Scattering of radial solutions to the Inhomogeneous Nonlinear Schr\"odinger Equation

TL;DR

This paper proves scattering for radial solutions of the focusing inhomogeneous nonlinear Schrödinger equation below the mass-energy threshold, extending previous results to a broader parameter range and including the classical case.

Contribution

It generalizes existing scattering results to a wider set of parameters for the inhomogeneous NLS and introduces a modified approach to handle inhomogeneity.

Findings

Established scattering below the threshold for a broader parameter range.

Extended the method to the classical NLS case ($b=0$).

Validated the approach for radial solutions in all intercritical cases.

Abstract

We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schr\"odinger equation \begin{equation} iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical case $0 < s_c <1$. This work generalizes the results of Farah and Guzm\'an [9], allowing a broader range of values for the parameters $p$ and $b$. We use a modified version of Dodson-Murphy's approach [6], allowing us to deal with the inhomogeneity. The proof is also valid for the classical nonlinear Schr\"odinger equation ($b = 0$), extending the work in [6] for radial solutions in all intercritical cases.

Equations

$$i{u}_t+\Delta u+|x{|}^{-b}|u{|}^{p-1}u=0,$$

$$\{\begin{array}{l}i{u}_t+\Delta u+|x{|}^{-b}|u{|}^{p-1}u=0,\\ u(x,0)=u_0(x)\in H^1({\mathbb{R}}^N),\end{array}$$

$$\{\begin{array}{l}i{u}_t+\Delta u+|u{|}^{p-1}u=0,\\ u(x,0)=u_0(x)\in H^1({\mathbb{R}}^N),\end{array}$$

$$1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}.$$

$$i{u}_t+\Delta u+V(x)|u{|}^{p-1}u=0,$$

$$b^{\ast }=\{\begin{array}{ll}\frac{N}{3},& N\le 3\\ 2,& N\ge 4,\end{array}$$

$$u_{\lambda }(x,t)={\lambda }^{\frac{2-b}{p-1}}u(\lambda x,{\lambda }^{2}t),\lambda >0,$$

$$\Vert u_{\lambda }(\cdot ,0){\Vert }_{{\dot{H}}^s}={\lambda }^{s-\frac{N}{2}-\frac{2-b}{p-1}}\Vert u_0{\Vert }_{{\dot{H}}^s}.$$

$$s_c=\frac{N}{2}-\frac{2-b}{p-1}.$$

$$M\left[u(t)\right]=\int |u(t){|}^{2}dx=M[u_0],$$

$$E\left[u(t)\right]=\frac{1}{2}\int |\nabla u(t){|}^{2}dx-\frac{1}{p+1}\int |x{|}^{-b}|u(t){|}^{p+1}dx=E[u_0].$$

$$\Delta \varphi -\varphi +|x{|}^{-b}|\varphi {|}^{p-1}\varphi =0.$$

$$\underset{t\in [0,+\infty )}{\sup }{\Vert u(t)\Vert }_{H_x^1}=E<+\infty .$$

$$\underset{t\to +\infty }{\mathrm{liminf}}{\int }_{B(0,R)}|u(x,t){|}^{2}dx\le {ϵ}^2,$$

$$\underset{t\to +\infty }{\lim }{\Vert u(t)-e^{it\Delta }{u}_{+}\Vert }_{H^1({\mathbb{R}}^n)}=0,$$

$$M[u_0{]}^{\frac{1-s_c}{s_c}}E[u_0]<M[Q{]}^{\frac{1-s_c}{s_c}}E[Q]$$

$$\Vert u_0{\Vert }_{L^2}^{\frac{1-s_c}{s_c}}\Vert \nabla u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}^{\frac{1-s_c}{s_c}}\Vert \nabla Q{\Vert }_{L^2}.$$

$$\frac{2}{q}=\frac{N}{2}-\frac{N}{r}-s,$$

$$2\le q,r\le \infty ,\text{ and }(q,r,N)\ne (2,\infty ,2).$$

$${\mathcal{A}}_0=\left\{(q,r)\text{ is }{L}^2\text{-admissible}|2\le r\le \frac{2N}{N-2}\right\}.$$

$${\mathcal{A}}_s=\left\{(q,r)\text{ is }{\dot{H}}^s\text{-admissible}|{\left(\frac{2N}{N-2s}\right)}^{+}\le r\le {\left(\frac{2N}{N-2}\right)}^{-}\right\}$$

$${\mathcal{A}}_{-s}=\left\{(q,r)\text{ is }{\dot{H}}^{-s}\text{-admissible}|{\left(\frac{2N}{N-2s}\right)}^{+}\le r\le {\left(\frac{2N}{N-2}\right)}^{-}\right\}.$$

