{c}-1)^{2}+1,3-s_{c}\}$ . We prove that if the solution has a prior bound in the critical Sobolev space, that is, $u\in L_{t}^{\infty}(I;\dot{H}_{x}^{s_{c}}(\mathbb{R}^{4}))$ , then $u$ is global and scatters. The proof of the main results is based on the concentration-compactness/rigidity framework developed by Kenig and Merle [Invent. Math. 166 (2006)], together with a long-time Strichartz estimate, a spatially localized Morawetz estimate, and a frequency-localized Morawetz estimate.

Keywords: Energy supcritical, inhonogeneous NLS, Global well-posedness, Scattering, Concentration-compactness.

1. Introduction

We consider the defocusing inhomogeneous nonlinear Schrödinger equation (INLS) in $\mathbb{R}^{4}$ :

\begin{cases}iu_{t}+\Delta u=|x|^{-b}|u|^{\alpha}u,\qquad(t,x)\in\mathbb{R}\times\mathbb{R}^{4},\\ u(0,x)=u_{0}(x),\end{cases}

(1.1)

where $\alpha>0$ , and $0<b<2$ . Note that the classical nonlinear Schrödinger (NLS) equation corresponds to the case $b=0$ . For the physical background and applications of the INLS model (1.1) in nonlinear optics, see Gill [24] and Liu-Tripathi [40].

For solutions $u$ that are sufficiently smooth and decay suitably at infinity, the energy

E(u)=\int_{\mathbb{R}^{4}}\frac{1}{2}|\nabla u(t,x)|^{2}+\frac{1}{\alpha+2}|x|^{-b}|u(t,x)|^{\alpha+2}\,dx

is conserved throughout the interval of existence. The Cauchy problem (1.1) is scale-invariant. Specifically, the scaling transformation

u(t,x)\longmapsto\lambda^{\frac{2-b}{\alpha}}u(\lambda^{2}t,\lambda x),\quad\lambda>0,

leaves the class of solutions to (1.1) invariant. This transformation also identifies the critical space $\dot{H}^{s_{c}}(\mathbb{R}^{4})$ , where the critical regularity $s_{c}$ is defined as $s_{c}:=2-\frac{2-b}{\alpha}$ . We classify (1.1) as mass-critical if $s_{c}=0$ , energy-critical if $s_{c}=1$ , inter-critical if $0<s_{c}<1$ , and energy-supercritical if $s_{c}>1$ .

The local well-posedness of (1.1) in $H^{s}(\mathbb{R}^{d})$ , for $0\leq s\leq\min\{1,\frac{d}{2}\}$ , under suitable conditions on $b$ and $\alpha$ , was first established by Guzman [25] using classical Strichartz estimates in Sobolev spaces. Subsequently, by employing techniques based on Lorentz spaces [1, 2] and weighted Strichartz estimates [8, 5, 6, 38], the local theory in $H^{s}(\mathbb{R}^{d})$ was extended to the range $0\leq s<\min\{d,1+\frac{d}{2}\}$ , under the condition $0<\alpha$ , $(d-2s)\alpha\leq 4-2b$ , along with suitable assumptions on $b$ .

In this paper, we focus on the global well-posedness and scattering theory for the Cauchy problem (1.1). A wealth of results has been established in this direction: for the mass-critical case, see [41]; for the inter-critical case, see [4, 10, 11, 12, 21, 22, 45, 49]; and for the energy-critical case, see [5, 6, 26, 27, 42]. However, to the best of our knowledge, no results are currently available for the energy-supercritical regime. This work aims to address precisely that case.

1.1. The nonlinear Schrödinger equation at critical regularity

To begin with, we review global well-posedness and scattering theory for the standard nonlinear Schrödinger equation (NLS) on $\mathbb{R}^{d}:$

iu_{t}+\Delta u=\lambda|u|^{\alpha}u,\qquad(t,x)\in\mathbb{R}\times\mathbb{R}^{d},

(1.2)

which has seen significant progress in recent years. Due to conserved quantities at critical regularity, mass- and energy-critical cases have received the most attention. For the defocusing ( $\lambda=+1$ ) energy-critical NLS, it is now known that initial data in $\dot{H}^{1}(\mathbb{R}^{d})$ leads to global solutions that scatter. This result was established first for radial data by Bourgain [3], Grillakis [28], Tao [53], and later for general data by Colliander et al. [7], Ryckman-Visan [51], and Visan [57, 58]. For the focusing ( $\lambda=-1$ ) case, see [18, 29, 32].

For mass-critical NLS, it has been shown that arbitrary data in $L^{2}(\mathbb{R}^{d})$ leads to global solutions that scatter, first for radial data in $d\geq 2$ (see [55, 31, 36]) and later for general data in all dimensions by Dodson [14, 15, 16, 17].

For cases where $s_{c}\neq 0,1$ , the energy- and mass-critical methods do not apply due to the absence of conserved quantities controlling the time growth of the $\dot{H}^{s_{c}}(\mathbb{R}^{d})$ norm of the solutions. However, it is conjectured that under a priori control of a critical norm, global well-posedness and scattering hold for all $s_{c}>0$ in any spatial dimension:

Conjection 1.1.

Let $d\geq 1$ , $\alpha\geq\frac{4}{d}$ , and $s_{c}=\frac{d}{2}-\frac{2}{\alpha}$ . Assume $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ is a maximal-lifespan solution to (1.2) such that

u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d}).

Then $u$ is global and scatters as $t\to\pm\infty$ .

The first progress on Conjecture 1.1 was made by Kenig and Merle [30], who addressed the case $d=3$ and $s_{c}=\frac{1}{2}$ using their concentration-compactness framework [29] and a scaling-critical Lin-Strauss Morawetz inequality. Subsequently, In the inter-critical regime ( $0<s_{c}<1$ ), some significant progress toward Conjecture 1.1 has been achieved by Murphy [46, 47, 48], Xie-Fang [59] ,Gao-Miao-Yang [20], Gao-Zhao [23] and Yu [60].

For energy-supercritical cases ( $s_{c}>1$ ), Killip and Visan [32] were the first to prove Conjecture 1.1 for $d\geq 5$ under certain conditions on $s_{c}$ . Murphy [48] later addressed the conjecture for radial initial data in $d=3$ with $s_{c}\in(1,\frac{3}{2})$ . By introducing long-time Strichartz estimates in the energy-supercritical setting, Miao-Murphy-Zheng [44] and Dodson-Miao-Murphy-Zheng [19] established Conjecture 1.1 for general initial data in $d=4$ with $1<s_{c}\leq\frac{3}{2}$ . For $d=4$ and $\frac{3}{2}<s_{c}<2$ , similar results for radial data were obtained by Lu and Zheng [43]. For related results in dimensions $d\geq 5$ , we refer the reader to Zhao [61] and Li–Li [39]. See Table 1 for a summary.

Table 1. Results for Conjecture 1.1 in the super-critical case:

1<s_{c}<\frac{d}{2}

$d=3$	$1<s_{c}<\frac{3}{2}$ , radial, Murphy [48]
$d=4$	$1<s_{c}<\frac{3}{2}$ , Miao-Murphy-Zheng[44]; $s_{c}=\frac{3}{2}$ , Dodson-Miao-Murphy-Zheng[19]; $\frac{3}{2}<s_{c}<2$ , radial, Lu-Zheng[43]
$d\geq 5$	$1<s_{c}<\frac{d}{2}$ , and assume $\alpha$ is even when $d\geq 8$ , Killip-Visan[32], Zhao[61], Li-Li[39]

1.2. Main results

Analogous to Conjecture 1.1, it is conjectured that for the inhomogeneous NLS (1.1):

Conjection 1.2.

Let $d\geq 1$ , $0<b<2$ , $\alpha\geq\frac{4-2b}{d}$ , and $s_{c}:=\frac{d}{2}-\frac{2-b}{\alpha}$ . Assume $u:I\times\mathbb{R}^{d}\rightarrow\mathbb{C}$ is a maximal-lifespan solution to (1.1) such that

u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d}),

then $u$ is global and scatters as $t\to\pm\infty$ .

Wang and the second author of this paper [56] first addressed Conjecture 1.2 in the subcritical case ( $d\geq 5,s_{c}=\frac{1}{2}$ ) by employing the Lin–Strauss Morawetz inequality. In this work, we establish Conjecture 1.2 in the energy-supercritical regime in four-dimensional case. Our main result is stated as follows.

Theorem 1.1.

Let $1<s_{c}<2$ , $0<b<\min\left\{(s_{c}-1)^{2}+1,3-s_{c}\right\}$ and $\alpha>0$ be such that $s_{c}=2-\frac{2-b}{\alpha}$ . Suppose $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ is a maximal-lifespan solution to (1.1) such that

u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4}).

(1.3)

Then $u$ is global and scatters.

Remark 1.2.

To avoid technical complications, in Theorem 1.1, we assume the condition $b<(s_{c}-1)^{2}+1$ , which is equivalent to $s_{c}<\alpha$ . Under this assumption, the proof of the stability theorem becomes fairly straightforward (see Lemma 2.17 below), since the derivative of the nonlinearity possesses enough regularity when applying the operator $|\nabla|^{s_{c}}$ . When $b\geq(s_{c}-1)^{2}+1$ , the stability theory becomes considerably more delicate, as this corresponds to the regime of low-power nonlinearities. In such cases, one needs to refine the function spaces used in the stability framework. We refer the reader to [32, Section 3.4] for further discussions and relevant references.

Remark 1.3.

The constraint $b<3-s_{c}$ comes from the Strichartz estimate (Proposition 2.15) and the singularity of the inhomogeneous coefficient $|x|^{-b}$ . Using the weighted Strichartz estimates as in [38], it may be possible to extend the admissible range of $b$ in Theorem 1.1 to include the case $b\geq 3-s_{c}$ .

Similar results for the classical nonlinear Schrödinger equation in dimension four have been established previously by Miao-Murphy-Zheng [44], Dodson-Miao-Murphy-Zheng [19] and Lu-Zheng [43]. Notably, in the range $\frac{3}{2}<s_{c}<2$ , their results require the initial data to satisfy a radial symmetry assumption (c.f. Table 1). In this paper, we show that this radial symmetry condition can be removed for the INLS equation. The key reason for this improvement is the presence of the spatially decaying factor $|x|^{-b}$ in the nonlinearity. This decay at infinity prevents the minimal almost periodic solution from concentrating at infinity in either physical or frequency space (see Proposition 2.19 below). Consequently, both spatial and frequency translation parameters must remain bounded. Thus, after applying a suitable translation, these parameters can be taken to be zero. This effectively places us in the same scenario as in the radial case but without the need for a radial assumption.

1.3. Outline of the proof of Theorem 1.1

We need some definitions:

Definition 1.4 (Strong solution).

Let

\gamma=2(\alpha+1),\qquad m=\frac{4\alpha(\alpha+1)}{\alpha+2-b(\alpha+1)}.

A function $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ is a solution to (1.1) if, for any compact $J\subset I$ , $u\in C_{t}^{0}L_{x}^{2}(J\times\mathbb{R}^{4})\cap L_{t}^{\gamma}L_{x}^{m,2}(J\times\mathbb{R}^{4})$ , and satisfies the Duhamel formula for all $t,t_{0}\in I$ :

u(t)=e^{i(t-t_{0})\Delta}u(t_{0})-i\int_{t_{0}}^{t}e^{i(t-\tau)\Delta}|x|^{-b}|u(\tau)|^{\alpha}u(\tau)\,d\tau,

where $L_{x}^{m,2}$ is the Lorentz space (defined in Subsection 2.2). We refer to $I$ as the lifespan of $u$ . If $u$ cannot be extended to any strictly larger interval, it is called a maximal-lifespan solution. If $I=\mathbb{R}$ , $u$ is called a global solution.

It is natural to assume that solutions belong locally in time to the space $L_{t}^{\gamma}L_{x}^{m,2}(I\times\mathbb{R}^{4})$ , as the linear flow always lies in this space by the Strichartz estimate (see proposition 2.15 below). Furthermore, any global solution $u$ of (1.1) satisfying $\|u\|_{L_{t}^{\gamma}L_{x}^{m,2}(\mathbb{R}\times\mathbb{R}^{4})}<+\infty$ necessarily scatters in both time directions in the sense that there exist unique scattering states $u_{\pm}\in\dot{H}^{s_{c}}(\mathbb{R}^{4})$ such that

\|u(t)-e^{it\Delta}u_{\pm}\|_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{4})}\rightarrow 0\quad\text{as}\quad t\rightarrow\pm\infty.

In view of this, we introduce the scattering size of the solution $u$ as

S_{I}(u)=\|u\|_{L_{t}^{\gamma}L_{x}^{m,2}(I\times\mathbb{R}^{4})}.

Closely associated with the notion of scattering is the notion of blow-up:

Definition 1.5 (Blow-up).

A solution $u$ to (1.1) is said to blow up forward in time if there exists $t_{0}\in I$ such that

\|u\|_{S_{[t_{0},\sup(I))}}=\infty.

Similarly, $u$ blows up backward in time if there exists $t_{0}\in I$ such that

\|u\|_{S_{(\inf(I),t_{0}]}}=\infty.

The local theory for (1.1) has been worked out in [2].

Theorem 1.6 (Local well-posedness).

Let $1<s_{c}<2,0<b<2$ and $\alpha>0$ be such that $s_{c}=2-\frac{2-b}{\alpha}$ . Given $u_{0}\in\dot{H}^{s_{c}}(\mathbb{R}^{4})$ and $t_{0}\in\mathbb{R}$ , there exists a unique maximal-lifespan solution $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ to (1.1) with $u(t_{0})=u_{0}$ . Moreover, this solution satisfies the following:

(1)

(Local existence) $I$ is an open neighborhood of $t_{0}$ .
(2)

(Blowup criterion) If $\sup I$ is finite, then $u$ blows up forward in time (in the sense of Definition 1.5). Similarly, if $\inf I$ is finite, then $u$ blows up backward in time.
(3)

(Scattering) If $\sup I=+\infty$ and $u$ does not blow up forward in time, then $u$ scatters forward in time. Specifically, there exists a unique $u_{+}\in\dot{H}^{s_{c}}_{x}(\mathbb{R}^{4})$ such that

$\lim_{t\to\infty}\|u(t)-e^{it\Delta}u_{+}\|_{\dot{H}^{s_{c}}_{x}(\mathbb{R}^{4})}=0.$

Conversely, given $u_{+}\in\dot{H}^{s_{c}}_{x}(\mathbb{R}^{4})$ , there is a unique solution to (1.1) in a neighborhood of $t=\infty$ such that (3) holds. Analogous statements hold backward in time.
(4)

(Small-data global existence and scattering) If $\|u_{0}\|_{\dot{H}^{s_{c}}_{x}}$ is sufficiently small, then $u$ is global and scatters.

