Exponential Scattering for a Damped Hartree Equation

Abstract

This note studies the linearly damped generalized Hartree equation $i\dot{u}-{(-\mathsf{\Delta })}^{s}u+iau=\pm {\left|u\right|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p)u,0<s<1,a>0,p\ge 2.$ Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem $i\dot{u}-{(-\mathsf{\Delta })}^{s}u+iau=0$, in $H^s$ norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that $v:=e^{at}u$ scatters in $H^s$, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when $p=2$ and the last one is a damping term $iau$. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms.

1. Introduction

This manuscript is concerned with the initial value problem for a fractional generalized Hartree equation with a linear damping

$$\left\{\begin{array}{c}i\dot{u}-{(-\mathsf{\Delta })}^{s}u+iau={\mu \left|u\right|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p)u;\hfill \\ u(0,.)=u_0.\hfill \end{array}\right.$$

(1)

Here and hereafter, u is a complex valued function of the variable $(t,x)\in \mathbb{R}\times {\mathbb{R}}^N$, denotes an eventual solution to (1), where $N\ge 2$ is the spatial dimension. $\mu =\pm 1$ refers to the defocusing or focusing regime. The damping coefficient is $a>0$ and the Riesz-potential $J_{\gamma }$ is defined on ${\mathbb{R}}^N$ by

$$J_{\gamma }:=\frac{\mathsf{\Gamma }\left(\frac{N-\gamma }{2}\right)}{\mathsf{\Gamma }\left(\frac{\gamma }{2}\right){\pi }^{\frac{N}{2}}{2}^{\gamma }{|·|}^{N-\gamma }},0<\gamma <N.$$

The fractional Laplacian operator stands for

$${(-\mathsf{\Delta })}^s·:={\mathcal{F}}^{-1}{\left(\right|\xi |}^{2s}\mathcal{F}·),0<s<1.$$

The fractional Laplacian model was proposed by Laskin [1,2] by extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical. Fractional non-linear Schrödinger equation also arise in the continuum limit of discrete models with long range interaction [3], in some models of water wave dynamics [4,5]. Fractional Schrödinger equations have been studied in the physics literature [6,7], with heuristic arguments justifying the derivation of these non-local continuum dynamics from the underlying biophysics, which is understood to be modeled by a discrete nonlinear Schrödinger equation, with interactions (e.g., between different base pairs in a strand of DNA) that decay like inverse power laws. Recently, this heuristic derivation has been rigorously justified for classes of fractional NLS equations, that they arise from microscopic (or properly speaking, mesoscopic) lattice systems with long-range interactions when passing to the continuum limit [3]. The above equation models the dynamical evolution of pseudo-relativistic boson stars [8,9,10], but also as the mean field limit description of a quantum relativistic Bose gas [11].

The concentration (finite time blow-up) phenomena of the mass super-critical Schrödinger equation may be prevented with a damped term. Indeed, global well-posedness and scattering for a damped Shrödinger equation in the energy space were obtained in different papers. Precisely, in [12], the existence of energy global solutions for the non-linear damped Schrödinger equation (NLDS), was proved depending on the damping size. Later on, the last author established a similar result in the case of a fractional Schrödinger equation with a pure power type source term [13]. Some global existence and blow-up results which depend on the size of a time depending linear damping coefficient were obtained in [14]. Recently, the author of [15] discussed when the global solutions to NLDS exponentially scatter to linear damped solutions.

It is the aim of this note to establish the global well-posedness and exponential scattering of the fractional Choquard problem (1) in the energy space. This means that the damping guarantees the existence of an energy solution for all time; moreover, this solution converges exponentially (quickly) to the solution of the free equation associated to (1), namely, $\mu =0$. This linear equation is much more understood because one can obtain an explicit solution via Fourier transform, see Definition 1.

This result is presented because of the gap caused by the lack of scattering in the mass sub-critical regime, which seems to not be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when $p=2$ and the last one is a damping term $iau$. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kinds of damping terms.

To the authors’ knowledge, this note is the first one dealing with the damped fractional generalized Hartree equation. Indeed, the previous works investigated the non-damped case, namely, $a=0$ in (1). In such a case, there is a global energy solution in the mass-sub-critical regime [16]. The novelty here is the exponential convergence of the solution, for large time, to the solution of the linear equation associated to (1), namely $\mu =0$. This gives much more information about energy solutions to the above fractional Schrödinger problem because the free solution is known explicitly, see Definition 1.