$$\Vert u{\Vert }_{S({\dot{H}}^s,I)}=\underset{(q,r)\in {\mathcal{A}}_s}{\sup }\Vert u{\Vert }_{L_I^q{L}_x^r},$$

$$\Vert u{\Vert }_{S^{\prime }({\dot{H}}^{-s},I)}=\underset{(q,r)\in {\mathcal{A}}_{-s}}{\inf }\Vert u{\Vert }_{L_I^{q^{\prime }}{L}_x^{r^{\prime }}}.$$

$$\Vert e^{it\Delta }f{\Vert }_{S(L^2)}\lesssim \Vert f{\Vert }_{L^2},$$

$$\Vert e^{it\Delta }f{\Vert }_{S({\dot{H}}^s)}\lesssim \Vert f{\Vert }_{{\dot{H}}^s},$$

$${\Vert {\int }_{\mathbb{R}}{e}^{i(t-\tau )\Delta }g(\cdot ,\tau )d\tau \Vert }_{S(L^2,I)}+{\Vert {\int }_0^t{e}^{i(t-\tau )\Delta }g(\cdot ,\tau )d\tau \Vert }_{S(L^2,I)}\lesssim \Vert g{\Vert }_{S^{\prime }(L^2,I)}.$$

$${\Vert {\int }_{\mathbb{R}}{e}^{i(t-\tau )\Delta }g(\cdot ,\tau )d\tau \Vert }_{S({\dot{H}}^s,I)}+{\Vert {\int }_0^t{e}^{i(t-\tau )\Delta }g(\cdot ,\tau )d\tau \Vert }_{S({\dot{H}}^s,I)}\lesssim \Vert g{\Vert }_{S^{\prime }({\dot{H}}^{-s},I)}.$$

$${\Vert {\int }_a^b{e}^{i(t-\tau )\Delta }g(\cdot ,\tau )d\tau \Vert }_{S({\dot{H}}^s,\mathbb{R})}\lesssim \Vert g{\Vert }_{S({\dot{H}}^{-s},[a,b])}.$$

$$\Vert e^{it\Delta }f{\Vert }_{L_x^p}\lesssim \frac{1}{|t{|}^{\frac{N}{2}\left(\frac{1}{p^{\prime }}-\frac{1}{p}\right)}}\Vert f{\Vert }_{L_x^{p^{\prime }}},p\ge 2,$$

Full text

[TABLE]

when $b\geq 0$ and $N>2$ in the intercritical case $0<s_{c}<1$. This work generalizes the results of Farah and Guzmán [FG_Scat], allowing a broader range of values for the parameters $p$ and $b$. We use a modified version of Dodson-Murphy’s approach [MD_New], allowing us to deal with the inhomogeneity. The proof is also valid for the classical nonlinear Schrödinger equation ($b=0$), extending the work in [MD_New] for radial solutions in all intercritical cases.

The author thanks Luiz Gustavo Farah (UFMG) and Svetlana Roudenko (FIU) for their valuable comments and suggestions which helped improve the manuscript. This work was done when the first author was visiting Florida International University in 2018-19 under the support of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES), for which the author is very grateful as it boosted the energy into the research project. L. C. was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

1. Introduction

In this work, we consider the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)

[TABLE]

as well as its homogeneous version (NLS)

[TABLE]

where $u:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{C}$, $N>2$, $0\leq b<2$, and

[TABLE]

The homogeneous case $b=0$ has been extensively studied over the past decades (for a textbook treatment, we refer the reader to Bourgain [Bo99], Cazenave [cazenave], Linares-Ponce [LiPo15], Tao [TaoBook]).

The inhomogeneous version of the nonlinear Schrödinger equation arises as a model in optics, in the form

[TABLE]

The potential $V(x)$ accounts for the inhomogeneity of the medium. We refer to Gill [Gill], Liu and Tripathi [Liu] for the physical motivation. The particular case $V(x)=|x|^{-b}$ appears naturally as a limiting case of potentials $V(x)$ that decay as $|x|^{-b}$ at infinity (Genoud and Stuart [g_8]).

We briefly review the literature about (1.1) and (1.2). It is well-known that the Cauchy problem for (1.2) is locally well-posed in $H^{1}(\mathbb{R}^{N})$, $N\geq 1$ (Ginibre and Velo [GV79], Kato [Kato87]). More precisely, given $u_{0}\in H^{1}(\mathbb{R}^{N})$, there exists $T>0$ and a unique solution $u\in C([0,T],H^{1}(\mathbb{R}^{N}))\cap S(L^{2},[0,T])$ to the NLS equation (1.2), where $S(L^{2},[0,T])$ is the intersection of all $L^{2}$-admissible spaces (see Definition 2.1 below).