To prove Theorem 1.1, following the road map in [29], we argue by contradiction. First, define $L:[0,\infty)\to[0,\infty]$ by

L(E):=\sup\{S_{I}(u)\mid u:I\times\mathbb{R}\to\mathbb{C}\text{ solving }(\ref{INLS})\text{ with }\|u\|^{2}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}\leq E\}.

From the stability theory (see Theorem 2.16 below), we know that $L$ is continuous. Note also that $L(E)$ is a nonnegative, nondecreasing function, and by Theorem 1.6 $L(E)\leq E^{\alpha}$ for $E$ sufficiently small. Then there must exist a unique critical threshold $E_{c}\in(0,\infty]$ such that

L(E)<\infty\quad\text{if }E<E_{c},\qquad L(E)=\infty\quad\text{if }E\geq E_{c}.

The failure of main theorem implies that $0<E_{c}<\infty$ . Moreover,

Theorem 1.7 (existence of minimal counterexamples).

Suppose Theorem 1.1 fails to be true. Then, there exists a maximal life-span solution $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\rightarrow\mathbb{C}$ such that the following hold:

(1)

$u$ blows up its scattering norm forward in time, i.e. $\mathrm{S}_{[0,T_{\max})}(u)=+\infty$ ;
(2)

there exists a frequency scale function $N:[0,T_{\max})\to(0,\infty)$ such that

$\left\{N(t)^{s_{c}-2}u(t,N(t)^{-1}x):t\in[0,T_{\max})\right\}$ (1.4)

is precompact in $\dot{H}^{s_{c}}(\mathbb{R}^{4})$ .

In addition, we may assume $\displaystyle\inf_{t\in[0,T_{\text{max}})}N(t)\geq 1$ .

Remark 1.8.

By Arzala-Ascoli theorem, (1.4) implies that for any $\eta>0,$ there exists $C(\eta)>0$ such that

\int_{|x|\geq\frac{C(\eta)}{N(t)}}||\nabla|^{s_{c}}u(t,x)|^{2}dx+\int_{|\xi|\geq C(\eta)N(t)}|\xi|^{2s_{c}}|\hat{u}(t.\xi)|^{2}d\xi\leq\eta

(1.5)

In the rest of this paper, we will call the solutions found in Theorem 1.7 as almost periodic solutions. The detailed construction of such solutions in the context of the standard $\dot{H}^{s_{c}}$ -critical NLS was carried out in [32, 46]. In the $\dot{H}^{1}$ -critical setting of the (INLS), a similar construction was established by Guzman and Murphy [26]. Following the arguments developed in [32, 46, 26], the key ingredients in the construction are the linear profile decomposition, as developed by Shao [52], and the embedding of nonlinear profiles (will be presented in detail in Proposition 2.19).

To complete the proof of Theorem 1.1, it sufficies to rule out the existence of almost periodic solutions as described in Theorem 1.7. We first note that for the maximal-lifespan solution found in Theorem 1.7, $N(t)$ enjoys the following properties: Lemma 1.9, Corollary 1.10 and Lemma 1.11 (see [35] for details).

Lemma 1.9 (Local constancy).

Let $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ be a maximal-lifespan almost periodic solution to (1.1). Then there exists $\delta=\delta(u)>0$ such that for all $t_{0}\in I$ ,

[t_{0}-\delta N(t_{0})^{-2},t_{0}+\delta N(t_{0})^{-2}]\subset I.

Moreover, $N(t)\sim_{u}N(t_{0})$ for $|t-t_{0}|\leq\delta N(t_{0})^{-2}$ .

Corollary 1.10 ( $N(t)$ at blow-up).

Let $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ be a maximal-lifespan almost periodic solution to (1.1). If $T$ is a finite endpoint of $I$ , then $N(t)\gtrsim_{u}|T-t|^{-1/2}$ . In particular, $\lim_{t\to T}N(t)=\infty$ . If $I$ is infinite or semi-infinite, then for any $t_{0}\in I$ , we have $N(t)\gtrsim_{u}\langle t-t_{0}\rangle^{-1/2}$ .

Lemma 1.11 (Spacetime bounds).

Let $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ be an almost periodic solution to (1.1). Then, the following bound holds:

\int_{I}N(t)^{2}\,dt\lesssim_{u}\||\nabla|^{s_{c}}u\|_{L_{t}^{q}L_{x}^{r,2}(I\times\mathbb{R}^{4})}^{q}\lesssim_{u}1+\int_{I}N(t)^{2}\,dt,

where $(q,r)$ is an admissible pair (see Definition 2.2 below) with $q<\infty$ .

In Section 3, using Lemma 1.9, Corollary 1.10, Lemma 1.11 and the space-localized Morawetz inequality employed in Bourgain [3] in the study of the radial energy-critical NLS, we rule out the almost periodic solutions in the case $1<s_{c}<3/2$ .

In the case $\frac{3}{2}\leq s_{c}<2$ , we first refine the class of almost periodic solutions. Using rescaling arguments as in [31, 33, 55], we can ensure that the almost periodic solutions under consideration do not escape to arbitrarily low frequencies on at least half of their maximal lifespan, say $[0,T_{\text{max}})$ . By applying Lemma 1.5 to partition $[0,T_{\text{max}})$ into characteristic subintervals $J_{k}$ , we obtain the following theorem.

Theorem 1.12 (Two scenarios for blow-up).

If Theorem 1.1 does not hold, then there exists an almost periodic solution $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ that blows up forward in time and satisfies

u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}([0,T_{\text{max}})\times\mathbb{R}^{4}).

Additionally, $[0,T_{\text{max}})$ can be decomposed as $[0,T_{\text{max}})=\bigcup_{k}J_{k}$ , where

N(t)\equiv N_{k}\geq 1\quad\text{for}\quad t\in J_{k},\quad\text{with}\quad|J_{k}|\sim_{u}N_{k}^{-2}.

The solution $u$ falls into one of the following two cases:

\int_{0}^{T_{\text{max}}}N(t)^{3-2s_{c}}\,dt<\infty\quad\text{(rapid frequency cascade solution)},

\int_{0}^{T_{\text{max}}}N(t)^{3-2s_{c}}\,dt=\infty\quad\text{(quasi-soliton solution)}.

In view of this theorem, to prove Theorem 1.1 in the case $\frac{3}{2}\leq s_{c}<2$ , it suffices to rule out all the possible scenarios described in Theorem 1.12.

In Section 4, we exclude the possibility of ”rapid frequency-cascade solutions.” Two key analytical tools are employed to achieve this. The first is the long-time Strichartz estimate, originally introduced by Dodson [14] in the context of mass-critical nonlinear Schrödinger equations. For further applications of this method to the energy-critical, energy-subcritical, and energy-supercritical regimes, see [58, 46, 48]. See also [41], where a long-time Strichartz estimate was established specifically for the mass-critical inhomogeneous NLS, which is more closely related to the supercritical INLS considered in this paper. The second tool is the following ”No-waste Duhamel formula” (see Proposition 5.23 in [35]):

Lemma 1.13 (No-waste Duhamel formula).

If $u$ is an almost periodic solution on the maximal lifespan $I$ , then for any $t\in I$ , we have

u(t)=\lim_{T\to T_{\text{max}}}i\int_{t}^{T}e^{i(t-s)\Delta}|x|^{-b}|u(s,x)|^{\alpha}u(s,x)\,ds,

where the equality holds in the weak limit sense in $\dot{H}^{s_{c}}(\mathbb{R}^{4})$ .

Using the long-time Strichartz estimate and the ”No-waste Duhamel formula,” we deduce that ”rapid frequency-cascade solutions” must have zero mass, which contradicts the assumption that the solution blows up. Hence, ”rapid frequency-cascade solutions” can be ruled out.

In Section 5, we exclude the existence of ”quasi-soliton solutions”. The primary tool employed here is the frequency-localized Morawetz inequality (Theorem 5.1). The key idea in Theorem 5.1 is to apply a high-frequency truncation to the standard Morawetz inequality and then estimate the error terms. For these error estimates, the long-time Strichartz estimate established earlier plays a crucial role. The frequency-localized Morawetz inequality provides a uniform bound on $\int_{I}N(t)^{3-2s_{c}}\,dt$ for any compact interval $I\subset[0,T_{\text{max}})$ . By choosing $I\subset[0,T_{\text{max}})$ sufficiently large, we arrive at a contradiction, thereby excluding the existence of ”quasi-soliton solutions.” This completes the proof of Theorem 1.1.

2. Preliminaries

2.1. Some notations

We use the standard notation for mixed Lebesgue space-time norms and Sobolev spaces. We write $A\lesssim B$ to denote $A\leq CB$ for some $C>0$ . If $A\lesssim B$ and $B\lesssim A$ , then we write $A\approx B$ . We write $A\ll B$ to denote $A\leq cB$ for some small $c>0$ . If $C$ depends upon some additional parameters, we will indicate this with subscripts; for example, $X\lesssim_{u}Y$ denotes that $X\leq C_{u}Y$ for some $C_{u}$ depending on $u$ . We use $O(Y)$ to denote any quantity $X$ such that $|X|\lesssim Y$ . We use $\langle x\rangle$ to denote $(1+|x|^{2})^{\frac{1}{2}}$ . We write $L_{t}^{q}L_{x}^{r}$ to denote the Banach space with norm

\|u\|_{L_{t}^{q}L_{x}^{r}(\mathbb{R}\times\mathbb{R}^{d})}:=\left(\int_{\mathbb{R}}\left(\int_{\mathbb{R}^{d}}|u(t,x)|^{r}\,dx\right)^{q/r}\,dt\right)^{1/q},

with the usual modifications when $q$ or $r$ are equal to infinity, or when the domain $\mathbb{R}\times\mathbb{R}^{d}$ is replaced by spacetime slab such as $I\times\mathbb{R}^{d}$ . When $q=r$ we abbreviate $L_{t}^{q}L_{x}^{q}$ as $L_{t,x}^{q}$ .

Let $\phi(\xi)$ be a radial bump function supported in the ball $\{\xi\in\mathbb{R}^{d}:|\xi|\leq 2\}$ and equal to $1$ on the ball $\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1\}$ . For each number $N>0$ , we define the Fourier multipliers

\widehat{P_{\leq N}f}(\xi):=\phi(\xi/N)\widehat{f}(\xi),\quad\widehat{P_{>N}f}(\xi):=(1-\phi(\xi/N))\widehat{f}(\xi),

\widehat{P_{N}f}(\xi)=:\psi(\xi/N)\widehat{f}(\xi):=(\phi(\xi/N)-\phi(2\xi/N))\widehat{f}(\xi).

We similarly define $P_{<N}$ and $P_{\geq N}$ . For convenience of notation, let $u_{N}:=P_{N}u$ , $u_{\leq N}:=P_{\leq N}u$ , and $u_{>N}:=P_{>N}u$ . We will normally use these multipliers when $M$ and $N$ are dyadic numbers (that is, of the form $2^{n}$ for some integer $n$ ); in particular, all summations over $N$ or $M$ are understood to be over dyadic numbers.

2.2. Lorentz spaces and useful lemmas

Let $f$ be a measurable function on $\mathbb{R}^{d}$ . The distribution function of $f$ is defined by

d_{f}(\lambda):=|\{x\in\mathbb{R}^{d}:|f(x)|>\lambda\}|,\quad\lambda>0,

where $|A|$ is the Lebesgue measure of a set $A$ in $\mathbb{R}^{d}$ . The decreasing rearrangement of $f$ is defined by

f^{*}(s):=\inf\left\{\lambda>0:d_{f}(\lambda)\leq s\right\},\quad s>0.

Definition 2.1 (Lorentz spaces).

Let $0<r<\infty$ and $0<\rho\leq\infty$ . The Lorentz space $L^{r,\rho}(\mathbb{R}^{d})$ is defined by

L^{r,\rho}(\mathbb{R}^{d}):=\left\{f\text{ is measurable on }\mathbb{R}^{d}:\|f\|_{L^{r,\rho}}<\infty\right\},

where

\|f\|_{L^{r,\rho}}:=\left\{\begin{array}[]{cl}(\frac{\rho}{r}\int_{0}^{\infty}(s^{1/r}f^{*}(s))^{\rho}\frac{1}{s}ds)^{1/\rho}&\text{ if }\rho<\infty,\\ \sup_{s>0}s^{1/r}f^{*}(s)&\text{ if }\rho=\infty.\end{array}\right.

We collect the following basic properties of $L^{r,\rho}(\mathbb{R}^{d})$ in the following lemmas.

Lemma 2.2 (Properties of Lorentz spaces [50]).

•

For $1<r<\infty$ , $L^{r,r}(\mathbb{R}^{d})\equiv L^{r}(\mathbb{R}^{d})$ and by convention, $L^{\infty,\infty}(\mathbb{R}^{d})=L^{\infty}(\mathbb{R}^{d})$ .
•

For $1<r<\infty$ and $0<\rho_{1}<\rho_{2}\leq\infty$ , $L^{r,\rho_{1}}(\mathbb{R}^{d})\subset L^{r,\rho_{2}}(\mathbb{R}^{d})$ .
•

For $1<r<\infty$ , $0<\rho\leq\infty$ , and $\theta>0$ , $\||f|^{\theta}\|_{L^{r,\rho}}=\|f\|^{\theta}_{L^{\theta r,\theta\rho}}$ .
•

For $b>0$ , $|x|^{-b}\in L^{\frac{d}{b},\infty}(\mathbb{R}^{d})$ and $\||x|^{-b}\|_{L^{\frac{d}{b},\infty}}=|B(0,1)|^{\frac{b}{d}}$ , where $B(0,1)$ is the unit ball of $\mathbb{R}^{d}$ .

Lemma 2.3 (Hölder’s inequality [50]).

•

Let $1<r,r_{1},r_{2}<\infty$ and $1\leq\rho,\rho_{1},\rho_{2}\leq\infty$ be such that

$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}},\quad\frac{1}{\rho}\leq\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}.$

Then for any $f\in L^{r_{1},\rho_{1}}(\mathbb{R}^{d})$ and $g\in L^{r_{2},\rho_{2}}(\mathbb{R}^{d})$

$\|fg\|_{L^{r,\rho}}\lesssim\|f\|_{L^{r_{1},\rho_{1}}}\|g\|_{L^{r_{2},\rho_{2}}}.$
•

Let $1<r_{1},r_{2}<\infty$ and $1\leq\rho_{1},\rho_{2}\leq\infty$ be such that

$1=\frac{1}{r_{1}}+\frac{1}{r_{2}},\quad 1\leq\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}.$

Then for any $f\in L^{r_{1},\rho_{1}}(\mathbb{R}^{d})$ and $g\in L^{r_{2},\rho_{2}}(\mathbb{R}^{d})$

$\|fg\|_{L^{1}}\lesssim\|f\|_{L^{r_{1},\rho_{1}}}\|g\|_{L^{r_{2},\rho_{2}}}.$

Lemma 2.4 (Convolution inequality [50]).

Let $1<r,r_{1},r_{2}<\infty$ and $1\leq\rho,\rho_{1},\rho_{2}\leq\infty$ be such that

1+\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}},\quad\frac{1}{\rho}\leq\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}.

