The fractional in time Schr\"{o}dinger equation with a Hartree perturbation

TL;DR

This paper investigates the mathematical properties of a nonlinear fractional Schrödinger equation with a fractional time derivative and Hartree-type nonlinear term, focusing on existence, uniqueness, and regularity of solutions.

Contribution

It establishes foundational results on the existence, uniqueness, and regularity for a fractional Schrödinger equation with a Hartree perturbation, extending classical analysis to fractional derivatives.

Findings

Proves existence of solutions under certain conditions.

Demonstrates uniqueness of solutions.

Analyzes regularity properties of solutions.

Abstract

The aim of this work is to show existence, uniqueness and regularity properties of nonlinear fractional Schr\"{o}dinger equation with fractional time derivative of order $\alpha\in (0,1)$ and with a Hartree-type nonlinear term.

Equations

$$\{\begin{array}{ll}{i}^{\alpha }{D}_t^{\alpha }u(t,x)=(-\Delta {)}^{\beta /2}u(t,x)+\lambda J_t^{1-\alpha }{K}_{\gamma }(|u{|}^2)(x)u(t,x),& (t,x)\in [0,T]\times {\mathbb{R}}^n,\\ u(0,x)=u_0(x),u_0\in H^{\beta }({\mathbb{R}}^n),& \end{array}$$

$$K_{\gamma }(u)(x)={\int }_{{\mathbb{R}}^n}\frac{\psi (x-y)}{|x-y{|}^{\gamma }}u(y)dy,$$

$$K_{\gamma }(u)(x)=({\psi }_{\gamma }\ast u)(x);$$

$$g_{\alpha }(t)=\frac{t^{\alpha -1}}{\Gamma (\alpha )},\text{mbox}for\alpha >0,t>0.$$

$$J_t^{\alpha }u(t)=\frac{1}{\Gamma (\alpha )}{\int }_0^t(t-s{)}^{\alpha -1}u(s)ds,$$

$$J_t^{\alpha }u(t)=(g_{\alpha }\ast u)(t),t>0.$$

$$D_t^{\alpha }u(t)=\frac{d}{dt}(g_{1-\alpha }\ast u)(t)-u(0)g_{1-\alpha }(t)=\frac{1}{\Gamma (1-\alpha )}\left[\frac{d}{dt}\left({\int }_0^t(t-s{)}^{-\alpha }u(s)ds\right)-\frac{u(0)}{t^{\alpha }}\right].$$

$$D_t^{\alpha }u(t)=J_t^{1-\alpha }{u}^{\prime }(t)\text{mbox}for\alpha \in (0,1);$$

$$E_{\alpha ,\eta }(z):=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (\alpha n+\eta )},\alpha >0,\eta \in \mathbb{C},z\in \mathbb{C}.$$

$$|E_{\alpha }((-it{)}^{\alpha }|\xi {|}^{\beta })|\le M_0,t>0,\xi \in {\mathbb{R}}^n.$$

$$H^{\beta ,p}=H^{\beta ,p}({\mathbb{R}}^n)=\{u\in L^p:{\mathcal{F}}^{-1}[(1+|\xi {|}^2{)}^{\beta /2}\hat{u}(\xi )]\in L^p\},$$

$$\Vert u{\Vert }_{H^{\beta ,p}}=\Vert {\mathcal{F}}^{-1}[(1+|\xi {|}^2{)}^{\beta /2}\hat{u}(\xi )]{\Vert }_{L^p},$$

$$(I-\Delta {)}^{\beta /2}(u)={\mathcal{F}}^{-1}[(1+|\xi {|}^2{)}^{\beta /2}\hat{u}(\xi )],\text{mbox}foru\in H^{\beta ,p}.$$

$$(-\Delta {)}^{\beta /2}u={\mathcal{F}}^{-1}[|\xi {|}^{\beta }\hat{u}(\xi )],u\in \mathcal{D}((-\Delta {)}^{\beta /2}),\xi \in {\mathbb{R}}^n,$$

$$\mathcal{D}((-\Delta {)}^{\beta /2})=\left\{u\in L^2:{\int }_{{\mathbb{R}}^n}|\xi {|}^{2\beta }|\hat{u}(\xi ){|}^{2}d\xi <\infty \right\}.$$

$$\mathcal{D}((-\Delta {)}^{\beta /2})=H^{\beta }.$$

$$\Vert u{\Vert }_{{\dot{H}}^{\beta ,p}}=\Vert {\mathcal{F}}^{-1}[|\xi {|}^{\beta }\hat{u}(\xi )]{\Vert }_{L^p}.$$

$$H^{{\gamma }_2,p}\hookrightarrow H^{{\gamma }_1,p}.$$

$$H^{\gamma ,p}\hookrightarrow L^{np/n-\gamma p},$$

$$H^{\gamma /2}\hookrightarrow L^{2n/n-\gamma }.$$

$$H^{\beta }\hookrightarrow H^{\gamma /2}\hookrightarrow L^{2n/n-\gamma }.$$

$$\Vert u{\Vert }_{(-\Delta {)}^{\beta /2}}=\Vert u{\Vert }_{L^2}+\Vert (-\Delta {)}^{\beta /2}u{\Vert }_{L^2},u\in H^{\beta }.$$