For the case $b>0$, Genoud and Stuart [g_8] proved that (1.1) is locally well-posed in $H^{1}(\mathbb{R}^{N})$, $N\geq 1$ for $0<b<\min\{2,N\}$. More recently, Guzmán [Boa] established the local well-posedness of (1.1) based on Strichartz estimates. In particular, defining

[TABLE]

he proved that, for $N\geq 2$ and $0<b<b^{*}$, the initial value problem (1.1) is locally well-posed in $H^{1}(\mathbb{R}^{N})$. Dinh [Boa_Dinh] extended Guzmán’s results in dimension $N=3$ for $0<b<\frac{3}{2}$ and $1+\frac{4-2b}{N}<p<\frac{5-2b}{2b-1}$. Note that, in the results of Guzmán [Boa] and Dinh [Boa_Dinh], the ranges of $b$ are more restricted than those in the results of Genoud and Stuart [g_8] (mainly due to the natural restrictions on Sobolev embeddings). However, Guzmán and Dinh give more detailed information on the solutions, showing that there exists $T(\|u_{0}\|_{H^{1}})>0$ such that $u\in S(L^{2},[0,T])$.

These equations are invariant under scaling. Indeed, if $u(x,t)$ is a solution to (1.1), then

[TABLE]

is also a solution. Computing the homogeneous Sobolev norm, we obtain

[TABLE]

The Sobolev index which leaves the scaling symmetry invariant is called the critical index and is defined as

[TABLE]

Note that the condition (1.3) is equivalent to $0<s_{c}<1$.

Solutions to the Cauchy problem (1.1) conserve mass $M[u]$ and energy $E[u]$, defined by

[TABLE]

[TABLE]

Note that mass and energy are not scale-invariant quantities when $0<s_{c}<1$. However, the interpolation quantity $M[u_{0}]^{1-s_{c}}E[u_{0}]^{s_{c}}$ defined by Holmer and Roudenko [holmer2007blow] is invariant under scaling, and plays a crucial role in the description of global behavior of solutions to (1.1).

The global behavior of $H^{1}(\mathbb{R}^{N})$ solutions to (1.1) is related to the existence of standing waves $u(x,t)=e^{it}\phi(x)$, where $\phi\in H^{1}(\mathbb{R}^{N})$ satisfies the elliptic equation

[TABLE]

Standing waves of particular interest are given by solutions of (1.11) which are positive and radial, also known as ground states. Questions about existence and uniqueness of ground states were answered in Berestycki and Lions [BL83], Gidas et al. [Gidas81], Kwong [Kwong89] for the case $b=0$. For the inhomogeneous case, existence of ground state was proved in Genoud [g_5, g_6], Genoud and Stuart [g_8], while uniqueness was handled in Yanagida [g_19], Genoud [g_7]. Existence and uniqueness of $Q$, the radial, positive solution to (1.11) hold for $N\geq 1$ and $0\leq b<\min\{2,N\}$.

Remark 1.1*.*

It is worth mentioning that $E(Q)>0$ if $0<s_{c}<1$ and $Q$ decays exponentially.

Before stating our main result, we give the scattering criterion, which was first proved for the $3d$ cubic NLS equation by Tao [Tao_Scat].

Theorem 1.2** (Scattering criterion).**

Let $N>2$, $1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}$ and $0\leq b<2$. Consider a spherically symmetric $H^{1}(\mathbb{R}^{N})$ solution $u$ to (1.1) defined on $[0,+\infty)$ and assume the a priori bound

[TABLE]

There exist constants $R>0$ and $\epsilon>0$ depending only on $E$, $N$, $p$ and $b$ (but never on $u$ or $t$) such that if

[TABLE]

then there exists a function $u_{+}\in H^{1}(\mathbb{R}^{N})$ such that

[TABLE]

i.e., $u$ scatters forward in time in $H^{1}(\mathbb{R}^{N})$.

Remark 1.3*.*

The notation $N>2$ instead of $N\geq 3$ is intentional, since we allow $N$ to be arbitrarily close to $2$. At least in the radial case, it is possible to define Sobolev spaces with non-integer $N$, as in this case the dimension becomes just a parameter. It is also mathematically convenient, as this flexibility is useful in some harder proofs. We mention here the work of Kopell and Landman [KL95] in which they constructed a blow-up profile for equation (1.2) in the cubic case when the dimension $N$ is exponentially asymptotically close to 2. In [MRS10], Merle, Raphael and Szeftel constructed stable blow-up solutions in the cubic case when $d\gtrapprox 2$. Later, Rottshafer and Kaper [RK02] improved the construction in [KL95] to allow the dimension to be polynomially close to 2.