Then for any $f\in L^{r_{1},\rho_{1}}(\mathbb{R}^{d})$ and $g\in L^{r_{2},\rho_{2}}(\mathbb{R}^{d})$

\|f\ast g\|_{L^{r,\rho}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{r_{1},\rho_{1}}(\mathbb{R}^{d})}\|g\|_{L^{r_{2},\rho_{2}}(\mathbb{R}^{d})}.

Using the above convolution inequality, one can easily obtain Bernstein’s inequality in Lorentz spaces.

Lemma 2.5 (Bernstein’s inequality).

Let $N>0$ , $1<r_{1}<r_{2}<\infty$ and $1\leq\rho_{1}\leq\rho_{2}\leq\infty$ . Then

\|P_{N}f\|_{L^{r_{2},\rho_{2}}(\mathbb{R}^{d})}\lesssim N^{d(\frac{1}{r_{1}}-\frac{1}{r_{2}})}\|f\|_{L^{r_{1},\rho_{1}}(\mathbb{R}^{d})}.

Next, in Lemma 2.7–Lemma 2.10, we recall the Sobolev embedding, Hardy’s inequality, product rule, and chain rule in Lorentz spaces. We start by introducing the following definition.

Definition 2.6.

Let $s\geq 0,1<r<\infty$ and $1\leq\rho\leq\infty$ . We define the Sobolev-Lorentz spaces

W^{s}L^{r,\rho}(\mathbb{R}^{d}):=\left\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d}):(1-\Delta)^{s/2}f\in L^{r,\rho}(\mathbb{R}^{d})\right\},

\dot{W}^{s}L^{r,\rho}(\mathbb{R}^{d}):=\left\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d}):(-\Delta)^{s/2}f\in L^{r,\rho}(\mathbb{R}^{d})\right\},

where $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is the space of tempered distributions on $\mathbb{R}^{d}$ and

(1-\Delta)^{s/2}f=\mathcal{F}^{-1}\left((1+|\xi|^{2})^{s/2}\mathcal{F}(f)\right),\qquad(-\Delta)^{s/2}f=\mathcal{F}^{-1}(|\xi|^{s}\mathcal{F}(f))

with $\mathcal{F}$ and $\mathcal{F}^{-1}$ the Fourier and its inverse Fourier transforms respectively. The spaces $W^{s}L^{r,\rho}(\mathbb{R}^{d})$ and $\dot{W}^{s}L^{r,\rho}(\mathbb{R}^{d})$ are endowed respectively with the norms

\|f\|_{W^{s}L^{r,\rho}}=\|f\|_{L^{r,\rho}}+\|(-\Delta)^{s/2}f\|_{L^{r,\rho}},\qquad\|f\|_{\dot{W}L^{r,\rho}}=\|(-\Delta)^{s/2}f\|_{L^{r,\rho}}.

For simplicity, when $s=1$ we write $WL^{r,\rho}:=WL^{r,\rho}(\mathbb{R}^{d})$ and $\dot{W}L^{r,\rho}:=\dot{W}L^{r,\rho}(\mathbb{R}^{d})$ .

Lemma 2.7 (Sobolev embedding [13]).

Let $1<r<\infty,1\leq\rho\leq\infty$ and $0<s<\frac{d}{r}$ . Then

\|f\|_{L^{\frac{dr}{d-sr},\rho}(\mathbb{R}^{d})}\lesssim\|(-\Delta)^{s/2}f\|_{L^{r,\rho}(\mathbb{R}^{d})}\quad\text{for any}\quad f\in\dot{W}^{s}L^{r,\rho}(\mathbb{R}^{d}).

Lemma 2.8 (Hardy’s inequality [13]).

Let $1<r<\infty$ , $1\leq\rho\leq\infty$ and $0<s<\frac{d}{r}$ . Then

\||x|^{-s}f\|_{L^{r,\rho}(\mathbb{R}^{d})}\lesssim\|(-\Delta)^{s/2}f\|_{L^{r,\rho}(\mathbb{R}^{d})}\quad\text{for any}\quad f\in\dot{W}^{s}L^{r,\rho}(\mathbb{R}^{d}).

Lemma 2.9 (Product rule I [9]).

Let $s\geq 0,1<r,r_{1},r_{2},r_{3},r_{4},<\infty$ , and $1\leq\rho,\rho_{1},\rho_{2},\rho_{3},\rho_{4}\leq\infty$ be such that

\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}=\frac{1}{r_{3}}+\frac{1}{r_{4}},\qquad\frac{1}{\rho}=\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}=\frac{1}{\rho_{3}}+\frac{1}{\rho_{4}}.

Then for any $f\in\dot{W}^{s}L^{r_{1},\rho_{1}}(\mathbb{R}^{d})\cap L^{r_{3},\rho_{3}}(\mathbb{R}^{d})$ and $g\in\dot{W}^{s}L^{r_{4},\rho_{4}}(\mathbb{R}^{d})\cap L^{r_{2},\rho_{2}}(\mathbb{R}^{d})$ ,

\|(-\Delta)^{s/2}(fg)\|_{L^{r,\rho}}\lesssim\|(-\Delta)^{s/2}f\|_{L^{r_{1},\rho_{1}}}\|g\|_{L^{r_{2},\rho_{2}}}+\|f\|_{L^{r_{3},\rho_{3}}}\|(-\Delta)^{s/2}g\|_{L^{r_{4},\rho_{4}}}.

Lemma 2.10 (Chain rule [1]).

Let $s\in[0,1],F\in C^{1}(\mathbb{C},\mathbb{C})$ and $1<p,p_{1},p_{2}<\infty$ , $1\leq q,q_{1},q_{2}<\infty$ be such that

\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\qquad\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}.

Then

\|(-\Delta)^{s/2}F(f)\|_{L^{p,q}(\mathbb{R}^{d})}\lesssim\|F^{\prime}(f)\|_{L^{p_{1},q_{1}}(\mathbb{R}^{d})}\|(-\Delta)^{s/2}f\|_{L^{p_{2},q_{2}}(\mathbb{R}^{d})}.

Finally, we record several nonlinear estimates in Sobolev spaces that will be used in subsequent analysis.

Lemma 2.11 (Product rule II [8]).

Let $b>0$ , $1<p<\frac{4}{b},$ and $0\leq s<\frac{4}{p}-b$ . For sufficiently small $\epsilon>0$ , we have

\||\nabla|^{s}(|x|^{-b}f)\|_{L^{p}(\mathbb{R}^{4})}\lesssim_{\epsilon}\Big{[}\||\nabla|^{s}f\|_{L^{q^{+}}(\mathbb{R}^{4})}\||\nabla|^{s}f\|_{L^{q^{-}}(\mathbb{R}^{4})}\Big{]}^{\frac{1}{2}},

where $\frac{1}{q^{\pm}}:=\frac{1}{p}-\frac{b\pm\epsilon}{4}$ .

Lemma 2.12 (Paraproduct estimate [44]).

Let $0<s<1$ . If $1<r<r_{1}<\infty$ and $1<r_{2}<\infty$ satisfy

\frac{1}{r_{1}}+\frac{1}{r_{2}}=\frac{1}{r}+\frac{s}{4}<1,

then

\||\nabla|^{-s}(fg)\|_{L^{r}(\mathbb{R}^{4})}\lesssim\||\nabla|^{-s}f\|_{L^{r_{1}}(\mathbb{R}^{4})}\||\nabla|^{s}g\|_{L^{r_{2}}(\mathbb{R}^{4})}.

Lemma 2.13.

Let $F$ be a Hölder continuous function of order $0<\alpha<1$ . Then, for every $0<\sigma<\alpha$ , $1<p<\infty$ , and $\sigma/\alpha<s<1$ , we have

\||\nabla|^{\sigma}F(u)\|_{L^{p}}\lesssim\||u|^{\alpha-\sigma/s}\|_{L^{p_{1}}}\||\nabla|^{s}u\|_{L^{(\sigma/s)p_{2}}}^{\sigma/s},

provided $1/p=1/p_{1}+1/p_{2}$ and $(1-\sigma/(\alpha s))p_{1}>1$ .

2.3. Strichartz estimate and stability.

In this subsection, we recall Strichartz estimate in the Lorentz space and the stability results of the Cauchy probelm (1.1).

Definition 2.14 (Admissibility).

A pair $(q,r)$ is said to be Schrödinger admissible, for short $(q,r)\in\Lambda$ , where

\Lambda=\left\{(q,r):2\leq q,r\leq\infty,\ \frac{2}{q}+\frac{4}{r}=2\right\}.

The following result is a key tool for our work. It is established in [37, Theorem 10.1].

Proposition 2.15 (Strichartz estimates).

•

Let $(q,r)\in\Lambda$ with $r<\infty$ . Then for any $f\in L^{2}(\mathbb{R}^{4})$

\displaystyle\|e^{it\Delta}f\|_{L^{q}_{t}L^{r,2}_{x}(\mathbb{R}\times\mathbb{R}^{4})}\lesssim\|f\|_{L^{2}_{x}(\mathbb{R}^{4})}.

•

Let $(q_{1},r_{1}),(q_{2},r_{2})\in\Lambda$ with $r_{1},r_{2}<\infty$ , $t_{0}\in\mathbb{R}$ and $I\subset\mathbb{R}$ be an interval containing $t_{0}$ . Then for any $F\in L_{t}^{q_{2}^{\prime}}L^{r_{2}^{\prime},2}_{x}(I\times\mathbb{R}^{4})$

\displaystyle\left\|\int_{t_{0}}^{t}e^{i(t-\tau)\Delta}F(\tau)d\tau\right\|_{L^{q_{1}}_{t}L^{r_{1},2}_{x}(I\times\mathbb{R}^{4})}\lesssim\|F\|_{L^{q_{2}^{\prime}}_{t}L^{r_{2}^{\prime},2}_{x}(I\times\mathbb{R}^{4})}.

Using Strichartz estimates and standard continuous argument, we can get the classical stability theory for equation (1.1) (see [7] for more details).

Theorem 2.16 (Stability).

Let $1<s_{c}<2$ , $0<b<\min\left\{(s_{c}-1)^{2}+1,3-s_{c}\right\}$ and $\alpha>0$ be such that $s_{c}=2-\frac{2-b}{\alpha}$ . Suppose $I$ is a compact time interval and $\tilde{u}:I\times\mathbb{R}^{4}\to\mathbb{C}$ is an approximate solution to (1.1) in the sense that

(i\partial_{t}+\Delta)\tilde{u}=F(\tilde{u})+e

for some function $e$ . Assume that

\|\tilde{u}\|_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}\leq E,\quad\|\widetilde{u}\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4\alpha(\alpha+1)}{\alpha+2-b(\alpha+1)},2}(I\times\mathbb{R}^{4})}\leq L

for some $E>0$ and $L>0$ . Let $t_{0}\in I$ and $u_{0}\in\dot{H}^{s_{c}}(\mathbb{R}^{4})$ . Then there exists $\varepsilon_{0}=\varepsilon_{0}(E,L)$ such that if

\|u_{0}-\tilde{u}(t_{0})\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}\leq\varepsilon,\quad\||\nabla|^{s_{c}}e\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}\leq\varepsilon

for some $0<\varepsilon<\varepsilon_{0}$ , then there exists a solution $u:I\times\mathbb{R}^{4}\to\mathbb{C}$ to (1.1) with $u(t_{0})=u_{0}$ satisfying

\||\nabla|^{s_{c}}(u-\widetilde{u})\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4(\alpha+1)}{2\alpha+1},2}(I\times\mathbb{R}^{4})}\lesssim_{E,L}\varepsilon.

Following the standard arguments presented in [7], to establish the stability theorem 2.16, it suffices to prove a short-time perturbation result. The proof of this short-time perturbation result, in turn, relies on the following nonlinear estimate:

Lemma 2.17.

Let $1<s_{c}<2,0<b<\min\left\{(s_{c}-1)^{2}+1,3-s_{c}\right\}$ and $\alpha>0$ be such that $s_{c}=2-\frac{2-b}{\alpha}$ . Let $F(u)=|u|^{\alpha}u$ and $(\gamma,\rho)=(2(\alpha+1),\frac{4(\alpha+1)}{2\alpha+1})$ . Then for any $u,v\in L_{t}^{\gamma}\dot{W}^{s_{c}}L_{x}^{\rho,2}(I\times\mathbb{R}^{d})$ , the following estimate holds:

\displaystyle\||\nabla|^{s_{c}}[|x|^{-b}(F(u+v)-F(u))]\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}\lesssim M_{s_{c}}(u,v),

where we have set

M_{s_{c}}(u,v):=\left(\||\nabla|^{s_{c}}u\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}^{\alpha}+\||\nabla|^{s_{c}}v\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}^{\alpha}\right)\||\nabla|^{s_{c}}v\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}.

Remark 2.18.

The assumption $b<(s_{c}-1)^{2}+1$ is equivalent to $s_{c}<\alpha$ . This condition ensures that $F^{\prime}(u)$ possesses enough regularity when applying the operator $|\nabla|^{s_{c}}$ .

Proof.

By direct computation, we have

	$\displaystyle\nabla[\|x\|^{-b}(F(u+v)-F(u))]$
	$\displaystyle=\|x\|^{-b-1}[F(u+v)-F(u)]+\|x\|^{-b}[F^{\prime}(u+v)-F^{\prime}(u)]\nabla u+\|x\|^{-b}F^{\prime}(u+v)\nabla v$
	$\displaystyle:=A_{1}(t,x)+A_{2}(t,x)+A_{3}(t,x).$

We first estimate $A_{1}(t,x)$ . Note that $\rho=\frac{4(\alpha+1)}{2\alpha+1}$ satisfies $\frac{3}{4}=\alpha(\frac{1}{\rho}-\frac{s_{c}}{4})+\frac{1}{\rho}+\frac{b}{4}.$ Using Lemma 2.9, Lemma 2.10, and Sobolev embedding, we obtain

	$\displaystyle\\|\|\nabla\|^{s_{c}-1}A_{1}(t,x)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\\|F(u+v)-F(u)\\|_{L_{t}^{2}L_{x}^{l_{1},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-s_{c}}\\|_{L_{x}^{\frac{4}{b+s_{c}},\infty}}$
	$\displaystyle\quad+\\|\|\nabla\|^{s_{c}-1}\int_{0}^{1}F^{\prime}(u+\lambda v)vd\lambda\\|_{L_{t}^{2}L_{x}^{l_{2},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-1}\\|_{L_{x}^{\frac{4}{b+1},\infty}}$
	$\displaystyle\lesssim M_{s_{c}}(u,v),$

where $\frac{1}{l_{1}}:=\frac{3-b-s_{c}}{4}=(\alpha+1)(\frac{1}{\rho}-\frac{s_{c}}{4})$ and $\frac{1}{l_{2}}:=\frac{2-b}{4}=\alpha(\frac{1}{\rho}-\frac{s_{c}}{4})+\frac{1}{\rho}-\frac{1}{4}$ .