The rest of the paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. Section three is devoted to proving the main result. In the Appendix A, a local well-posedness result is given.

This section is closed with some notations. The Lebesgue and Sobolev spaces endowed with usual norms are

$$\begin{array}{cc}\hfill L^r& :=L^r\left({\mathbb{R}}^N\right),W^{s,p}:=W^{s,p}\left({\mathbb{R}}^N\right),H^s:=W^{s,2};\hfill \\ \hfill H_{rd}^s& :=\{f\in H^s,f(·)=f\left(\right|·\left|\right)\};\hfill \\ \hfill {\parallel ·\parallel }_r& :={\parallel ·\parallel }_{L^r},\parallel ·\parallel :={\parallel ·\parallel }_2;\hfill \\ \hfill {\parallel ·\parallel }_{H^s}& :={\left({\parallel ·\parallel }^2+{\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}·\parallel }^2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Finally, for an eventual solution to (1), $T^{*}>0$ is its lifespan and $0<T<T^{*}$.

2. Background and Main Results

In this section, we recall some preliminary results, and then present the main contribution of this note.

2.1. Preliminary

This subsection contains some notations and definitions. Here and hereafter, one defines for $u\in H^s$ the quantities

$$\begin{array}{cc}\hfill p_c& :=1+\frac{\gamma +2s}{N},p^c:=1+\frac{\gamma +2s}{N-2s};\hfill \\ \hfill B& :=\frac{Np-N-\gamma }{s},A:=2p-B;\hfill \\ \hfill E_1\left(u\right)& {:=\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\parallel }^2+\frac{\mu }{p}{\int }_{{\mathbb{R}}^N}{\left|u\left(x\right)\right|}^p(J_{\gamma }{\ast \left|u\right|}^p)dx;\hfill \\ \hfill K\left(u\right)& {:=\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\parallel }^2+\mu {\int }_{{\mathbb{R}}^N}{\left|u\left(x\right)\right|}^p(J_{\gamma }{\ast \left|u\right|}^p)dx.\hfill \end{array}$$

If $u\in C_T\left(H^s\right)$, one defines two functions on $[0,T]$ by

$$E_1\left(t\right):=E_1\left(u\left(t\right)\right)andK\left(t\right):=K\left(u\left(t\right)\right).$$

A direct computation gives the identity

$$E_1^{\prime }=-2aK.$$

Thus, it follows that the following energy is conserved.

$$\begin{array}{ccc}\hfill E\left(u\right)& :=& e^{2at}{E}_1\left(u\right)-2a{\int }_0^t{e}^{2as}(E_1\left(u\right)-K\left(u\right))ds\hfill \\ & =& e^{2at}{E}_1\left(u\right)+2a\mu (1-\frac{1}{p}){\int }_0^t{\int }_{{\mathbb{R}}^N}|u\left(x\right){|}^p(J_{\gamma }{\ast \left|u\right|}^p)dx.\hfill \end{array}$$

Moreover, the mass satisfies the decay identity

$$M^{\prime }\left(t\right):={\partial }_t{\parallel u\left(t\right)\parallel }^2=-2aM\left(t\right).$$

(2)

The damped Choquard problem (1) is locally well-posed in the energy space.

Proposition 1.

Let $N\ge 2$, $\frac{N}{2N-1}<s<1$, $0<\gamma <N$ such that $\gamma \ge N-4s$, $2\le p<p^c$ and $u_0\in H_{rd}^s$. Then, there exist $T>0$ and a unique $u\in C([0,T),H^s)$ solution to (1). Moreover,

1. 

$u\in L_{loc}^q((0,T),W^{s,r})$, for any admissible couple $(q,r)$;

2. 

$T^{*}<\infty$ implies that ${lim\; sup}_{t\to T^{*}}\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\left(t\right)\parallel =\infty$.

Remark 1.

1. 

If $1+\frac{\gamma }{N}<p<p^c$, the energy is well-defined, by use of the inequality (4). Therefore, it seems that the previous theorem also holds;

2. 

this condition $p\ge 2$, which seems to be technical, gives the restriction $\gamma \ge N-4s$;

3. 

for the sake of completeness, the previous result is proved in the Appendix A.