The criterion above is used to prove scattering in $H^{1}$ below the mass-energy threshold, as in the following theorem. We emphasize that the main aim of this paper is to show that a different approach, based on Dodson-Murphy’s method, instead of the classic Kenig-Merle’s concentration-compactness-rigidity technique, can be applied to the INLS equation. Moreover, our method extends the range of parameters in which scattering can be proved.

Theorem 1.4**.**

Let $N>2$, $1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}$, $0\leq b<2$, and $u_{0}\in H^{1}_{rad}$ be such that

[TABLE]

and

[TABLE]

Then the solution $u(t)$ to (1.1) is defined on $\mathbb{R}$ and scatters in $H^{1}$ in both time directions.

Remark 1.5*.*

The above result is known for $b=0$ and proved in Holmer and Roudenko [HR_Scat] Duyckaerts et al. [DHR_Scat], Fang et al. [FXC_Scat], Guevara [Guevara].

The case $b>0$ is considered by Farah and Guzmán [FG_Scat] with the assumption $0<b<\min\{N/3,1\}$, for $N\geq 2$. In the theorem above, not only we employ a new method to prove scattering, but we actually extend the range of $b$ in dimensions $N>2$, allowing $0<b<]min\{N/2,2\}$ in this case. Moreover, we extend the range of $p$ in the case $N=3$. Indeed, the result proved in Farah and Guzmán [FG_Scat] considered $p<4-2b$, while here we allow $p$ to be in all the intercritical range for the 3d case.

Remark 1.6*.*

The proofs in [HR_Scat, DHR_Scat, FXC_Scat, FG_Scat, Guevara] use the so-called concentration-compactness-rigidity approach, pionereed by Kenig and Merle [KM_Glob] in the context of the energy-critical ($s_{c}=1$) NLS equation. More recently, Dodson and Murphy [MD_New] developed a new approach, based on Tao’s scattering criterion in [Tao_Scat] and on Virial/Morawetz estimates. We develop here a modification of Dodson-Murphy’s approach, replacing $L_{t}^{2}W^{1,2^{*}}_{x}$ estimates by local-in-time Strichartz estimates which, together with small data theory, makes it possible to handle the inhomogeneity. Since our estimates also hold in the case $b=0$, we immediately extend the proof in [MD_New], to $0<s_{c}<1$, $N>2$ (see also Arora [AndyScat]). In lower dimensions, this approach fails due to the slow decay on time of the Schrödinger operator $e^{it\Delta}$.

This paper is organized as follows: in the next section, we introduce some notation and basic estimates. In Section $3$, we prove the scattering criterion (Theorem 1.2). In Section $4$, we apply this criterion, together with Morawetz/Virial estimates to prove Theorem 1.4.

2. Notation and basic estimates

We denote by $p^{\prime}$ the Holder’s conjugate of $p\geq 1$. We use $X\lesssim Y$ to denote $X\leq CY$, where the constant $C$ only depends on the parameters (such as $N$, $p$, $b$, as well as $E$ in (1.12)) and exponents, but never on $u$ or on $t$. The notations $a^{+}$ and $a^{-}$ denote, respectively, $a+\eta$ and $a-\eta$, for a fixed $0<\eta\ll 1$. We use $p^{*}$ to denote the critical exponent of the Sobolev embedding $H^{1}\hookrightarrow L^{p^{*}}$, that is, $p^{*}=2N/(N-2)$ if $N>2$, and $p^{*}=+\infty$ if $N\leq 2$.

Definition 2.1**.**

If $N\geq 1$ and $s\in(-1,1)$, the pair $(q,r)$ is called $\dot{H}^{s}$-admissible if it satisfies the condition

[TABLE]

where

[TABLE]

In particular, if $s=0$, we say that the pair is $L^{2}$-admissible.

Definition 2.2**.**

Given $N>2$, consider the set

[TABLE]

For $N>2$ and $s\in(0,1)$, consider also

[TABLE]

and

[TABLE]

We define the following Strichartz norm

[TABLE]

and the dual Strichartz norm

[TABLE]

If $s=0$, we shall write $S(\dot{H}^{0},I)=S(L^{2},I)$ and $S^{\prime}(\dot{H}^{0},I)=S^{\prime}(L^{2},I)$. If $I=\mathbb{R}$, we will often omit $I$.