Next, we consider $A_{2}(t,x)$ . By Lemma 2.9, Lemma 2.10, and the Sobolev embedding again, we have

	$\displaystyle\\|\|\nabla\|^{s_{c}-1}A_{2}(t,x)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\\|[F^{\prime}(u+v)-F^{\prime}(u)]\nabla u\\|_{L_{t}^{2}L_{x}^{l_{3},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-s_{c}+1}\\|_{L_{x}^{\frac{4}{b+s_{c}-1},\infty}}$
	$\displaystyle\quad+\\|\|\nabla\|^{s_{c}-1}\int_{0}^{1}F^{\prime\prime}(u+\lambda v)vd\lambda\\|_{L_{t}^{\frac{\gamma}{\alpha}}L_{x}^{l_{4}}(I\times\mathbb{R}^{4})}\,\\|\|\nabla\|^{s_{c}}u\\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-4}\\|_{L_{x}^{\frac{4}{b},\infty}},$		(2.1)

where $\frac{1}{l_{3}}:=\frac{4-b-s_{c}}{4}=\alpha(\frac{1}{\rho}-\frac{s_{c}}{4})+\frac{1}{\rho}-\frac{s_{c}-1}{4}$ and $\frac{1}{l_{4}}:=\frac{2-b+s_{c}}{4}-\frac{1}{\rho}=(\alpha-1)(\frac{1}{\rho}-\frac{s_{c}}{4})+\frac{1}{\rho}-\frac{1}{4}$ .

When $\alpha\geq 2$ , $F^{\prime\prime}(u)$ is Lipschitz continuous. Therefore, by applying Lemma 2.10, Hölder and Sobolev embedding to estimate the right-hand side of (2.1), we obtain

\||\nabla|^{s_{c}-1}A_{2}(t,x)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}\lesssim M_{s_{c}}(u,v).

(2.2)

In the case $1<\alpha<2$ , $F^{\prime\prime}(u)$ is only Hölder continuous of order $\alpha-1$ . Note that

\frac{s_{c}-1}{\alpha-1}<s_{c}\iff b<2s_{c}+\frac{2}{s_{c}}-3,

and $2s_{c}+\frac{2}{s_{c}}-3>1+(s_{c}-1)^{2}$ for $s_{c}\in(1,2)$ . Therefore, by applying Lemma 2.13, Hölder and Sobolev embedding to estimate (2.1), we obtain (2.2).

Similarly, by the same argument as that used to estimae $A_{2}(t,x)$ , we get

\displaystyle\||\nabla|^{s_{c}-1}A_{3}(t,x)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}\lesssim M_{s_{c}}(u,v).

Combining the above estimates, we obtain the desired estimate in Lemma 2.17. ∎

2.4. The construction of scattering solutions away from the origin.

In this section, as explained in Introduction, to prove Theorem 1.7, we only need to establish a key proposition about the existence of scattering solutions to (1.1) associated to initial data living sufficiently far from the origin.

Proposition 2.19.

Let $\lambda_{n}\in(0,\infty)$ , $x_{n}\in\mathbb{R}^{4}$ , satisfy $\lim_{n\rightarrow\infty}\frac{|x_{n}|}{\lambda_{n}}=\infty$ . Let $\phi\in\dot{H}^{s_{c}}(\mathbb{R}^{4})$ . Then for sufficiently large $n$ , there exists a global scattering solution $u_{n}(t,x)$ to equation (1.1) such that

u_{n}(0)=\lambda_{n}^{s_{c}-2}\phi(\frac{x-x_{n}}{\lambda_{n}})\quad\text{and}\quad\||\nabla|^{s_{c}}u_{n}\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}\lesssim\|\phi\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})},

(2.3)

where $(\gamma,\rho)=(2(\alpha+1),\frac{4(\alpha+1)}{2\alpha+1})$ .

Proof.

By scalling, it sufficies to consider the case $\lambda_{n}\equiv 1$ . Define $v_{n}(t,x):=e^{it\Delta}\phi(x-x_{n})$ . Then $v_{n}(t,x)$ solves the perturbed equation

i\partial_{t}v_{n}(t,x)+\Delta v_{n}(t,x)=|x|^{-b}|v_{n}(t,x)|^{\alpha}v_{n}(t,x)+e_{n}(t,x),

where $e_{n}(t,x):=-|x|^{-b}|v_{n}(t,x)|^{\alpha}v_{n}(t,x)$ .

By Strichartz, it is easy to see that $e_{n}(t,x)$ is bounded in $L_{t}^{\infty}\dot{H}^{s_{c}}_{x}(I\times\mathbb{R}^{4})\cap L_{t}^{\gamma}\dot{W}^{s_{c}}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})$ . To apply the stability theorem 2.16, it suffices to show that

\lim_{n\rightarrow\infty}\||\nabla|^{s_{c}}e_{n}(t,x)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}=0.

(2.4)

Once this is established, the existence of the desired solution $u_{n}(t,x)$ satisfying (2.3) follows directly from the stability theorem 2.16.

We now proceed to prove estimate (2.4). For any $\varepsilon>0$ , we first choose a function $\psi(t,x)\in C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{4})$ such that

\||\nabla|^{s_{c}}[\psi(t,x)-e^{it\Delta}\phi(x)]\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}<\varepsilon.

Fix $T>0$ . We first estimate $e_{n}$ on the time interval $[-T,T]$ . Let

\Phi_{n}(t,x):=|x|^{-b}|\psi(t,x-x_{n})|^{\alpha}\psi(t,x-x_{n}).

Then, by Hölder, Strichartz and Sobolev embedding, we have

	$\displaystyle\\|\|\nabla\|^{s_{c}}[e_{n}(t,x)-\Phi_{n}(t,x)]\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([-T,T]\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\left(\\|\|\nabla\|^{s_{c}}e^{it\Delta}\phi\\|_{L_{t}^{\gamma}L_{x}^{\rho,2}}^{\alpha}+\\|\|\nabla\|^{s_{c}}\psi(t,x)\\|_{L_{t}^{\gamma}L_{x}^{\rho,2}}^{\alpha}\right)\\|\|\nabla\|^{s_{c}}[\psi(t,x)-e^{it\Delta}\phi(x)]\\|_{L_{t}^{\gamma}L_{x}^{\rho,2}}\lesssim\varepsilon.$		(2.5)

On the other hand, since the function $\psi(t,x)$ has compact support in space and $|x_{n}|\rightarrow\infty$ as $n\rightarrow\infty$ , it follows that

\lim_{n\rightarrow\infty}\||\nabla|^{s_{c}}\Phi_{n}(t,x)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([-T,T]\times\mathbb{R}^{4})}=0.

(2.6)

Combining (2.5) and (2.6), we obtain

\limsup_{n\rightarrow\infty}\||\nabla|^{s_{c}}e_{n}(t,x)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([-T,T]\times\mathbb{R}^{4})}\lesssim\varepsilon.

(2.7)

We now estimate $e_{n}$ on the set $\{t:|t|>T\}$ . Again by Hölder’s inequality, we have

\||\nabla|^{s_{c}}e_{n}\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(\{|t|>T\}\times\mathbb{R}^{4})}\lesssim\||\nabla|^{s_{c}}e^{it\Delta}\phi\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(\{|t|>T\}\times\mathbb{R}^{4})}^{\alpha+1},

(2.8)

which tends to zero as $T\to\infty$ , by the Strichartz estimate and the monotone convergence theorem. The desired estimate (2.4) then follows from (2.7) and (2.8). This completes the proof of Proposition 2.19. ∎

3. The case $1<s_{c}<\frac{3}{2}$

In this section, we rule out the existence of almost periodic solutions as described in Theorem 1.7 for the case $1<s_{c}<3/2$ . The argument relies on a space-localized Morawetz inequality, similar to the method employed by Bourgain [3] in the study of the radial energy-critical NLS.

Lemma 3.1 (Morawetz inequality).

Let $1<s_{c}<\frac{3}{2}$ , and let $u$ be a solution to (1.1) on the time interval $I$ . Then, for any $A\geq 1$ , the following estimate holds:

\int_{I}\int_{|x|\leq A|I|^{1/2}}\frac{|u(t,x)|^{\alpha+2}}{|x|^{1+b}}\,dx\,dt\lesssim(A|I|^{\frac{1}{2}})^{2s_{c}-1}\left\{\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}^{2}+\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}^{\alpha+2}\right\}.

(3.1)

Proof.

Let $\phi(x)$ be a smooth, radial bump function such that $\phi(x)=1$ for $|x|\leq 1$ and $\phi(x)=0$ for $|x|>2$ . Let $R>1$ and $a(x):=|x|\phi\left(\frac{x}{R}\right)$ .

Define the Morawetz potential as

M(t):=2\text{Im}\int\partial_{j}a(x)\overline{u}(t,x)\partial_{j}u(t,x)\,dx,

where we sum the repeated indices. Using equation (1.1), direct computation gives

	$\displaystyle\frac{d}{dt}M(t)$	$\displaystyle=4\text{Re}\int\partial_{j}\partial_{k}a\partial_{j}\overline{u}\partial_{k}u\,dx-\int\|u\|^{2}\Delta^{2}a\,dx$
		$\displaystyle\qquad+\frac{2\alpha}{\alpha+2}\int\|x\|^{-b}\|u\|^{\alpha+2}\Delta a\,dx+\frac{4b}{\alpha+2}\int\|x\|^{-b-2}\|u\|^{\alpha+2}x_{j}\partial_{j}a\,dx.$

For $|x|\leq R$ , note that $\partial_{j}\partial_{k}a$ is positive definite, and $\Delta a(x)=3/|x|$ . This implies

$\displaystyle\frac{d}{dt}M(t)$	$\displaystyle\geq\frac{6\alpha+4b}{\alpha+2}\int_{\|x\|<R}\|x\|^{-b-1}\|u\|^{\alpha+2}\,dx+4\text{Re}\int_{\|x\|>R}\partial_{j}\partial_{k}a\partial_{j}\overline{u}\partial_{k}u\,dx$
	$\displaystyle\quad-\int_{\|x\|>R}\|u\|^{2}\Delta^{2}a\,dx+\frac{2\alpha}{\alpha+2}\int_{\|x\|>R}\|x\|^{-b}\|u\|^{\alpha+2}\Delta a\,dx$
	$\displaystyle\quad+\frac{4b}{\alpha+2}\int_{\|x\|>R}\|x\|^{-b-2}\|u\|^{\alpha+2}x_{j}\partial_{j}a\,dx.$	(3.2)

Using Hölder’s inequality and Sobolev embedding, we estimate

	$\displaystyle\int_{\|x\|>R}\|\partial_{j}\partial_{k}a\partial_{j}\overline{u}\partial_{k}u\|\,dx+\int_{\|x\|>R}\|u\|^{2}\|\Delta^{2}a\|\,dx$
	$\displaystyle\lesssim\\|\Delta a\\|_{L_{x}^{\frac{2}{s_{c}-1}}(\|x\|>R)}\\|\nabla u\\|_{L_{x}^{\frac{4}{3-s_{c}}}(\mathbb{R}^{4})}^{2}+\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{2}\\|\Delta^{2}a\\|_{L_{x}^{\frac{2}{s_{c}}}(\mathbb{R}^{4})}$
	$\displaystyle\lesssim R^{2s_{c}-3}\\|u\\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}^{2},$		(3.3)

and

	$\displaystyle\int_{\|x\|>R}\|x\|^{-b}\|u\|^{\alpha+2}\|\Delta a\|\,dx+\int_{\|x\|>R}\|x\|^{-b-2}\|u\|^{\alpha+2}\|x_{j}\partial_{j}a\|\,dx$
	$\displaystyle\lesssim R^{-b}\\|\Delta a\\|_{L_{x}^{\frac{4}{2(s_{c}-1)+b}}(\|x\|>R)}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}+R^{-b-1}\\|\nabla a\\|_{L_{x}^{\frac{4}{2(s_{c}-1)+b}}(\|x\|>R)}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}$
	$\displaystyle\lesssim R^{2s_{c}-3}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}\lesssim R^{2s_{c}-3}\\|u\\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}^{\alpha+2}.$		(3.4)

On the other hand, Hölder’s inequality together with Sobolev embedding yields the following bound for $M(t)$ :

|M(t)|\lesssim\|u\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}\|\nabla u\|_{L_{x}^{\frac{4}{3-s_{c}}}(\mathbb{R}^{4})}\|\nabla a\|_{L_{x}^{\frac{4}{2s_{c}-1}}(\mathbb{R}^{4})}\lesssim R^{2s_{c}-1}\|u\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}^{2}.

(3.5)

Substituting (3.3)–(3.5) into (3.2) and integrating over the time interval $I$ , we obtain

\int_{I}\int_{|x|\leq R}\frac{|u(t,x)|^{\alpha+2}}{|x|^{1+b}}\,dx\,dt\lesssim R^{2s_{c}-1}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}^{2}+R^{2s_{c}-3}|I|\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{4})}^{\alpha+2}.

Taking $R=A|I|^{\frac{1}{2}}$ gives the desired estimate (3.1). ∎

Now, we use the spatially localized Morawetz inequality (Lemma 3.1) to rule out the existence of the almost periodic solutions in Theorem 1.7 with $1<s_{c}<\frac{3}{2}$ .

Theorem 3.2.

Let $1<s_{c}<\frac{3}{2}$ . There are no global radial almost periodic solutions in Theorem 1.7 satisfying $T_{\text{max}}=+\infty$ .

Proof.

We argue by contradiction. Assume that $u$ is a almost periodic solution in Theorem 1.7 and satisfies $T_{\text{max}}=+\infty$ . For the almost periodic solution $u$ , we can use the compactness property (1.5) to deduce the following inequality (a detailed proof can be found in [44, (7.4)]):

\inf_{t\in[0,+\infty)}N(t)^{2s_{c}}\int_{|x|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{2}\,dx\gtrsim_{u}1.