Let us recall the free damped Schrödinger kernel.

Definition 1.

Denote the free operator associated to the damped Schrödinger equation

$$e^{-at}{e}^{-it{(-\mathsf{\Delta })}^s}u:={\mathcal{F}}^{-1}\left(e^{-(a+i|·{|}^{2s})t}\right)\ast u.$$

With the Duhamel formula, solutions to (1) will be considered as fixed points of

$$e^{-a.}{e}^{-i.{(-\mathsf{\Delta })}^s}{u}_0+\mu {\int }_0^{.}{e}^{-a(.-\tau )}{e}^{-i(.-\tau ){(-\mathsf{\Delta })}^s}{\left[\right|u|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p\left)u\right]d\tau .$$

Definition 2.

One says that a global solution u to (1) scatters exponentially, if there exists $u_{+}\in H^s$ such that

$$\underset{t\to +\infty }{lim}{\parallel e^{at}{e}^{it{(-\mathsf{\Delta })}^s}u\left(t\right)-u_{+}\parallel }_{H^s}=0,$$

or with an equivalent way

$$\underset{t\to +\infty }{lim}{e}^{at}{\parallel u\left(t\right)-e^{-at}{e}^{-it{(-\mathsf{\Delta })}^s}{u}_{+}\parallel }_{H^s}=0.$$

This means that the solution to (1) goes exponentially to the solution of the associated free problem.

Remark 2.

Take the change $v(t,x):=e^{at}u(t,x)$. The exponential scattering of u is equivalent to the classical scattering of v. This means that

$$\underset{t\to +\infty }{lim}{\parallel v\left(t\right)-e^{-it{(-\mathsf{\Delta })}^s}{v}_{+}\parallel }_{H^s}=0.$$

2.2. Main Result

The contribution of this manuscript reads as follows.

Theorem 1.

Let $N\ge 2$, $\frac{N}{2N-1}<s\le 1$, $0<\gamma <N$ such that $N-4s\le \gamma$, $2\le p<p^c$ and $u_0\in H_{rd}^s$. Then, the Schrödinger problem (1) is globally well-posed and exponentially scatters in the energy space in each of the following cases.

1. 

$\mu =1$;

2. 

$\mu =-1$ and $p<p_c$;

3. 

$\mu =-1$, $p=p_c$ and $\parallel u_0\parallel <\parallel \varphi \parallel$, where ϕ is the ground state of

$${(-\mathsf{\Delta })}^s\varphi +\varphi ={\left|\varphi \right|}^{p-2}(J_{\gamma }{\ast \left|\varphi \right|}^p)\varphi .$$

(3)

In view of the results stated in the above theorem, some comments are in order.

Remark 3.

1. 

Let us present a comparison with the classical (non-damped) non-linear Schrödinger problem.

a. 

It seems that there is no scattering result in the mass sub-critical regime [17];

b. 

A decay result, which is weaker than the scattering, is available in the defocusing mass-sub-critical regime [18];

2. 

The uniqueness of ground states for the above problem was obtained in [19];

3. 

The spherically symmetric assumption avoids a loss of regularity in Strichartz estimates [20];

4. 

To the authors’ knowledge, there is no existing work dealing with the damped fractional generalized Hartree equation;

5. 

The above result indicates that the solution to (1) is close to the linear one given in Definition 1. Therefore, the graph of such a solution can be seen with a numerical approach for the linear fractional Schrödinger equation, see ([21], Section 4.2);

6. 

To the authors’ knowledge, there is no explicit form of the ground state solution of (3). See ([22], Figure 2) for a numerical approximation of such a ground state for $N=1$ and some particular cases of $s\in (0,1)$.

In the next subsection, one collects some standard estimates needed in the paper.

2.3. Tools

Strichartz estimate is a classical tool to control eventual solutions to Schrödinger equations.

Definition 3.

A pair $(q,r)$ of positive real numbers is said to be s-admissible if

$$2\le q,r\le \infty ,(q,r,N)\ne (2,\infty ,2)and\frac{2s}{q}=N\left(\frac{1}{2}-\frac{1}{r}\right).$$

Let us present a standard Strichartz estimate in the radial case [20].

Proposition 2.