2.1. Strichartz Estimates

In this work, we use the following versions of the Strichartz estimates:

The standard Strichartz estimates* (Cazenave [cazenave], Keel and Tao [KT98], Foschi [Foschi05]*)

[TABLE]

[TABLE]

[TABLE]

The Kato-Strichartz estimate* (Kato [Kato94], Foschi [Foschi05]*)

[TABLE]

And a local-in-time estimate

[TABLE]

These relations are obtained from the decay of the linear operator (see, for instance, Linares and Ponce [LiPo15, Lemma 4.1])

[TABLE]

combined with Sobolev inequalities and interpolation. The inequalities (2.7)-(2.10) are standard in the theory [cazenave]. To prove (2.11), we recall the following definition.

Definition 2.3**.**

If $f\in L^{p}(\mathbb{R})$, and $0<\alpha<1$, define the Riesz potential of order $\alpha$ as

[TABLE]

The next theorem is well-known, and we refer the reader to Stein [steinbook, Page 119, Theorem 1] for a complete proof.

Theorem 2.4** (Hardy-Littlewood-Sobolev).**

If $p$, $q>1$, $0<\alpha<1$ and $\frac{1}{q}+\frac{1}{p}=\alpha$, then

[TABLE]

Proof of (2.11).

For $s\in[0,1)$, let $q$, $\tilde{q}$ and $r$ be such that $(q,r)$ is an $\dot{H}^{s}$-admissible pair, and $(\tilde{q},r)$ is an $\dot{H}^{-s}$-admissible pair. If $s=0$, assume additionally that $2<q<\infty$. Consider $\alpha:=(N/2)(1/r^{\prime}-1/r)=2/\tilde{q}+s=2/q-s$ and note that $0<\alpha<1$ and $\frac{1}{q}+\frac{1}{\tilde{q}}=\alpha$. From Minkowski’s inequality, and the decay of the linear Schrödinger operator (2.12):

[TABLE]

From the Hardy-Littlewood-Sobolev Theorem, we get

[TABLE]

In particular, if $s=0$, then $q=\tilde{q}$ and

[TABLE]

Note that (2.20) also immediately holds in the case $(s,q,r)=(0,\infty,2)$. Now observe that, if $s=0$ and $g\in C^{\infty}_{0}(\mathbb{R}^{N+1})$,

[TABLE]

Therefore, as in Kato [Kato94, Theorem 2.1], we can interpolate (2.19) and (2.25) and use a density argument to obtain (2.11). ∎

2.2. Other useful estimates

We start recalling a couple of useful estimates for radial functions. The first one is the so-called Strauss lemma. The second estimate is a Gagliardo-Nirenberg-type estimate, which is an immediate consequence of the first inequality.

Lemma 2.5** (Strauss [Strauss77]).**

If $f\in H^{1}_{rad}(\mathbb{R}^{N})$, $N\geq 2$, then, for any $R>0$,

[TABLE]

Corollary 2.6**.**

If $f\in H^{1}_{rad}(\mathbb{R}^{N})$, $N\geq 2$, then, for any $R>0$,

[TABLE]

In what follows we also use the following standard estimates.

Lemma 2.7** (See Guzmán [Boa, Section 4]).**

Let $N>2$, $u,v\in C^{\infty}_{0}(\mathbb{R}^{N+1})$, $1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}$ and $0\leq b<\min\{N/2,2\}$. Then there exists $0\leq\theta=\theta(N,p,b)\ll p-1$ such that the following inequalities hold

[TABLE]

Proof.

Inequality (2.28) follows immediately from Hölder and Sobolev inequalities. To prove the remaining inequalities, consider the exponents

[TABLE]

Choosing $\theta=0$ if $b=0$, and $0<\theta\ll 1$ if $b>0$, we have that $(\hat{q},\hat{r})\in\mathcal{A}_{0}$, $(\hat{a},\hat{r})\in\mathcal{A}_{s_{c}}$ and $(\tilde{a},\hat{r})\in\mathcal{A}_{-s_{c}}$. By Hölder and Sobolev inequalities (see [Boa, Lemmas 4.1 and 4.2] for details), we have

[TABLE]

so that (2.29) and (2.30) follow.