(3.6)

Let $I\subset[0,\infty)$ be the union of contiguous characteristic subspaces $J_{k}$ . Choose $C(u)$ sufficiently large. Using the assumption (1.3), the spatially localized Morawetz inequality (Lemma 3.1), Hölder’s inequality, and (3.6), we obtain:

	$\displaystyle\|I\|^{s_{c}-\frac{1}{2}}$	$\displaystyle\gtrsim_{u}\sum_{J_{k}\subset I}\int_{J_{k}}\int_{\|x\|\leq C(u)\|J_{k}\|^{\frac{1}{2}}}\frac{\|u(t,x)\|^{\alpha+2}}{\|x\|^{1+b}}\,dx\,dt$
		$\displaystyle\gtrsim_{u}\sum_{J_{k}\subset I}N_{k}^{1+b}\int_{J_{k}}\int_{\|x\|\leq C(u)N_{k}^{-1}}\|u(t,x)\|^{\alpha+2}\,dx\,dt$
		$\displaystyle\gtrsim_{u}\sum_{J_{k}\subset I}\int_{J_{k}}N_{k}^{1+b+2\alpha}\left(\int_{\|x\|\leq C(u)N_{k}^{-1}}\|u(t,x)\|^{2}\,dx\right)^{\frac{\alpha+2}{2}}\,dt$
		$\displaystyle\gtrsim_{u}\sum_{J_{k}\subset I}\int_{J_{k}}N_{k}^{1+b+2\alpha}\left(N(t)^{-2s_{c}}\right)^{\frac{\alpha+2}{2}}\,dt$
		$\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-2s_{c}}\,dt\gtrsim_{u}\|I\|,$

where, in the last inequality, we use the fact that $s_{c}<\frac{3}{2}$ and $N(t)\geq 1$ . If we take the length of the interval $I$ to be sufficiently large in the inequalities above, we reach a contradiction. Thus, Theorem 3.2 is proved. ∎

4. The case $\frac{3}{2}\leq s_{c}<2$ : Rapid frequency-cascade

This section is devoted to ruling out the rapid frequency-cascade solutions as described in Theorem 1.12 for the case $\frac{3}{2}\leq s_{c}<2$ . The main technical tools used are the long-time Strichartz estimate and a frequency-localized Morawetz estimate.

4.1. Long-time Strichartz estimates

Here, we establish long-time Strichartz estimates tailored to the Lin-Strauss Morawetz inequality. These estimates were originally introduced by Dodson [14] in the context of the mass-critical NLS. See also [41, 43, 48] for further developments that are more closely related to our setting.

Theorem 4.1 (Long-time Strichartz estimate).

u\in L_{t}^{\infty}([0,T_{\text{max}}),\dot{H}_{x}^{s}(\mathbb{R}^{4})),\quad\text{for some }s_{c}-\frac{1}{2}<s\leq s_{c}.

Then, on any compact time interval $I\subset[0,T_{\text{max}})$ , which is a union of characteristic subintervals $J_{k}$ , and for any $N>0$ , the following holds:

\||\nabla|^{s}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}\lesssim_{u}1+N^{\sigma(s)}K_{I}^{1/2},

(4.1)

where $K_{I}:=\int_{I}N(t)^{3-2s_{c}}\,dt$ and $\sigma(s):=2s_{c}-s-\frac{1}{2}$ . Specifically, for $s=s_{c}$ , we have

\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}\lesssim_{u}1+N^{s_{c}-\frac{1}{2}}K_{I}^{1/2}.

(4.2)

Moreover, for any $\eta>0$ , there exists $N_{0}=N_{0}(\eta)$ such that for all $N\leq N_{0}$ ,

\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}\lesssim_{u}\eta(1+N^{s_{c}-\frac{1}{2}}K_{I}^{1/2}).

(4.3)

The implicit constants in (4.1) and (4.3) are independent of $I$ .

By Lemma 1.11, it is straightforward to verify that Theorem 4.1 holds for sufficiently large $N$ . For general $N>0$ , we proceed by induction. Following the argument in the proof of [Proposition 4.1, [48]] or [Theorem 3.1, [43]], it suffices to prove the following iterative lemma:

Lemma 4.2 (Iterative formula of $B(N)$ ).

Let $B(N):=\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}$ . For any $\epsilon,\ \epsilon_{0}>0$ , there exists a constant $C(\varepsilon,\varepsilon_{0})>0$ such that

	$\displaystyle B(N)\lesssim_{u}$	$\displaystyle\inf_{t\in I}\\|\|\nabla\|^{s_{c}}u_{\leq N}(t)\\|_{L_{x}^{2}}+C(\varepsilon,\varepsilon_{0})N^{\sigma(s)}K_{I}^{\frac{1}{2}}$
		$\displaystyle\quad+\varepsilon^{\alpha}B(N/\varepsilon_{0})+\sum_{M>N/\varepsilon_{0}}\Big{(}\frac{N}{M}\Big{)}^{s_{c}}B(M)$		(4.4)

holds uniformly in $N$ .

Proof.

First, by Strichartz estimate, we have

B(N)\lesssim\inf_{t\in I}\||\nabla|^{s_{c}}u_{\leq N}(t)\|_{L_{x}^{2}}+\||\nabla|^{s_{c}}P_{\leq N}|x|^{-b}F(u)\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([0,T_{\max})\times\mathbb{R}^{4})}.

(4.5)

To estimate the nonlinear term, we decompose

F(u)=F(u_{\leq N/\epsilon_{0}})+(F(u)-F(u_{\leq N/\epsilon_{0}})).

Using Bernstein’s inequality, Hölder’s inequality, Sobolev embedding, and the bound (1.3), we obtain

	$\displaystyle\\|\|\nabla\|^{s_{c}}P_{\leq N}\|x\|^{-b}(F(u)-F(u_{\leq N/\epsilon_{0}}))\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}$
$\displaystyle\lesssim$	$\displaystyle N^{s_{c}}\\|\|x\|^{-b}\\|_{L_{x}^{\frac{4}{b},\infty}(\mathbb{R}^{4})}\\|(F(u)-F(u_{\leq N/\epsilon_{0}}))\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}(I\times\mathbb{R}^{4})}$
$\displaystyle\lesssim$	$\displaystyle N^{s_{c}}\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{2-b}}(I\times\mathbb{R}^{4})}^{\alpha}\\|u_{>N/\epsilon_{0}}\\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}$
$\displaystyle\lesssim$	$\displaystyle\sum_{M>N/\epsilon_{0}}\Big{(}\frac{N}{M}\Big{)}^{s_{c}}B(M).$	(4.6)

We now turn to estimating $F(u_{\leq N/\varepsilon_{0}})$ . To begin, we focus on a single characteristic interval $J_{k}$ . By applying Lemma 2.9, the Sobolev embedding, and the bound (1.3), we obtain the following estimate

		$\displaystyle\\|\|\nabla\|^{s_{c}}P_{\leq N}\|x\|^{-b}F(u_{\leq N/\epsilon_{0}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(J_{k}\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|\|x\|^{-b}\\|_{L_{x}^{\frac{4}{b},\infty}(\mathbb{R}^{4})}\\|\|\nabla\|^{s_{c}}F(u_{\leq N/\epsilon_{0}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}(J_{k}\times\mathbb{R}^{4})}$
		$\displaystyle+\\|\|x\|^{-b-s_{c}}\\|_{L_{x}^{\frac{4}{b+s_{c}},\infty}(\mathbb{R}^{4})}\\|F(u_{\leq N/\epsilon_{0}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b-s_{c}},2}(J_{k}\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{2-b}}(I\times\mathbb{R}^{4})}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{2}L_{x}^{4,2}(J_{k}\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|\|\nabla\|^{s_{c}}P_{\leq c(\epsilon)N_{k}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{4})}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{2}L_{x}^{4,2}(J_{k}\times\mathbb{R}^{4})}$
		$\displaystyle+\\|\|\nabla\|^{s_{c}}P_{>c(\epsilon)N_{k}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{4})}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{2}L_{x}^{4,2}(J_{k}\times\mathbb{R}^{4})}$
	$\displaystyle:=$	$\displaystyle I_{k}+II_{k}.$

By the compactness property, there exists $c=c(\epsilon)$ such that $\||\nabla|^{s_{c}}u_{\leq c(\epsilon)N(t)}\|_{L_{t}^{\infty}L_{x}^{2}}<\epsilon$ , and thus

I_{k}\lesssim\epsilon^{\alpha}\||\nabla|^{s_{c}}u_{\leq N/\epsilon_{0}}\|_{L_{t}^{2}L_{x}^{4,2}(J_{k}\times\mathbb{R}^{4})}.

For the term $II_{k}$ , we only need to consider the case $c(\epsilon)N_{k}\leq N/\epsilon_{0}$ . By Bernstein’s inequality and Lemma 1.11, we obtain

	$\displaystyle II_{k}\lesssim$	$\displaystyle\Big{(}\frac{N}{\epsilon_{0}c(\epsilon)N_{k}}\Big{)}^{s_{c}-\frac{1}{2}}\\|\|\nabla\|^{s_{c}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\Big{(}\frac{N}{\epsilon_{0}c(\epsilon)N_{k}}\Big{)}^{s_{c}-\frac{1}{2}}\Big{(}\frac{N}{\epsilon_{0}}\Big{)}^{s_{c}-s}\\|u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}\dot{H}^{s}(J_{k}\times\mathbb{R}^{4})}.$

Summing over all characteristic intervals, we deduce

		$\displaystyle\\|\|\nabla\|^{s_{c}}P_{\leq N}\|x\|^{-b}F(u_{\leq N/\epsilon_{0}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}^{2}$
	$\displaystyle\lesssim$	$\displaystyle\sum_{J_{k}\subset I}\Big{[}\epsilon^{2\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{2}L_{x}^{4,2}(J_{k}\times\mathbb{R}^{4})}^{2}$
		$\displaystyle\qquad\quad+\Big{(}\frac{N}{\epsilon_{0}c(\epsilon)N_{k}}\Big{)}^{2s_{c}-1}\Big{(}\frac{N}{\epsilon_{0}}\Big{)}^{2s_{c}-2s}\\|u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}\dot{H}^{s}(J_{k}\times\mathbb{R}^{4})}^{2}\Big{]}$
	$\displaystyle\lesssim$	$\displaystyle\epsilon^{2\alpha}B^{2}(N/\epsilon_{0})+c(\varepsilon)^{-2s_{c}+1}\epsilon_{0}^{-2\sigma(s)}\sup_{J_{k}}\\|u_{\leq N/\epsilon_{0}}\\|_{L_{t}^{\infty}\dot{H}^{s}(J_{k}\times\mathbb{R}^{4})}^{2}N^{2\sigma}K_{I},$

which together with (4.5) yields (4.2). This completes the proof of the lemma. ∎

4.2. The rapid frequency-cascade scenario

In this subsection, we use the long-time Strichartz estimate proved in the previous section to rule out the existence of rapid frequency-cascade solutions, i.e. the almost periodic solutions as described in Theorem 1.12 such that

\int_{0}^{T_{\text{max}}}N(t)^{3-2s_{c}}\,dt<\infty.

(4.7)

Theorem 4.3 (No rapid frequency-cascades).

Let $s_{c}\geq\frac{3}{2}$ . Then there are no almost periodic solutions $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{\text{max}})$ such that $u$ blows up forward in time and

K:=\int_{0}^{T_{\text{max}}}N(t)^{3-2s_{c}}dt<+\infty.

(4.8)

The proof proceeds by contradiction. Suppose such a solution $u$ exists. From (4.8) and Corollary 1.7, it follows that

\lim_{t\to T_{\text{max}}}N(t)=\infty,

(4.9)

whether $T_{\text{max}}$ is finite or infinite. Together with the compactness property, this implies

\displaystyle\lim_{t\to T_{\text{max}}}\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{x}^{2}(\mathbb{R}^{4})}=0,

(4.10)

for any $N>0$ .

To proceed, we require the following Proposition 4.4 and Proposition 4.5. These results show that an almost periodic solution satisfying (4.8) must exhibit negative regularity. In the final part of this subsection, we will exploit this negative regularity to derive a contradiction, thereby concluding the proof of Theorem 4.3.

Proposition 4.4 (Lower regularity).

Let $\frac{3}{2}\leq s_{c}<2$ and let $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ be an almost periodic solution to (1.1) satisfying (4.8). Suppose that

u\in L_{t}^{\infty}([0,T_{\text{max}}),\dot{H}_{x}^{s}(\mathbb{R}^{4})),\quad\text{for some }s_{c}-\frac{1}{2}<s\leq s_{c}.

Then it follows that

u\in L_{t}^{\infty}([0,T_{\text{max}}),\dot{H}_{x}^{\beta}(\mathbb{R}^{4})),\quad\forall s-\sigma(s)<\beta\leq s_{c},

where $\sigma(s)=2s_{c}-s-\frac{1}{2}$ .

The proof of Proposition 4.4 follows a similar strategy to that of [Lemma 4.3, [48]] and [Proposition 4.2, [43]], where their arguments crucially rely on the negative derivative estimates provided by Lemma 2.12. However, such estimates have not yet been established in the Lorentz space setting due to the lack of a Coifman–Meyer multiplier theorem in this framework. To circumvent this issue, we first apply embedding theorems to reduce the Lorentz norms to Lebesgue norms, and then invoke the negative derivative estimates from Lemma 2.12. Subsequently, to handle the inhomogeneous coefficient in the Lebesgue space setting, we will make use of the nonlinear estimates in Lemma 2.11.

Proof of Proposition 4.4.

Let $I_{n}\subset[0,T_{max})$ be unions of characteristic subintervals, each consisting of a finite number of contiguous characteristic subintervals. We first claim that for any $N>0$ , the following estimate holds:

\displaystyle B_{n}(N)\lesssim_{u}\inf_{t\in I_{n}}\||\nabla|^{s_{c}}u_{\leq N}(t)\|_{L_{x}^{2}}+N^{\sigma(s)},

(4.11)

where $B_{n}(N):=\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I_{n}\times\mathbb{R}^{4})}.$

In fact, by the same argument as that used to derive (4.2), we obtain

	$\displaystyle B_{n}(N)\lesssim_{u}$	$\displaystyle\inf_{t\in I_{n}}\\|\|\nabla\|^{s_{c}}u_{\leq N}(t)\\|_{L_{x}^{2}}+C(\varepsilon,\varepsilon_{0})N^{\sigma(s)}$
		$\displaystyle\quad+\varepsilon^{\alpha}B_{n}(N/\varepsilon_{0})+\sum_{M>N/\varepsilon_{0}}\Big{(}\frac{N}{M}\Big{)}^{s_{c}}B_{n}(M).$

The estimate (4.11) then follows from a standard iterative argument.

Letting $n\to\infty$ in (4.11) and applying (4.10), we get

\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}([0,T_{\max})\times\mathbb{R}^{4})}\lesssim_{u}N^{\sigma(s)},\quad\forall\ N\geq 0.

(4.12)

In what follows, we use (4.12) to prove that

\||\nabla|^{s}u_{\leq N}\|_{L_{t}^{\infty}L_{x}^{2}([0,T_{\max})\times\mathbb{R}^{4})}\lesssim N^{\sigma(s)}.

(4.13)

First, by the no-waste Duhamel formula (Lemma 1.13) and the Strichartz estimate, we obtain

\displaystyle\||\nabla|^{s}u_{\leq N}\|_{L_{t}^{\infty}L_{x}^{2}([0,T_{\max})\times\mathbb{R}^{4})}\lesssim\||\nabla|^{s}P_{\leq N}(|x|^{-b}F(u))\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([0,T_{\max})\times\mathbb{R}^{4})}.

Then we decompose $F(u)$ as

F(u)=F(u_{\leq N})+(F(u)-F(u_{\leq N})).