Let $N\ge 2$, $\frac{N}{2N-1}<s\le 1$ and two admissible pairs $(q_i,r_i)$, $i\in \{1,2\}$. Then, there exists a positive real number $C:=C_{N,s,q_i}$ such that for any $u_0\in L_{rd}^2$ and $T>0,$

$${\parallel u\parallel }_{L_T^{q_1}\left(L^{r_1}\right)}\le C\left(\parallel u_0\parallel +\parallel i\dot{u}-{(-\mathsf{\Delta })}^s{u\parallel }_{L_T^{q_2^{\prime }}\left(L^{r_2^{\prime }}\right)}\right).$$

Let us give a sharp Gagliardo–Nirenberg-type inequality related to (1) established in [23].

Proposition 3.

Let $s\in (0,1)$, $N\ge 2$, $0<\gamma <N$ and $1+\frac{\gamma }{N}<p<p^c$. Then,

1. 

there exists a (best) constant $C(N,p,s,\gamma )>0$, such that for any $u\in H^s$,

$${\int }_{{\mathbb{R}}^N}{\left|u\right|}^p(J_{\gamma }{\ast \left|u\right|}^p{)dx\le C(N,p,s,\gamma )\parallel u\parallel }^A{\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\parallel }^B;$$

(4)

2. 

moreover

$$C(N,p,s,\gamma )=\frac{2p}{A}{\left(\frac{A}{B}\right)}^{\frac{B}{2}}{\parallel \varphi \parallel }^{-2(p-1)},$$

where ϕ is the ground state solution to (3).

Recall a Hardy–Littlewood–Sobolev inequality [24].

Lemma 1.

Let $0<\gamma <N\ge 1$ and $1<s,r<\infty$ be such that $\frac{1}{r}+\frac{1}{s}=1+\frac{\gamma }{N}$. Then,

$$\parallel f(g\ast J_{\gamma }){\parallel }_1\le C(N,s,\gamma ){\parallel f\parallel }_r{\parallel g\parallel }_s,\forall (f,g)\in L^r\times L^s.$$

Finally, the following fractional chain rule [25] will be useful.

Lemma 2.

Let $s\in (0,1]$ and $1<p,p_i,q_i<\infty$ satisfying $\frac{1}{p}=\frac{1}{p_i}+\frac{1}{q_i}$. Then,

1. 

if $F\in C^1\left(\mathbb{C}\right)$, then

$${\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{F\left(u\right)\parallel }_p\lesssim \parallel F^{\prime }{\left(u\right)\parallel }_{p_1}{\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\parallel }_{q_1};$$

2. 

$${\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{\left(fg\right)\parallel }_p{\lesssim \parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{f\parallel }_{p_1}{\parallel g\parallel }_{q_1}{+\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{g\parallel }_{p_2}{\parallel f\parallel }_{q_2}.$$

Using the previous estimates, one can establish the essential contribution.

3. Proof of Theorem 1

This section is devoted to proving the main result of this note. Let the admissible couple $(q,r)$ be defined as follows

$$q:=\frac{4sp}{N(p-1)-\gamma },r:=\frac{2Np}{\gamma +N}and\theta :=\frac{2q(p-1)}{q-2}.$$

Let us give an auxiliary result.

Lemma 3.

Take $N\ge 2$, $\frac{N}{2N-1}<s<1$ and $1<p<p^c$. If $u\in C({\mathbb{R}}_{+},H^s)$ is a global solution to (1) satisfying $u\in L^{\theta }({\mathbb{R}}_{+},L^r)$, then, u exponentially scatters.

Proof. 

Recall the integral formula

$$u=e^{-a.}{e}^{-i.{(-\mathsf{\Delta })}^s}{u}_0+\mu {\int }_0^{.}{e}^{-a(.-\tau )}{e}^{-i(.-\tau ){(-\mathsf{\Delta })}^s}{\left[\right|u|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p\left)u\right]d\tau .$$

Then, using Strichartz estimate and Sobolev injection, one gets for any time slab $I:=(t,t^{\prime })$,

$$\begin{array}{ccc}\hfill \parallel e^{a.}{u\parallel }_{L^q(I,L^r)}& \lesssim & \parallel u\left(t\right)\parallel +\parallel e^{a.}{\left|u\right|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p{)u\parallel }_{L^{q^{\prime }}(I,L^{r^{\prime }})}\hfill \\ & \lesssim & \parallel u\left(t\right)\parallel +{\parallel u\parallel }_{L^{\theta }(I,L^r)}^{2(p-1)}{\parallel e^{a.}u\parallel }_{L_T^q\left(L^r\right)}.\hfill \end{array}$$