Consider now (2.31). If $b=0$, then it follows directly from (2.35). For $b>0$, define the pairs

[TABLE]

It is immediate to check that $(2,2N/(N-2)$, $(\bar{q},\bar{r})\in\mathcal{A}_{0}$, and that $(\bar{a},\bar{r})\in\mathcal{A}_{s_{c}}$. Let $B$ be the unit ball centered at the origin, $B^{c}=\mathbb{R}^{N}\backslash B$ and let $A$ denote $B$ or $B^{c}$. Since

[TABLE]

we estimate, by Hölder inequality

[TABLE]

where we choose

[TABLE]

and

[TABLE]

Since $1<\frac{2N}{N+2-2b}<N$ for $N>2$ and $0<b<N/2$, if we choose $\theta$ (and thus $l$) small enough, we conclude that $\||x|^{-b}\|_{L^{r_{1}}_{A}}<+\infty$, and that $1<r_{2}<N$. In view of Hardy’s inequality (see [kufner1990hardy]),

[TABLE]

we have

[TABLE]

Therefore, (2.40) becomes

[TABLE]

Now, by splitting

[TABLE]

it is easy to see that $2\leq\theta r_{3}\leq 2N/(N-2)$. By Hölder and Sobolev inequalities

[TABLE]

Therefore, by Hölder inequality on the time variable:

[TABLE]

which finishes the proof of the lemma. ∎

Remark 2.8*.*

Inequalities (2.29)-(2.31) were proved in [Boa] for $0<b<b^{*}$ and with the additional restriction $p<4-2b$ instead of $p<5-2b$ in the 3d case. The proof we give here extends the range of $p$ and $b$ to the whole range where local well-posedness is proved. We expect that Lemma 2.7 can be used to extend the results in [Boa] using the concentration-compactness-rigidity tecnique.

The next lemma was proved in [Boa] with the same restrictions mentioned in Remark 2.8. In view of Lemma 2.7, the proof in [Boa] immediately extends to the new range of $p$ and $b$.

Lemma 2.9** (Small data theory, see Guzmán [Boa, Theorem 1.8]).**

Let $N\geq 1$, $1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}$ and $0\leq b<\min\{N/2,2\}$. Suppose $\|u_{0}\|_{H^{1}}\leq E$. Then there exists $\delta_{sd}=\delta_{sd}(E)>0$ such that if

[TABLE]

then the solution $u$ to (1.1) with initial condition $u_{0}\in H^{1}(\mathbb{R}^{N})$ is globally defined on $[0,+\infty)$. Moreover,

[TABLE]

and

[TABLE]

3. Proof of the scattering criterion

We start this section with a remark.

Remark 3.1*.*

Under Definition 2.2, there exists a small $\delta>0$ (possibly depending on $N$, $p$, $s$ and $b$) such that, for a fixed $0<s<1$

[TABLE]

[TABLE]

for any pair $(q,r)\in\mathcal{A}_{s}$.

For $N>2$, fix the parameters

[TABLE]

and

[TABLE]

Where $0\leq\theta\ll p-1$ is given in Lemma 2.7. The following result is the key to prove Theorem 1.2.

Lemma 3.2**.**

Let $N>2$, $1+\frac{4-2b}{N}<p<1+\frac{4-2b}{N-2}$, $0\leq b<2$ and $u$ be a radial $H^{1}(\mathbb{R}^{N})$-solution to (1.1) satisfying (1.12). If $u$ satisfies (1.13) for some $0<\epsilon<1$, then there exists $T>0$ such that the following estimate is valid

[TABLE]

Proof.

From (2.8), there exists $T_{0}>\epsilon^{-\alpha}$ such that

[TABLE]

For $T\geq T_{0}$ to be chosen later, define $I_{1}:=\left[T-\epsilon^{-\alpha},T\right]$, $I_{2}:=[0,T-\epsilon^{-\alpha}]$ and let $\eta$ denote a smooth, spherically symmetric function which equals $1$ on $B(0,1/2)$ and [math] outside $B(0,1)$. For any $R>0$ use $\eta_{R}$ to denote the rescaling $\eta_{R}(x):=\eta(x/R)$.

From Duhamel’s formula

[TABLE]

we obtain

[TABLE]

where, for $i=1,2,$

[TABLE]

We refer to $F_{1}$ as the “recent past”, and to $F_{2}$ as the “distant past”. By (3.2), it remains to estimate $F_{1}$ and $F_{2}$.

Step 1. Estimate on recent past.

By hypothesis (1.13), we can fix $T\geq T_{0}$ such that

[TABLE]

Given the relation (obtained by multiplying (1.1) by $\eta_{R}\bar{u}$ , taking the imaginary part and integrating by parts, see Tao [Tao_Scat, Section 4] for details)

[TABLE]

we have, from (1.12), for all times,

[TABLE]

so that, by (3.5), for $t\in I_{1}$,

[TABLE]

If $R>\epsilon^{-(\alpha+2)}$, then we have $\left\|\eta_{R}u\right\|_{L^{\infty}_{I_{1}}L^{2}_{x}}\lesssim\epsilon$.