For the first term, using Bernstein, Hölder and estimate (4.12), we obtain

		$\displaystyle\\|\|\nabla\|^{s}P_{\leq N}\|x\|^{-b}F(u_{\leq N})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|\|x\|^{-b}\\|_{L_{x}^{\frac{4}{b},\infty}}\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{s_{c}-s+2-b}}([0,T_{\max})\times\mathbb{R}^{4})}^{\alpha}\\|\|\nabla\|^{s}u_{\leq N}\\|_{L_{t}^{2}L_{x}^{\frac{4}{1-s_{c}+s},2}([0,T_{\max})\times\mathbb{R}^{4})}$
		$\displaystyle+\\|\|x\|^{-b}\\|_{L_{x}^{\frac{4}{b},\infty}}\\|\|x\|^{-s}F(u_{\leq N})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|u\\|_{L_{t}^{\infty}\dot{H}^{s_{1}}}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N}\\|_{L_{t}^{2}L_{x}^{4,2}([0,T_{\max})\times\mathbb{R}^{4})}+\\|\|\nabla\|^{s}F(u_{\leq N})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle\\|u\\|_{L_{t}^{\infty}\dot{H}^{s_{1}}}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N}\\|_{L_{t}^{2}L_{x}^{4,2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle N^{\sigma(s)},$

where $s_{1}:=2-\frac{s_{c}-s+2-b}{\alpha}=s_{c}-\frac{s_{c}-s}{\alpha}\in(s,s_{c}]$ , and in the second inequality we used the Hardy’s inequality and the embedding $\dot{H}^{s_{1}}(\mathbb{R}^{4})\hookrightarrow L^{\frac{4\alpha}{s_{c}-s+2-b}}(\mathbb{R}^{4})$ .

For the second term, we consider two cases. When $s=s_{c}$ , by arguing as in (4.6) and applying (4.12), we obtain

	$\displaystyle\\|\|\nabla\|^{s}P_{\leq N}\|x\|^{-b}(F(u)-F(u_{\leq N}))\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\sum_{M>N}(\frac{N}{M})^{s_{c}}\\|\|\nabla\|^{s_{c}}u_{\leq M}\\|_{L_{t}^{2}L_{x}^{4,2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\sum_{M>N}(\frac{N}{M})^{s_{c}}M^{\sigma(s)}\lesssim N^{\sigma(s)}.$

When $s<s_{c}$ , we have $s_{1}=s_{c}-\frac{s_{c}-s}{\alpha}\in(s,s_{c})$ . By employing the Bernstein inequality, the embedding $L_{x}^{\frac{4}{3}}\hookrightarrow L_{x}^{\frac{4}{3},2}$ , Lemma 2.12, Lemma 2.11, and the Sobolev embedding theorem, we obtain

		$\displaystyle\\|\|\nabla\|^{s}P_{\leq N}\|x\|^{-b}(F(u)-F(u_{\leq N}))\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}([0,T_{\max})\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim$	$\displaystyle N^{s_{c}}\left\\|\|\nabla\|^{s-s_{c}}\int_{0}^{1}\|x\|^{-b}F^{\prime}(u_{\leq N}+\lambda u_{\geq N})u_{\geq N}d{\lambda}\right\\|_{L_{t}^{2}L_{x}^{\frac{4}{3}}}$
	$\displaystyle\lesssim$	$\displaystyle N^{s_{c}}\int_{0}^{1}\\|\|\nabla\|^{s_{c}-s}\|x\|^{-b}F^{\prime}(u_{\leq N}+\lambda u_{\geq N})\\|_{L_{t}^{\infty}L_{x}^{\frac{2}{s_{c}-s+1}}}d\lambda\\|\|\nabla\|^{s-s_{c}}u_{\geq N}\\|_{L_{t}^{2}L_{x}^{\frac{4}{1-s_{c}+s}}}$
	$\displaystyle\lesssim$	$\displaystyle N^{s_{c}}\\|\|\nabla\|^{s_{c}-s}u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{(\alpha+1)(s_{c}-s)+2-b}}}\left(\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{s_{c}-s+2-b}-}}^{\alpha-1}\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{s_{c}-s+2-b}+}}^{\alpha-1}\right)^{\frac{1}{2}}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
	$\displaystyle\lesssim$	$\displaystyle\\|u\\|_{L_{t}^{\infty}\dot{H}^{s_{1}}}\left(\\|u\\|_{L_{t}^{\infty}\dot{H}^{s_{1}^{-}}}^{\alpha-1}\\|u\\|_{L_{t}^{\infty}\dot{H}^{s_{1}^{+}}}^{\alpha-1}\right)^{\frac{1}{2}}\sum_{M>N}\Big{(}\frac{N}{M}\Big{)}^{s_{c}}\\|\|\nabla\|^{s_{c}}u_{M}\\|_{L_{t}^{2}L_{x}^{4,2}}$
	$\displaystyle\lesssim$	$\displaystyle N^{\sigma(s)},$

In the final two steps, we have used the Sobolev embeddings $\dot{H}^{s}(\mathbb{R}^{4})\cap\dot{H}^{s_{c}}(\mathbb{R}^{4})\hookrightarrow\dot{H}^{s_{1}\pm}(\mathbb{R}^{4})\hookrightarrow L^{\frac{4\alpha}{s_{c}-s+2-b}\pm}(\mathbb{R}^{4})$ along with the bound (4.12). This completes the proof of (4.13).

Finally, we apply (4.13) to complete the proof of Proposition 4.4. By the Bernstein inequality and (4.13), we obtain:

	$\displaystyle\\|\|\nabla\|^{\beta}u\\|_{L_{t}^{\infty}L_{x}^{2}}$	$\displaystyle\lesssim\\|\|\nabla\|^{\beta}u_{\geq 1}\\|_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N\leq 1}N^{\beta-s}\\|\|\nabla\|^{s}u_{N}\\|_{L_{t}^{\infty}L_{x}^{2}}$
		$\displaystyle\lesssim\\|\|\nabla\|^{s_{c}}u_{\geq 1}\\|_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N\leq 1}N^{\beta-s+\sigma(s)}\lesssim 1,$

provided that $s-\sigma(s)<\beta\leq s_{c}$ . This complets the proof of Proposition 4.4. ∎

Proposition 4.5 (Negative regularity).

Let $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ be an almost periodic solution to (1.1) as in Theorem 1.12 with (4.8). Suppose that

\inf_{t\in[0,T_{\text{max}})}N(t)\geq 1.

Then, for any $0<\varepsilon<\frac{1}{2}$ , we have

u\in L_{t}^{\infty}([0,T_{\text{max}});\dot{H}_{x}^{-\varepsilon}(\mathbb{R}^{4})).

(4.14)

Proof.

First, applying Proposition 4.4 with $s_{c}=s$ , we obtain

u\in L_{t}^{\infty}([0,T_{\text{max}});\dot{H}_{x}^{\beta}),\quad\text{for all }\frac{1}{2}<\beta\leq s_{c}.

Then, applying Proposition 4.4 again with $s=(s_{c}-\frac{1}{2})_{+}$ yields (4.14). ∎

Proof of Theorem 4.3.

Since $\left\{N(t)^{s_{c}-2}u(t,N(t)^{-1}x):t\in[0,T_{\max})\right\}$ is precompact in $\dot{H}^{s_{c}}(\mathbb{R}^{4})$ , by Arzala-Ascoli theorem, for any $\eta>0,$ there exists $c(\eta)>0$ such that

\int_{|\xi|\leq c(\eta)N(t)}|\xi|^{2s_{c}}|\hat{u}(t.\xi)|^{2}d\xi\leq\eta

(4.15)

Interpolating between the inequalities (4.15) and (4.14), we obtain that for any $\eta>0$ , there holds:

\int_{|\xi|\leq c(\eta)N(t)}|\hat{u}(t,\xi)|^{2}d\xi\lesssim_{u}\eta^{\frac{\epsilon}{s_{c}+\epsilon}}.

Therefore, using (4.10), we obtain

	$\displaystyle M[u_{0}]=M[u(t)]$	$\displaystyle=\int_{\|\xi\|\leq c(\eta)N(t)}\|\hat{u}(t,\xi)\|^{2}d\xi+\int_{\|\xi\|\geq c(\eta)N(t)}\|\hat{u}(t,\xi)\|^{2}d\xi$
		$\displaystyle\lesssim_{u}\eta^{\frac{\epsilon}{s_{c}+\epsilon}}+(c(\eta)N(t))^{-2s_{c}}\\|\|\nabla\|^{s_{c}}u\\|_{L_{t}^{\infty}L_{x}^{2}}^{2}$
		$\displaystyle\lesssim_{u}\eta^{\frac{\epsilon}{s_{c}+\epsilon}}+(c(\eta)N(t))^{-2s_{c}}.$

Letting $t\to T_{\text{max}}$ , by (4.9) we deduce $M(u_{0})=0$ , hence $u_{0}=0$ , which contradicts the fact that $u$ blows up forward in time. This completes the proof of Theorem 4.3. ∎

5. The Case $\frac{3}{2}\leq s_{c}<2$ : Soliton

In this section, we rule out soliton solutions as described in Theorem 1.12 for the case $\frac{3}{2}\leq s_{c}<2$ . Subsection 5.1 focuses on establishing a frequency-localized Lin-Strauss Morawetz inequality, which will be utilized in Subsection 5.2 to rule out soliton solutions.

5.1. The Frequency-Localized Morawetz Inequality

This subsection develops spacetime estimates for the high-frequency components of almost periodic solutions to (1.1). These estimates will be applied in the next section to preclude the existence of quasi-soliton solutions as described in Theorem 1.12.

Theorem 5.1 (Frequency-Localized Morawetz Inequality).

Let $s_{c}\geq\frac{3}{2}$ , and let $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ be an almost periodic solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{\text{max}})$ . Let $I\subset[0,T_{\text{max}})$ be a compact interval composed of consecutive characteristic subintervals $J_{k}$ . Then, for any $\eta>0$ , there exists $N_{0}=N_{0}(\eta)$ such that for all $N\leq N_{0}$ , the following holds:

\int_{I}\int_{\mathbb{R}^{4}}\frac{|u_{>N}(t,x)|^{\alpha+2}}{|x|}\,dx\,dt\lesssim_{u}\eta\big{(}N^{1-2s_{c}}+K_{I}\big{)},

where $K_{I}:=\int_{I}N(t)^{3-2s_{c}}\,dt$ . Moreover, $N_{0}$ and the implicit constants in the inequality are independent of the interval $I$ .

We need the following lemma to prove Theorem 5.1.

Lemma 5.2 (Control of Low and High Frequencies).

Let $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\to\mathbb{C}$ be an almost periodic solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{\text{max}})$ . Then, on any compact interval $I\subset[0,T_{\text{max}})$ , composed of consecutive subintervals $J_{k}$ , the following estimates hold:
(i) Let $(q,r)$ be an admissible pair. For any $N>0$ and $0\leq s<1/2$ , we have

\||\nabla|^{s}u_{\geq N}\|_{L_{t}^{q}L_{x}^{r,2}(I\times\mathbb{R}^{4})}\lesssim_{u}N^{s-s_{c}}\left(1+N^{2s_{c}-1}K_{I}\right)^{\frac{1}{q}}.

(5.1)

(ii) For any $\eta>0$ , there exists $N_{1}=N_{1}(\eta)$ such that for all $N\leq N_{1}$ , we have

\||\nabla|^{\frac{1}{2}}u_{\geq N}\|_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{4})}\lesssim_{u}\eta N^{\frac{1}{2}-s_{c}}.

(iii) Using (4.3), it follows that for any $\eta>0$ , there exists $N_{2}=N_{2}(\eta)$ such that for all $N\leq N_{2}$ ,

\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}(I\times\mathbb{R}^{4})}\lesssim_{u}\eta\left(1+N^{2s_{c}-1}K_{I}\right)^{\frac{1}{2}}.

(5.2)

Proof.

The proof follows from the same method outlined in [48, Lemma 4.7] and [43, Lemma 5.2], and we omit the details here for brevity. ∎

Proof of Theorem 5.1 Throughout the proof, the spacetime norms are taken over $I\times\mathbb{R}^{4}$ .

We set $0<\eta\ll 1$ and choose

N<\min\left\{N_{1}(\eta),\eta^{2}N_{2}(\eta^{2s_{c}})\right\},

where $N_{1}$ and $N_{2}$ are constants determined in Lemma 5.2. Notably, from (5.1), we have

\|u_{>N/\eta^{2}}\|_{L_{t}^{2}L_{x}^{4,2}}\lesssim_{u}\eta N^{-s_{c}}(1+N^{2s_{c}-1}K_{I})^{1/2}.

(5.3)

Additionally, since $N/\eta^{2}<N_{2}(\eta^{2s_{c}})$ , we can apply (5.2) to obtain

\||\nabla|^{s_{c}}u_{\leq N/\eta^{2}}\|_{L_{t}^{2}L_{x}^{4,2}}\lesssim_{u}\eta(1+N^{2s_{c}-1}K_{I})^{1/2}.

(5.4)

We define the Morawetz functional as

M(t):=2\text{Im}\int_{\mathbb{R}^{4}}\frac{x}{|x|}\cdot\nabla u_{>N}(t,x)\overline{u}_{>N}(t,x)dx.

By computing using the equation $(i\partial_{t}+\Delta)u_{>N}=P_{>N}F(u)$ , we have

\partial_{t}M\gtrsim\int_{\mathbb{R}^{4}}\frac{x}{|x|}\cdot\{P_{>N}F(u),u_{>N}\}_{p}dx,

where the momentum bracket $\{\cdot,\cdot\}_{p}$ is defined as $\{f,g\}_{p}:=\text{Re}(f\nabla\overline{g}-g\nabla\overline{f})$ . Thus, by the fundamental theorem of calculus, we obtain

\int_{I}\int_{\mathbb{R}^{4}}\frac{x}{|x|}\cdot\{P_{>N}F(u),u_{>N}\}_{p}dxdt\lesssim\|M(t)\|_{L_{t}^{\infty}(I)}.