Using the chain rules in Lemma 2 and Strichartz estimate, one obtains

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_{L_T^{q^{\prime }}\left({\dot{W}}^{s,r^{\prime }}\right)}& \lesssim & {\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}u\left(t\right)\parallel +\parallel e^{a.}\left((J_{\gamma }\ast {(-\mathsf{\Delta })}^{\frac{s}{2}}{\left(\right|u|}^p{\left)\right)\left|u\right|}^{p-2}u+(J_{\gamma }{\ast \left|u\right|}^p{)(-\mathsf{\Delta })}^{\frac{s}{2}}{\left(\right|u|}^{p-2}u)\right){\parallel }_{L_T^{q^{\prime }}\left(L^{r^{\prime }}\right)}\hfill \\ & \lesssim & {\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}u\left(t\right)\parallel +\parallel e^{a.}\left({\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{\left(\right|u|}^p{)\parallel }_{\frac{r}{p}}{\parallel u\parallel }_r^{p-1}+{\parallel u\parallel }_r^p{\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{\left(\right|u|}^{p-2}u{)\parallel }_{\frac{r}{p-1}}\right){\parallel }_{L_T^{q^{\prime }}}\hfill \\ & \lesssim & {\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\left(t\right)\parallel +\parallel \parallel u\parallel }_r^{2p-2}\parallel e^{a.}u{{\parallel }_{{\dot{W}}^{s,r}}\parallel }_{L_T^{q^{\prime }}}\hfill \\ & \lesssim & {\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\left(t\right)\parallel +\parallel u\parallel }_{L^{\theta }(I,L^r)}^{2(p-1)}{\parallel e^{a.}u\parallel }_{L_T^q\left({\dot{W}}^{s,r}\right)}.\hfill \end{array}$$

Thus,

$$\parallel e^{a.}{u\parallel }_{L^q(I,W^{s,r})}\lesssim {\parallel u\left(t\right)\parallel }_{H^s}+{\parallel u\parallel }_{L^{\theta }(I,L^r)}^{2(p-1)}{\parallel e^{a.}u\parallel }_{L_T^q\left(W^{s,r}\right)}.$$

Thanks to the local well-posedness result, $u\in L_{loc}^q\left(W^{s,r}\right)$. Thus, Cauchy criterion gives

$$e^{a.}u\in L^q({\mathbb{R}}_{+},W^{s,r}).$$

Taking account of Strichartz estimate and Duhamel formula, one gets

$$\begin{array}{ccc}\hfill \parallel e^{at}{e}^{it{(-\mathsf{\Delta })}^s}u\left(t\right)-e^{a{t}^{\prime }}{e}^{i{t}^{\prime }{(-\mathsf{\Delta })}^s}u\left(t^{\prime }\right){\parallel }_{H^s}& \lesssim & \parallel e^{a.}{\left|u\right|}^{p-2}(J_{\gamma }{\ast \left|u\right|}^p{)u\parallel }_{L^{q^{\prime }}(I,W^{s,r^{\prime }})}\hfill \\ & \lesssim & {\parallel u\parallel }_{L^{\theta }(I,L^r)}^{2(p-1)}{\parallel e^{a.}u\parallel }_{L^q(I,W^{s,r})}.\hfill \end{array}$$

Since $e^{a.}u\in L^q({\mathbb{R}}_{+},W^{s,r}),$ using Cauchy criterion, it follows that there exists $u_{+}\in H^s$ such that

$$\underset{t\to +\infty }{lim}{\parallel e^{at}{e}^{it{(-\mathsf{\Delta })}^s}u\left(t\right)-u_{+}\parallel }_{H^s}=0.$$

□

3.1. The Defocusing Case $\mu =1$.

Taking account of the mass decay (2), one gets $\parallel u\parallel =e^{-a.}\parallel u_0\parallel$. Moreover, using the defocusing sign and Sobolev embedding via the fact that $p<p^c$, it follows that