Let $(q,r)\in\mathcal{A}_{s_{c}}$ . Recalling that $2+\delta\leq r\leq p^{*}-\delta$ (see Remark 3.1), using interpolation and Sobolev inequalities and the decay of the $L^{\infty}$ norm of radial functions outside the ball (2.26), we get

[TABLE]

if $R$ is large enough. Note that, in the penultimate step, we used the $H^{1}\hookrightarrow L^{p^{*}}$ embedding. Using the local-in-time Strichartz estimate (2.11), together with estimates (2.29) and (3), we bound

[TABLE]

where we used the definition of $\alpha>0$ and the fact that $q\geq 2+\delta$.

Step 2. Estimate on distant past.

Let $(q,r)\in\mathcal{A}_{s_{c}}$. Define

[TABLE]

and

[TABLE]

We claim that $(c,d)\in\mathcal{A}_{0}$. Indeed, it is immediate to check that $(c,d)$ satisfies (2.1) with $s=0$. Moreover, since

[TABLE]

we see, since $\delta>0$ is small, that $2<c<+\infty$, so that the pair $(c,d)$ is $L^{2}$-admissible. We have

[TABLE]

Using Duhamel’s principle, write

[TABLE]

Thus, by the Strichartz estimate (2.7),

[TABLE]

since, by (2.12) and (2.28),

[TABLE]

Therefore, recalling that

[TABLE]

we have

[TABLE]

Hence, Lemma 3.2 is proved. ∎

Proof of Theorem 1.2.

Choose $\epsilon$ is small enough so that, by Lemma 3.2,

[TABLE]

where $\delta_{sd}$ is given in Lemma 2.9. Thus, by small data theory, we have

[TABLE]

Define $u_{+}=e^{-iT\Delta}u(T)+i\displaystyle\int_{T}^{+\infty}e^{-is\Delta}|x|^{-b}|u|^{p-1}u(s)\,ds$. Using (2.30) and (2.31), we estimate

[TABLE]

(Note that the same estimate ensures that $u_{+}\in H^{1}$). Hence, we conclude that

[TABLE]

as desired. ∎

4. Proof of scattering

We now turn to Theorem 1.4. The main idea behind the proof is to combine radial decay with a truncated Virial identity. By choosing the right weight, and using bounds given by coercivity in large balls around the origin, one can control a time-averaged $L^{p}$ norm on these balls. Averaging is necessary due to the lack uniform estimates in time, since we are not employing concentration-compactness as in Holmer-Roudenko [HR_Scat, DHR_Scat].

We start with the following “trapping” lemmas, which can be found in Farah and Guzmán [FG_Scat, Lemma 4.2].

Lemma 4.1** (Energy trapping).**

Let $N\geq 1$ and $0<s_{c}<1$. If

[TABLE]

for some $\delta>0$ and

[TABLE]

then there exists $\delta^{\prime}=\delta^{\prime}(\delta)>0$ such that

[TABLE]

for all $t\in I$, where $I\subset\mathbb{R}$ is the maximal interval of existence of the solution $u(t)$ to (1.1). Moreover, $I=\mathbb{R}$ and $u$ is uniformly bounded in $H^{1}$.

Lemma 4.2**.**

Suppose, for $f\in H^{1}(\mathbb{R}^{N})$, $N\geq 1$, that

[TABLE]

Then there exists $\delta^{\prime}=\delta^{\prime}(\delta)>0$ so that

[TABLE]

From now on, we consider $u$ to be a solution to (1.1) satisfying the conditions

[TABLE]

and

[TABLE]

In particular, by Lemma 4.1, $u$ is global and uniformly bounded in $H^{1}$. Moreover, there exists $\delta>0$ such that

[TABLE]

In the spirit of Dodson and Murphy [MD_New], we prove a local coercivity estimate. We start with a preliminary result.

Lemma 4.3**.**

For $N\geq 1$, let $\phi$ be a smooth cutoff to the set $\{|x|\leq\frac{1}{2}\}$ and define $\phi_{R}(x)=\phi\left(\frac{x}{R}\right)$. If $f\in H^{1}(\mathbb{R}^{N})$ , then

[TABLE]

In particular,

[TABLE]

Proof.

We first calculate directly

[TABLE]

Now, integrating by parts, we have

[TABLE]

Using the last two identities, we conclude (4.4). To obtain (4.5), we note that

[TABLE]

∎

Lemma 4.4** (Local coercivity).**

For $N\geq 1$, let $u$ be a globally defined $H^{1}(\mathbb{R}^{N})$-solution to (1.1) satisfying (4.3). There exists $\bar{R}=\ \bar{R}(\delta,M[u_{0}],Q,s_{c})>0$ such that, for any $R\geq\bar{R}$,

[TABLE]

In particular, by Lemma 4.2, there exists $\delta^{\prime}=\delta^{\prime}(\delta)>0$ such that

[TABLE]

Proof.