(5.5)

Observing that $\{F(u),u\}_{p}=-\frac{\alpha}{\alpha+2}\nabla(|x|^{-b}|u|^{\alpha+2})-\frac{2}{\alpha+2}\nabla(|x|^{-b})|u|^{\alpha+2}$ , we can write

	$\displaystyle\{P$	${}_{>N}F(u),u_{>N}\}_{p}$
		$\displaystyle=\{F(u),u\}_{p}-\{F(u_{\leq N},u_{\leq N})\}_{p}-\{F(u)-F(u_{\leq N}),u_{\leq N}\}_{p}-\{P_{\leq N}F(u),u_{>N}\}_{p}$
		$\displaystyle=-\frac{\alpha}{\alpha+2}\nabla[\|x\|^{-b}(\|u\|^{\alpha+2}-\|u_{\leq N}\|^{\alpha+2})]-\frac{2}{\alpha+2}\nabla(\|x\|^{-b})(\|u\|^{\alpha+2}-\|u_{\leq N}\|^{\alpha+2})$
		$\displaystyle\qquad-\{F(u)-F(u_{\leq N}),u_{\leq N}\}_{p}-\{P_{\leq N}F(u),u_{>N}\}_{p}$
		$\displaystyle:=I+II+III.$

For $I$ , integrating by parts shows that it contributes to the LHS of (5.5) a multiple of

\int_{I}\int_{\mathbb{R}^{4}}\frac{|u_{>N}(t,x)|^{\alpha+2}}{|x|^{1+b}}dxdt

and to the RHS of (5.5) a multiple of

	$\displaystyle\\|\frac{1}{\|x\|^{1+b}}$	$\displaystyle(\|u\|^{\alpha+2}-\|u_{>N}\|^{\alpha+2}-\|u_{\leq N}\|^{\alpha+2})\\|_{L_{t,x}^{1}}$
		$\displaystyle\lesssim\\|\frac{1}{\|x\|^{1+b}}(u_{\leq N})^{\alpha+1}u_{>N}\\|_{L_{t,x}^{1}}+\\|\frac{1}{\|x\|^{1+b}}(u_{>N})^{\alpha+1}u_{\leq N}\\|_{L_{t,x}^{1}}:=I_{1}+I_{2}.$		(5.6)

For $II$ , integration by parts yields

	$\displaystyle II$	$\displaystyle\lesssim\\|\frac{1}{\|x\|^{1+b}}u_{\leq N}(\|u\|^{\alpha}u-\|u_{\leq N}\|^{\alpha}u_{\leq N})\\|_{L_{t,x}^{1}}+\\|\frac{1}{\|x\|^{b}}\nabla u_{\leq N}(\|u\|^{\alpha}u-\|u_{\leq N}\|^{\alpha}u_{\leq N})\\|_{L_{t,x}^{1}}$
		$\displaystyle:=II_{1}+II_{2}.$		(5.7)

Similarly, for $III$ , we have

III\lesssim\|\frac{1}{|x|^{1+b}}u_{>N}P_{\leq N}(|u|^{\alpha}u)\|_{L_{t,x}^{1}}+\|\frac{1}{|x|^{b}}u_{>N}\nabla P_{\leq N}(|u|^{\alpha}u)\|_{L_{t,x}^{1}}.

(5.8)

Thus, to complete the proof of Theorem 5.1, it suffices to show that

\|M(t)\|_{L_{t}^{\infty}(I)}\lesssim_{u}\eta N^{1-2s_{c}},

(5.9)

and that the error terms (5.6), (5.7), and (5.8) are all controlled by $\eta(N^{1-2s_{c}}+K_{I})$ .

We begin by proving (5.9). Using Bernstein’s inequality and (1.3), we obtain

	$\displaystyle\\|M(t)\\|_{L_{t}^{\infty}(I)}$	$\displaystyle\lesssim\\|\|\nabla\|^{-1/2}\nabla u_{>N}\\|_{L_{t}^{\infty}L_{x}^{2}}\\|\|\nabla\|^{1/2}\left(\frac{x}{\|x\|}u_{>N}\right)\\|_{L_{t}^{\infty}L_{x}^{2}}$
		$\displaystyle\lesssim\\|\|\nabla\|^{1/2}u_{>N}\\|_{L_{t}^{\infty}L_{x}^{2}}^{2}\lesssim\eta N^{1-2s_{c}}.$

Next, we estimate the error terms (5.6), (5.7), and (5.8).

First, by interpolation, (1.3), and Lemma 5.2, we have

\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4(\alpha+1)}{2\alpha+1},2}}^{\alpha+1}\lesssim\||\nabla|^{s_{c}}u\|_{L_{t}^{\infty}L_{x}^{2}}^{\alpha}\||\nabla|^{s_{c}}u_{\leq N}\|_{L_{t}^{2}L_{x}^{4,2}}\lesssim\eta(1+N^{2s_{c}-1}K_{I})^{\frac{1}{2}}.

Combining the above inequality with Hölder’s inequality, Hardy’s inequality, Bernstein’s inequality, Sobolev embedding, and Lemma 5.2, we obtain

	$\displaystyle I_{1}$	$\displaystyle\lesssim\\|\|x\|^{-\frac{1+b}{\alpha+1}}u_{\leq N}\\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4(\alpha+1)}{3},2}}^{\alpha+1}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
		$\displaystyle\lesssim\\|\|\nabla\|^{\frac{1+b}{\alpha+1}}u_{\leq N}\\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4(\alpha+1)}{3},2}}^{\alpha+1}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
		$\displaystyle\lesssim N^{1-s_{c}}\\|\|\nabla\|^{s_{c}}u_{\leq N}\\|_{L_{t}^{2(\alpha+1)}L_{x}^{\frac{4(\alpha+1)}{2\alpha+1},2}}^{\alpha+1}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
		$\displaystyle\lesssim\eta N^{1-2s_{c}}(1+N^{2s_{c}-1}K_{I}).$

For the estimate of $I_{2}$ , we consider two cases. If $|u_{\leq N}|\ll|u_{>N}|$ , then this term can be absorbed into the LHS of (5.5), provided we can show that

\|\frac{1}{|x|^{1+b}}|u_{>N}|^{\alpha+2}\|_{L_{t,x}^{1}}<\infty.

(5.10)

For the other cases of $I_{2}$ , we reduce the problem to estimating $I_{1}$ . Therefore, to complete the estimate of $I_{2}$ , it suffices to prove (5.10). For this, using Hardy’s inequality, Sobolev embedding, Bernstein’s inequality, and Lemma 1.11, we have

	$\displaystyle\\|\frac{1}{\|x\|^{1+b}}\|u_{>N}\|^{\alpha+2}\\|_{L_{t,x}^{1}}$	$\displaystyle\lesssim\\|\|x\|^{-\frac{1+b}{\alpha+2}}u_{>N}\\|_{L_{t,x}^{\alpha+2}}^{\alpha+2}\lesssim\\|\|\nabla\|^{\frac{1+b}{\alpha+2}}u_{>N}\\|_{L_{t}^{\alpha+2}L_{x}^{\alpha+2}}^{\alpha+2}$
		$\displaystyle\lesssim\\|\|\nabla\|^{\frac{2\alpha-1-b}{\alpha+2}}u_{>N}\\|_{L_{t}^{\alpha+2}L_{x}^{\frac{2(\alpha+2)}{\alpha+1},2}}$
		$\displaystyle\lesssim N^{1-2s_{c}}\\|\|\nabla\|^{s_{c}}u\\|_{L_{t}^{\alpha+2}L_{x}^{\frac{2(\alpha+2)}{\alpha+1},2}}<\infty,$

where we also used the embedding $L^{\frac{2(\alpha+2)}{\alpha+1},2}(\mathbb{R}^{4})\hookrightarrow L^{\frac{2(\alpha+2)}{\alpha+1}}(\mathbb{R}^{4})$ .

Next, we consider the estimate of (5.7). The term $II_{1}$ can be directly controlled by $I_{1}+I_{2}$ . For $II_{2}$ , using Hölder’s inequality, Hardy’s inequality, Sobolev embedding, (1.3), and Lemma 5.2, we have

	$\displaystyle II_{2}$	$\displaystyle\lesssim\\|\frac{1}{\|x\|^{b}}\nabla u_{\leq N}(\|u_{>N}\|^{\alpha}+\|u_{<N}\|^{\alpha})u_{>N}\\|_{L_{t,x}^{1}}$
		$\displaystyle\lesssim\left(\\|\|x\|^{-\frac{b}{\alpha}}u_{>N}\\|_{L_{t}^{\infty}L_{x}^{2\alpha}}^{\alpha}+\\|\|x\|^{-\frac{b}{\alpha}}u_{>N}\\|_{L_{t}^{\infty}L_{x}^{2\alpha}}^{\alpha}\right)\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\nabla u_{\leq N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
		$\displaystyle\lesssim\\|\|\nabla\|^{\frac{b}{\alpha}}u\\|^{\alpha}_{L_{t}^{\infty}L_{x}^{2\alpha}}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\nabla u_{\leq N}\\|_{L_{t}^{2}L_{x}^{4,2}}\lesssim\\|\|\nabla\|^{s_{c}}u\\|_{L_{t}^{\infty}L_{x}^{2}}^{\alpha}\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\nabla u_{\leq N}\\|_{L_{t}^{2}L_{x}^{4,2}}$
		$\displaystyle\lesssim\eta N^{1-2s_{c}}(1+N^{2s_{c}-1}K_{I}).$

Finally, we consider the estimate of (5.8). First, using Hölder and Hardy’s inequality, we have

	$\displaystyle III$	$\displaystyle\lesssim\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\frac{1}{\|x\|^{1+b}}P_{\leq N}(\|u\|^{\alpha}u)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}}+\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\frac{1}{\|x\|^{b}}\nabla P_{\leq N}(\|u\|^{\alpha}u)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}}$
		$\displaystyle\lesssim\\|u_{>N}\\|_{L_{t}^{2}L_{x}^{4,2}}\\|\|x\|^{-b}\\|_{L_{x}^{\frac{4}{b},\infty}}\\|\nabla P_{\leq N}(\|u\|^{\alpha}u)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}}.$

Thus, in light of (5.1), it suffices to prove

\|\nabla P_{\leq N}(|u|^{\alpha}u)\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}}\lesssim_{u}\eta N^{1-s_{c}}(1+N^{2s_{c}-1}K_{I})^{1/2}.

To prove the above estimate, we use Hölder’s inequality, Bernstein’s inequality, the fractional chain rule, (1.3), (5.3), and (5.4) to estimate

	$\displaystyle\\|\nabla P_{\leq N}(\|u\|^{\alpha}u)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}}$
	$\displaystyle\lesssim\\|\nabla P_{\leq N}(\|u\|^{\alpha}u-\|u_{\leq N/\eta^{2}}\|^{\alpha}u_{\leq N/\eta^{2}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}}+N^{1-s_{c}}\\|\|\nabla\|^{s_{c}}(\|u_{\leq N/\eta^{2}}\|^{\alpha}u_{\leq N/\eta^{2}})\\|_{L_{t}^{2}L_{x}^{\frac{4}{3-b},2}}$
	$\displaystyle\lesssim N\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{2-b},2}}^{\alpha}\\|u_{>N/\eta^{2}}\\|_{L_{t}^{2}L_{x}^{4,2}}+N^{1-s_{c}}\\|u\\|_{L_{t}^{\infty}L_{x}^{\frac{4\alpha}{2-b},2}}^{\alpha}\\|\|\nabla\|^{s_{c}}u_{\leq N/\eta^{2}}\\|_{L_{t}^{2}L_{x}^{4,2}}$
	$\displaystyle\lesssim_{u}\eta N^{1-s_{c}}(1+N^{2s_{c}-1}K_{I})^{1/2}.$

This completes the proof of Theorem 5.1.

5.2. Ruling Out the Soliton

In this subsection, we rule out the soliton solutions as in Theorem 1.12.

Theorem 5.3 (No quasi-solitons).

Let $\frac{3}{2}\leq s_{c}<2$ . There are no almost periodic solutions as in Theorem 1.12 such that

K_{[0,T_{\text{max}})}=\int_{0}^{T_{\text{max}}}N(t)^{3-2s_{c}}dt=\infty.

(5.11)

The proof of Theorem 1.12 relies on the frequency-localized Morawetz inequality established in Subsection 5.1 and the following lower bound for the Morawetz estimate:

Lemma 5.4 (Lower bound).

Let $\frac{3}{2}\leq s_{c}<2$ , and let $u:[0,T_{\text{max}})\times\mathbb{R}^{4}\rightarrow\mathbb{C}$ be an almost periodic solution as in Theorem 1.12. Then, there exists $N_{0}>0$ such that for any $N<N_{0}$ , we have

K_{I}\lesssim_{u}\int_{I}\int_{\mathbb{R}^{4}}\frac{|u_{>N}(t,x)|^{\alpha+2}}{|x|^{1+b}}dxdt,

(5.12)

where $K_{I}:=\int_{I}N(t)^{3-2s_{c}}dt$ .

Proof.

First, by the same argument as in [44, (7.3)], there exist a sufficiently large constant $C(u)$ and $N_{0}>0$ such that for all $N<N_{0}$ , we have

\inf_{t\in I}N(t)^{2s_{c}}\int_{|x|\leq\frac{C(u)}{N(t)}}|u_{>N}(t,x)|^{2}dx\gtrsim_{u}1.

This inequality, combined with Hölder’s inequality, directly yields (5.12):

	$\displaystyle\int_{I}\int_{\mathbb{R}^{4}}\frac{\|u_{>N}(t,x)\|^{\alpha+2}}{\|x\|^{1+b}}dxdt$	$\displaystyle\gtrsim_{u}\int_{I}N(t)^{1+b}\int_{\|x\|\leq\frac{C(u)}{N(t)}}\|u_{>N}\|^{\alpha+2}dxdt$
		$\displaystyle\gtrsim_{u}\int_{I}N(t)^{1+b+2\alpha}\left(\int_{\|x\|\leq\frac{C(u)}{N(t)}}\|u_{>N}\|^{2}dx\right)^{\frac{\alpha+2}{2}}dt$
		$\displaystyle\gtrsim_{u}\int_{I}N(t)^{1+b+2\alpha}\left(N(t)^{-2s_{c}}\right)^{\frac{\alpha+2}{2}}dt$
		$\displaystyle=\int_{I}N(t)^{3-2s_{c}}dt=K_{I}.$

This completes the proof of Lemma 5.4. ∎

Now, we return to the proof of Theorem 5.3. Assume that $u$ is an almost periodic solution satisfying (5.11). Combining Theorem 5.1 and Lemma 5.4, we obtain

K_{I}\lesssim_{u}\eta(N^{1-2s_{c}}+K_{I}).

By choosing $\eta>0$ sufficiently small, we deduce that $K_{I}\lesssim_{u}N^{1-2s_{c}}$ holds uniformly over the interval $I$ . Taking $I\subset[0,T_{\text{max}})$ to be sufficiently large leads to a contradiction. This completes the proof of Theorem 5.3. Therefore, we conclude Theorem 1.1.