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_{L^{\theta }({\mathbb{R}}_{+},L^r)}& \le & {\parallel u\parallel }_{L^{\theta }({\mathbb{R}}_{+},H^s)}\hfill \\ & \le & \parallel e^{a.}{u\parallel }_{L^{\infty }({\mathbb{R}}_{+},H^s)}{\left({\int }_0^{\infty }{e}^{-a\theta \tau }d\tau \right)}^{\frac{1}{\theta }}\hfill \\ & \lesssim & \parallel u_0\parallel +\parallel e^{a.}{(-\mathsf{\Delta })}^{\frac{s}{2}}{u\parallel }_{L^{\infty }({\mathbb{R}}_{+},L^2)}\hfill \\ & \lesssim & \parallel u_0\parallel +E\left(0\right).\hfill \end{array}$$

(5)

The proof of exponential scattering in the defocusing regime follows from Lemma 3.

3.2. The Focusing Case $\mu =-1$.

In this subsection, one considers the mass sub-critical case.

1.

First case: $1<p<p_c$.

Thanks to the energy conservation law and the Gagliardo–Nirenberg inequality (4), write

$$\begin{array}{ccc}\hfill E\left(t\right)& =& e^{2at}\left({\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\parallel }^2-\frac{1}{p}{\int }_{{\mathbb{R}}^N}{\left|u\left(x\right)\right|}^p(J_{\gamma }{\ast \left|u\right|}^p)dx\right)\hfill \\ & -& 2a(1-\frac{1}{p}){\int }_0^t{\int }_{{\mathbb{R}}^N}{e}^{2as}|u\left(x\right){|}^p(J_{\gamma }{\ast \left|u\right|}^p)dxds\hfill \\ & \ge & e^{2at}{\parallel (-\mathsf{\Delta })}^{\frac{s}{2}}{u\parallel }^2-\frac{C_{N,p,s,\gamma }}{p}{e}^{2at}{\parallel u\parallel }^A{\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\parallel }^B\hfill \\ & -& 2a{C}_{N,p,s,\gamma }(1-\frac{1}{p}){\int }_0^t{e}^{2as}{\parallel u\parallel }^A{\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\parallel }^{B}ds.\hfill \end{array}$$

Denoting the real function $X:=e^{a.}\parallel {(-\mathsf{\Delta })}^{\frac{s}{2}}u\parallel$ and using the mass decay (2), it follows that

$$\begin{array}{ccc}\hfill E\left(0\right)& \ge & X^2\left(t\right)-\frac{C_{N,p,s,\gamma }}{p}{e}^{a(2-B-A)t}{\parallel u_0\parallel }^A{X}^B\left(t\right)\hfill \\ & -& 2a{C}_{N,p,s,\gamma }(1-\frac{1}{p}){\parallel u_0\parallel }^A{\int }_0^t{e}^{a(2-B-A)s}{X}^B\left(s\right)ds\hfill \\ & \ge & X^2\left(t\right)-\frac{C_{N,p,s,\gamma }}{p}{e}^{-2a(p-1)t}{\parallel u_0\parallel }^A{X}^B\left(t\right)\hfill \\ & -& 2a{C}_{N,p,s,\gamma }(1-\frac{1}{p}){\parallel u_0\parallel }^A{\int }_0^t{e}^{-2a(p-1)s}{X}^B\left(s\right)ds.\hfill \end{array}$$

Since $p<p_c$, one obtains $B<2$. Then, by the Young inequality

$$\begin{array}{ccc}\hfill \frac{C_{N,p,s,\gamma }}{p}{\parallel u_0\parallel }^A{e}^{-2a(p-1)t}{X}^B\left(t\right)& \le & (2-B)\frac{[\frac{C_{N,p,s,\gamma }}{p}\parallel u_0{{\parallel }^A{e}^{-a(p-1)t}]}^{\frac{B}{2-B}}}{2}\hfill \\ & & +2\frac{{\left[e^{-a(p-1)t}{X}^B\left(t\right)\right]}^{\frac{2}{B}}}{B}\hfill \\ & \le & C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}t}+\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}{X}^2\left(t\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill E\left(0\right)& \ge & X^2\left(t\right)-C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}t}-\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}{X}^2\left(t\right)\hfill \\ & -& 2a(p-1){\int }_0^t\left(C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}s}+\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}{X}^2\left(s\right)\right)ds\hfill \\ & \ge & \left(1-\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}\right)X^2\left(t\right)-C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}t}\hfill \\ & -& 2a(p-1){\int }_0^t\left(C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}s}+\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}{X}^2\left(s\right)\right)ds.\hfill \end{array}$$