First note that

[TABLE]

for all $t\in\mathbb{R}$. Thus, we only need to control the $\dot{H}^{1}$ term. Using Lemma 4.3 and (4.3), we conclude

[TABLE]

Thus, by choosing $R$ large enough, depending on $\delta$, $M[u_{0}]$, $Q$ and $s_{c}$, we bound the last expression by $\left[(1-\delta)\|Q\|^{\frac{1-s_{c}}{s_{c}}}_{L^{2}}\|\nabla Q\|_{L^{2}}\right]^{2}$, which finishes the proof.

∎

We exploit the coercivity given by the previous lemma by making use of the Virial identity (see Dodson and Murphy [MD_New, Lemma 3.3], Farah and Guzmán [FG_Scat, Proposition 7.2])

Lemma 4.5** (Virial identity).**

Let $a:\mathbb{R}^{N}\rightarrow\mathbb{R}$ be a smooth weight. If $|\nabla a|\in L^{\infty}$, define

[TABLE]

Then, if $u$ is a solution to (1.1), we have the following identity

[TABLE]

[TABLE]

We now have all the basic tools needed to prove scattering. Let $R\gg 1$ to be determined below. We take $a$ to be a radial function satisfying

[TABLE]

In the intermediate region $\frac{R}{2}<|x|\leq R$, we impose that

[TABLE]

Here, $\partial_{r}$ denotes the radial derivative, i.e., $\partial_{r}a=\nabla a\cdot\frac{x}{|x|}$. Note that for $|x|\leq\frac{R}{2}$, we have

[TABLE]

while, for $|x|>R$, we have

[TABLE]

Proposition 4.6** (Virial/Morawetz estimate).**

For $N>2$, let $u$ be a radial $H^{1}$-solution to (1.1) satisfying (4.3). Then, for $R=R(\delta,M[u_{0}],Q)$ sufficiently large, and $T>0$,

[TABLE]

Proof.

Choose $R\geq\bar{R}(\delta,M[u_{0}],Q,s_{c})$ as in Lemma 4.4. We define the weight $a$ as above and define $Z(t)$ as in Lemma 4.5. Using Cauchy-Schwarz inequality, and the definition of $Z(t)$, we have

[TABLE]

As in Dodson and Murphy [MD_New, Proposition 3.4], we compute

[TABLE]

where we used the radiality of $u$ and $a$. By the definition of $a$, and the fact that $\partial_{r}^{2}a\geq 0$,

[TABLE]

Define $\phi^{A}$ , $A>0$, as a smooth cutoff to the set $\{|x|\leq\frac{1}{2}\}$ that vanishes outside the set $\{|x|\leq\frac{1}{2}+\frac{1}{A}\}$, and define $\phi_{R}^{A}(x)=\ \phi^{A}\left(\frac{x}{R}\right)$. We will now estimate the first term in the last inequality.

[TABLE]

Using Lemma 4.3, we can write

[TABLE]

The inequalities (4.19), (4) and (4) can be rewritten as

[TABLE]

By Corollary 2.6 and by Lemma 4.4, we can write (4.24) as

[TABLE]

We can now make $A\to+\infty$ to obtain $I_{A}+II_{A}\to 0$ by dominated convergence. Hence,

[TABLE]

We finish the proof integrating over time, and using (4.14). We have

[TABLE]

since $p>1+\frac{4-2b}{N}$. ∎

We are now able to prove the energy evacuation.

Proposition 4.7** (Energy evacuation).**

Under the hypotheses of Proposition 4.6, there exist a sequence of times $t_{n}\to+\infty$ and a sequence of radii $R_{n}\to+\infty$ such that

[TABLE]

Proof.

Using Proposition 4.6, choose $T_{n}\to+\infty$ and $R_{n}=T_{n}^{\frac{N}{3N-2+b}}$, so that

[TABLE]

Therefore, by the Mean Value Theorem, there is a sequence $t_{n}\to+\infty$ such that (4.28) holds. The proof is complete. ∎

Using Proposition 4.7, we can prove Theorem 1.4. We will prove only the case $t\to+\infty$, as the case $t\to-\infty$ is entirely analogous.

Proof of Theorem 1.4.

Take $t_{n}\to+\infty$ and $R_{n}\to+\infty$ as in Proposition 4.7. Fix $\epsilon>0$ and $R>0$ as in Theorem 1.2. Choosing $n$ large enough, such that $R_{n}\geq R$, Hölder’s inequality yields

[TABLE]

Therefore, by Theorem 1.2, $u$ scatters forward in time. ∎