References

[1] L. Aloui, S. Tayachi, Local well-posedness for the inhomogeneous nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst. 48 (2021), no. 11, 5409–5437.
[2] J. An, J. Kim, A note on the $H^{s}$ -critical inhomogeneous nonlinear Schrödinger equation, Z. Anal. Anwend., 42 (2023), no. 3-4, 403–433.
[3] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145-171.
[4] L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation. Nonlinear Anal. 202 (2021), 112118, 17 pp.
[5] Y. Cho, S. Hong, K. Lee, On the global well-posedness of focusing energy-critical inhomogeneous NLS. J. Evol. Equ. 20 (2020), no. 4, 1349-1380.
[6] Y. Cho, K. Lee, On the focusing energy-critical inhomogeneous NLS: weighted space approach. Nonlinear Anal. 205 (2021), Paper No. 112261, 21 pp.
[7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$ , Ann. of Math. 167 (2008), 767–865.
[8] L. Campos, S. Correia, L. G. Farah, Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation, Nonlinear Analysis: Real World Applications, 85 (2025): 104336.
[9] D. Cruz-Uribe, V. Naibo, Kato-Ponce inequalities on weighted and variable Lebesgue spaces, Differential Integral Equations 29 (2016), 801–836.
[10] V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions. J. Hyperbolic Differ. Equ. 18 (2021), no. 1, 1-28.
[11] V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation. J. Evol. Equ. 19 (2019), no. 2, 411–434.
[12] V. D. Dinh, Scattering theory in a weighted ${L}^{2}$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation. Adv. Pure Appl. Math. 12 (2021), no. 3, 38-72.
[13] V. D. Dinh, S. Keraani, Energy scattering for a class of inhomogeneous biharmonic nonlinear Schrödinger equations in low dimensions, arXiv:2211.11824v2.
[14] B. Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$ -critical nonlinear Schrödinger equation when $d\geq 3$ , J. Amer. Math. Soc. 25 (2012), 429–463.
[15] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285 (2015), 1589–1618.
[16] B. Dodson, Global well-posedness and scattering for the defocusing $L^{2}$ -critical nonlinear Schrödinger equation when $d=2$ , Duke Math. J. 165 (2016), 3435–3516.
[17] B. Dodson, Global well-posedness and scattering for the defocusing $L^{2}$ -critical nonlinear Schrödinger equation when $d=1$ , Am. J. Math. 138 (2016), 531–569.
[18] B. Dodson, Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension $d=4$ , Ann. Sci. Ec. Norm. Supér. (4), 52 (2019), no. 1, 139–180.
[19] B. Dodson, C. X. Miao, J. Murphy, J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 759–787.
[20] C. Gao, C. Miao, J. Yang, The Interctitical Defocusing Nonlinear Schrödinger Equations with Radial Initial Data in Dimensions Four and Higher, Anal. Theory Appl. 35 (2019), 205–234.
[21] L. G. Farah, C. M. Guzmán, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation. Journal of Differential Equations 262 (2017), 4175–4231.
[22] L. G. Farah, C. M. Guzmán, Scattering for the radial focusing inhomogeneous NLS equation in higher dimensions. Bull. Braz. Math. Soc. (N.S.) 51 (2020), 449–512.
[23] C. Gao, Z. Zhao, On scattering for the defocusing high dimensional inter-critical NLS, J. Differential Equations 267 (2019), 6198–6215.
[24] T.S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana, J. Phys. 55 (5–6) (2000) 835–842.
[25] C. M. Guzmán. On well posedness for the inhomogeneous nonlinear Schrödinger equation. Nonlinear Anal. Real World Appl. 37 (2017), 249–286.
[26] C. M. Guzmán, J. Murphy, Scattering for the non-radial energy-critical inhomogeneous NLS, J. Differ. Equ. 295 (2021), 187–210.
[27] C. M. Guzmán, C. Xu, Dynamics of the non-radial energy-critical inhomogeneous NLS, Potential Anal (2024). https://doi.org/10.1007/s11118-024-10183-z.
[28] M. Grillakis, On nonlinear Schrödinger equations, Comm. Part. Diff. Eqs. 25 (2000), 1827-1844.
[29] C. E. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645-675.
[30] C. E. Kenig, F. Merle, Scattering for $\dot{H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Am. Math. Soc. 362 (2010), 1937–1962.
[31] R. Killip, T. Tao, M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. 11 (2009), 1203–1258.
[32] R. Killip, M. Visan, Energy-supercritical NLS: Critical $\dot{H}^{s}$ -bounds imply scattering, Comm. Partial Differential Equations 35 (2010), 945–987.
[33] R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), 361–424.
[34] R. Killip, M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE 5 (2012), 855–885.
[35] R. Killip, M. Visan, Nonlinear Schrödinger equations at critical regularity, in: Evolution Equations, in: Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437.
[36] R. Killip, M. Visan, X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE 1 (2008), 229–266.
[37] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980.
[38] Y. Lee, The 3D energy-critical inhomogeneous nonlinear Schrödinger equation with strong singularity, arXiv:2501.02697.
[39] J. Li, K. Li, The Defocusing Energy-supercritical Nonlinear Schrödinger Equation in High Dimensions, SIAM J. Math. Anal., 54 (2022), no.3, 3253-3274.
[40] C.S. Liu, V.K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas 1(9) (1994) 3100–3103.
[41] X. Liu, C. Miao, J. Zheng, Global well-posedness and scattering for mass-critical inhomogeneous NLS when $d\geq 3$ , arxiv: 2412.04566v1.
[42] X. Liu, Y. Yang, T. Zhang, Dynamics of threshold solutions for the energy-critical inhomogeneous NLS, arXiv:2409.00073v1.
[43] C. Lu, J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differ. Equ. 262 (2017), 4390–4414.
[44] C. Miao, J. Murphy, J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal. 267 (2014), 1662–1724.
[45] C. Miao, J. Murphy, J. Zheng, Scattering for the non-radial inhomogeneous NLS. Math. Res. Lett. 28 (2021), no. 5, 1481-1504.
[46] J. Murphy, Inter-critical NLS: Critical $\dot{H}^{s}$ -bounds imply scattering, SIAM J. Math. Anal. 46 (2014), 939–997.
[47] J. Murphy, The defocusing $\dot{H}^{1/2}$ -critical NLS in high dimensions, Discrete Contin. Dyn. Syst. Ser. A 34 (2014), 733–748.
[48] J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations 40 (2015), 265–308.
[49] J. Murphy, A simple proof of scattering for the intercritical inhomogeneous NLS. Proc. Amer. Math. Soc. 150 (2022), no. 3, 1177-1186.
[50] R. O’Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963), 129–142.
[51] E. Ryckman, M. Visan, Global well-posedness and scattering for the defocusing energy critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$ , Am. J. Math. 129 (2007), 1–60.
[52] S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations 2009, No. 3, 13 pp.
[53] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. Math. 11 (2005), 57–80.
[54] T. Tao, M. Visan, X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 (2008), 881–919.
[55] T. Tao, M. Visan, X. Zhang, Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), 165–202.
[56] Y. Wang, C. Xu, Defocusing $\dot{H}^{1/2}$ -critical inhomogeneous nonlinear Schrödinger equations, J. Math. Anal. Appl., 521 (2023), 126913.
[57] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007), 281–374.
[58] M. Visan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not. IMRN 2012 (2012), 1037–1067.
[59] J. Xie, D. Fang, Global well-posedness and scattering for the defocusing $\dot{H}^{s}$ -critical NLS, Chin. Ann. Math. 34B (2013), 801–842.
[60] X. Yu, Global well-posedness and scattering for the defocusing $\dot{H}^{1/2}$ -critical nonlinear Schrödinger equation in $\mathbb{R}^{2}$ , Anal. PDE 14 (2021), 1037–1067.
[61] T. F. Zhao, The Defocusing Energy-supercritical NLS in Higher Dimensions, Acta Mathematica Sinica, English Series, 33 (2017), 911–925.

	$\displaystyle\\|\|\nabla\|^{s_{c}-1}A_{1}(t,x)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\\|F(u+v)-F(u)\\|_{L_{t}^{2}L_{x}^{l_{1},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-s_{c}}\\|_{L_{x}^{\frac{4}{b+s_{c}},\infty}}$
	$\displaystyle\quad+\\|\|\nabla\|^{s_{c}-1}\int_{0}^{1}F^{\prime}(u+\lambda v)vd\lambda\\|_{L_{t}^{2}L_{x}^{l_{2},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-1}\\|_{L_{x}^{\frac{4}{b+1},\infty}}$
	$\displaystyle\lesssim M_{s_{c}}(u,v),$

	$\displaystyle\\|\|\nabla\|^{s_{c}-1}A_{2}(t,x)\\|_{L_{t}^{2}L_{x}^{\frac{4}{3},2}(I\times\mathbb{R}^{4})}$
	$\displaystyle\lesssim\\|[F^{\prime}(u+v)-F^{\prime}(u)]\nabla u\\|_{L_{t}^{2}L_{x}^{l_{3},2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-b-s_{c}+1}\\|_{L_{x}^{\frac{4}{b+s_{c}-1},\infty}}$
	$\displaystyle\quad+\\|\|\nabla\|^{s_{c}-1}\int_{0}^{1}F^{\prime\prime}(u+\lambda v)vd\lambda\\|_{L_{t}^{\frac{\gamma}{\alpha}}L_{x}^{l_{4}}(I\times\mathbb{R}^{4})}\,\\|\|\nabla\|^{s_{c}}u\\|_{L_{t}^{\gamma}L_{x}^{\rho,2}(I\times\mathbb{R}^{4})}\,\\|\|x\|^{-4}\\|_{L_{x}^{\frac{4}{b},\infty}},$		(2.1)

$\displaystyle\frac{d}{dt}M(t)$	$\displaystyle\geq\frac{6\alpha+4b}{\alpha+2}\int_{\|x\|<R}\|x\|^{-b-1}\|u\|^{\alpha+2}\,dx+4\text{Re}\int_{\|x\|>R}\partial_{j}\partial_{k}a\partial_{j}\overline{u}\partial_{k}u\,dx$
	$\displaystyle\quad-\int_{\|x\|>R}\|u\|^{2}\Delta^{2}a\,dx+\frac{2\alpha}{\alpha+2}\int_{\|x\|>R}\|x\|^{-b}\|u\|^{\alpha+2}\Delta a\,dx$
	$\displaystyle\quad+\frac{4b}{\alpha+2}\int_{\|x\|>R}\|x\|^{-b-2}\|u\|^{\alpha+2}x_{j}\partial_{j}a\,dx.$	(3.2)

	$\displaystyle\int_{\|x\|>R}\|\partial_{j}\partial_{k}a\partial_{j}\overline{u}\partial_{k}u\|\,dx+\int_{\|x\|>R}\|u\|^{2}\|\Delta^{2}a\|\,dx$
	$\displaystyle\lesssim\\|\Delta a\\|_{L_{x}^{\frac{2}{s_{c}-1}}(\|x\|>R)}\\|\nabla u\\|_{L_{x}^{\frac{4}{3-s_{c}}}(\mathbb{R}^{4})}^{2}+\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{2}\\|\Delta^{2}a\\|_{L_{x}^{\frac{2}{s_{c}}}(\mathbb{R}^{4})}$
	$\displaystyle\lesssim R^{2s_{c}-3}\\|u\\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}^{2},$		(3.3)

	$\displaystyle\int_{\|x\|>R}\|x\|^{-b}\|u\|^{\alpha+2}\|\Delta a\|\,dx+\int_{\|x\|>R}\|x\|^{-b-2}\|u\|^{\alpha+2}\|x_{j}\partial_{j}a\|\,dx$
	$\displaystyle\lesssim R^{-b}\\|\Delta a\\|_{L_{x}^{\frac{4}{2(s_{c}-1)+b}}(\|x\|>R)}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}+R^{-b-1}\\|\nabla a\\|_{L_{x}^{\frac{4}{2(s_{c}-1)+b}}(\|x\|>R)}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}$
	$\displaystyle\lesssim R^{2s_{c}-3}\\|u\\|_{L_{x}^{\frac{4}{2-s_{c}}}(\mathbb{R}^{4})}^{\alpha+2}\lesssim R^{2s_{c}-3}\\|u\\|_{\dot{H}^{s_{c}}(\mathbb{R}^{4})}^{\alpha+2}.$		(3.4)

The defocusing energy-supercritical inhomogeneous NLS in four space dimension

Abstract.

1. Introduction

1.1. The nonlinear Schrödinger equation at critical regularity

Conjection 1.1.

1.2. Main results

Conjection 1.2.

Theorem 1.1.

Remark 1.2.

Remark 1.3.

1.3. Outline of the proof of Theorem 1.1

Definition 1.4 (Strong solution).

Definition 1.5 (Blow-up).

Theorem 1.6 (Local well-posedness).

Theorem 1.7 (existence of minimal counterexamples).

Remark 1.8.

Lemma 1.9 (Local constancy).

Corollary 1.10 (N​(t)N(t) at blow-up).

Lemma 1.11 (Spacetime bounds).

Theorem 1.12 (Two scenarios for blow-up).

Lemma 1.13 (No-waste Duhamel formula).

2. Preliminaries

2.1. Some notations

2.2. Lorentz spaces and useful lemmas

Definition 2.1 (Lorentz spaces).

Lemma 2.2 (Properties of Lorentz spaces [50]).

Lemma 2.3 (Hölder’s inequality [50]).

Lemma 2.4 (Convolution inequality [50]).

Lemma 2.5 (Bernstein’s inequality).

Definition 2.6.

Lemma 2.7 (Sobolev embedding [13]).

Lemma 2.8 (Hardy’s inequality [13]).

Lemma 2.9 (Product rule I [9]).

Lemma 2.10 (Chain rule [1]).

Lemma 2.11 (Product rule II [8]).

Lemma 2.12 (Paraproduct estimate [44]).

Lemma 2.13.

2.3. Strichartz estimate and stability.

Definition 2.14 (Admissibility).

Proposition 2.15 (Strichartz estimates).

Theorem 2.16 (Stability).

Lemma 2.17.

Remark 2.18.

Proof.

2.4. The construction of scattering solutions away from the origin.

Proposition 2.19.

Proof.

3. The case 1<sc<321<s_{c}<\frac{3}{2}

Lemma 3.1 (Morawetz inequality).

Proof.

Theorem 3.2.

Proof.

4. The case 32≤sc<2\frac{3}{2}\leq s_{c}<2: Rapid frequency-cascade

4.1. Long-time Strichartz estimates

Theorem 4.1 (Long-time Strichartz estimate).

Lemma 4.2 (Iterative formula of B​(N)B(N)).

Proof.

4.2. The rapid frequency-cascade scenario

Theorem 4.3 (No rapid frequency-cascades).

Proposition 4.4 (Lower regularity).

Proof of Proposition 4.4.

Proposition 4.5 (Negative regularity).

Proof.

Proof of Theorem 4.3.

5. The Case 32≤sc<2\frac{3}{2}\leq s_{c}<2: Soliton

5.1. The Frequency-Localized Morawetz Inequality

Theorem 5.1 (Frequency-Localized Morawetz Inequality).

Lemma 5.2 (Control of Low and High Frequencies).

Proof.

5.2. Ruling Out the Soliton

Theorem 5.3 (No quasi-solitons).

Lemma 5.4 (Lower bound).

Proof.

References

Corollary 1.10 ( $N(t)$ at blow-up).

3. The case $1<s_{c}<\frac{3}{2}$

4. The case $\frac{3}{2}\leq s_{c}<2$ : Rapid frequency-cascade

Lemma 4.2 (Iterative formula of $B(N)$ ).

5. The Case $\frac{3}{2}\leq s_{c}<2$ : Soliton