Since $B<2$, one gets

$$\begin{array}{ccc}\hfill X^2\left(t\right)& \le & \frac{B}{B-2}\left(1-\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}\right)X^2\left(t\right)\hfill \\ & \le & \frac{B}{B-2}E\left(0\right)+C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}t}\hfill \\ & +& \frac{B}{B-2}2a(p-1){\int }_0^t\left(C(N,p,s,\gamma ,\parallel u_0\parallel )e^{-\frac{a(p-1)B}{2-B}s}+\frac{2}{B}{e}^{-\frac{2a(p-1)}{B}t}{X}^2\left(s\right)\right)ds\hfill \\ & \le & \frac{B}{B-2}E\left(0\right)+C(N,p,s,\gamma ,\parallel u_0\parallel )+\frac{4}{B-2}a(p-1){\int }_0^t{e}^{-\frac{2a(p-1)}{B}s}{X}^2\left(s\right)ds.\hfill \end{array}$$

Using the Gronwall Lemma, it follows that

$$\begin{array}{ccc}\hfill X^2\left(t\right)& \le & \left(\frac{B}{B-2}E\left(0\right)+C(N,p,s,\gamma ,\parallel u_0\parallel )\right)exp\left(\frac{4}{B-2}a(p-1){\int }_0^t{e}^{-\frac{2a(p-1)}{B}s}ds\right)\hfill \\ & \le & \left(\frac{B}{B-2}E\left(0\right)+C(N,p,s,\gamma ,\parallel u_0\parallel )\right)exp\left(\frac{2B}{B-2}\right).\hfill \end{array}$$

This implies that

$$u\in C({\mathbb{R}}_{+},H^s).$$

Then, arguing as in (5), the proof of the exponential scattering follows by Lemma 3.

2.

Second case: $p=p_c$ and $\parallel u_0\parallel <\parallel \varphi \parallel$.

It is known that u is global [23]. Arguing as previously, and denoting $B_{*}:=B\left(p_c\right)=2$, $A_{*}:=A\left(p_c\right)=\frac{2(2s+\gamma )}{N}$, one obtains

$$\begin{array}{ccc}\hfill E\left(0\right)& \ge & X^2\left(t\right)-\frac{C_{N,p_c,s,\gamma }}{p_c}{\parallel u_0\parallel }^{A_{*}}{e}^{-2a(p_c-1)t}{X}^{B_{*}}\left(t\right)\hfill \\ & -& 2a{C}_{N,p_c,s,\gamma }(1-\frac{1}{p_c}){\parallel u_0\parallel }^{A_{*}}{\int }_0^t{e}^{-2a(p_c-1)\tau }{X}^{B_{*}}\left(\tau \right)d\tau \hfill \\ & \ge & \left(1-\frac{C_{N,p_c,s,\gamma }}{p_c}{\parallel u_0\parallel }^{2\frac{2s+\gamma }{N}}{e}^{-\frac{2a(2s+\gamma )}{N}t}\right)X^2\left(t\right)\hfill \\ & -& 2a{C}_{N,p_c,s,\gamma }(1-\frac{1}{p_c}){\parallel u_0\parallel }^{2\frac{2s+\gamma }{N}}{\int }_0^t{e}^{-\frac{2a(2s+\gamma )}{N}\tau }{X}^2\left(\tau \right)d\tau .\hfill \end{array}$$

Taking account of the Gronwall Lemma, one obtains for large time

$$\begin{array}{ccc}\hfill X^2\left(t\right)& \le & 2E\left(0\right)+4a{C}_{N,p_c,s,\gamma }(1-\frac{1}{p_c}){\parallel u_0\parallel }^{2\frac{2s+\gamma }{N}}{\int }_0^t{e}^{-\frac{2a(2s+\gamma )}{N}\tau }{X}^2\left(\tau \right)d\tau \hfill \\ & \le & 2E\left(0\right)exp\left(\frac{2N}{2s+\gamma }{C}_{N,p_c,s,\gamma }(1-\frac{1}{p_c}){\parallel u_0\parallel }^{2\frac{2s+\gamma }{N}}\right).\hfill \end{array}$$

The proof follows as previously.