$${\Vert \frac{u}{|\cdot {|}^s}\Vert }_{L^2}\le C\Vert u{\Vert }_{{\dot{H}}^s}.$$

$$\Vert (-\Delta {)}^{\sigma /2}(uv){\Vert }_{L^r}\le C\left(\Vert (-\Delta {)}^{\sigma /2}u{\Vert }_{L^{p_1}}\Vert v{\Vert }_{L^{q_1}}+\Vert u{\Vert }_{L^{p_2}}\Vert (-\Delta {)}^{\sigma /2}v{\Vert }_{L^{q_2}}\right),$$

$$\Vert (I-\Delta {)}^{\sigma /2}(uv){\Vert }_{L^r}\le C\left(\Vert (I-\Delta {)}^{\sigma /2}u{\Vert }_{L^{p_1}}\Vert v{\Vert }_{L^{q_1}}+\Vert u{\Vert }_{L^{p_2}}\Vert (I-\Delta {)}^{\sigma /2}v{\Vert }_{L^{q_2}}\right).$$

$$\frac{1}{q}=\frac{1}{p}-\frac{(n-\gamma )}{n},$$

$${\Vert u\ast \frac{1}{|\cdot {|}^{\gamma }}\Vert }_{L^q}\le C\Vert u{\Vert }_{L^p},$$

$$\Vert K_{\gamma }(u){\Vert }_{L^q}\le \Vert \psi {\Vert }_{\infty }{\Vert u\ast \frac{1}{|\cdot {|}^{\gamma }}\Vert }_{L^q}\le C\Vert \psi {\Vert }_{\infty }\Vert u{\Vert }_{L^p},$$

$$\underset{y\in {\mathbb{R}}^n}{\sup }{\int }_{{\mathbb{R}}^n}\frac{|u(x){|}^2}{|x-y{|}^{\gamma }}dx\le C\Vert u{\Vert }_{{\dot{H}}^{\gamma /2}}^2$$

$$\Vert {\tau }_{y}u{\Vert }_{{\dot{H}}^{\gamma /2}}^2={\int }_{{\mathbb{R}}^n}|\xi {|}^{\gamma }|\mathcal{F}({\tau }_{y}u)(\xi ){|}^{2}d\xi ={\int }_{{\mathbb{R}}^n}|\xi {|}^{\gamma }|\mathcal{F}(u)(\xi ){|}^{2}d\xi =\Vert u{\Vert }_{{\dot{H}}^{\gamma /2}}^2.$$

Full text

1 Introduction

The fractional in time linear Schrödinger equation has been studied in [6] in which the abstract fractional evolution equation has been investigated in the general setting of Hilbert spaces. We point out that the fractional in time Schrödinger equation has applications in the context of quantum fractional mechanics; see [11] and the references in there.

The main results of this research are strongly motivated by recent investigation on non-linear semi-relativistic Schrödinger equations e.g., [2],[3],[9]; This class of equations have interesting applications for a large systems of self-interactions, and the effective description of pseudo-relativistic boson stars via a Coulomb law; see e.g., [4],[5],[12] and the references given there. Nevertheless, to the best of our knowledge the analogous problem with a fractional time derivative has not been investigated so far. Thus, our main purpose in this paper is to study the following time fractional evolution non-linear problem with time fractional derivative in the Caputo sense,

[TABLE]

in which $K_{\gamma}(|u|^{2})\,u$ is the Hartree potential. We assume that $\beta>0$, $\alpha\in(0,1),$ $i^{\alpha}=e^{i\alpha\pi/2},$  and $\lambda\in\mathbb{R}\backslash\{0\}.$ The nonlocal operator $(-\Delta)^{\beta/2}$ is defined as a pseudo-differential operator with the symbol $|\xi|^{\beta}$ on $\mathbb{R}^{n}$, $J_{t}^{1-\alpha}$ is the fractional integral in the Riemann-Liouville sense of order $1-\alpha,$ and the non-linear term is defined defined for each $u\in L^{p}:=L^{p}(\mathbb{R}^{n}),$ by the convolution operator,

[TABLE]

where $\psi$ is assumed to be nonnegative and bounded, and $\gamma\in(0,n).$ Henceforth we denote $\psi_{\gamma}:=\displaystyle\frac{\psi}{|\cdot|^{\gamma}},$ and thus

[TABLE]

see (16) and (15) below for definitions and further properties.

2 Preliminaries

In this section we establish the basic notations, and the technical results which will be used thereafter.

2.1 The fractional derivative of Caputo

Hereafter, we denote

[TABLE]

Then we define the Riemmann Liouville integral as

[TABLE]

for a given locally integrable function $u$ defined on the half line $\mathbb{R}_{+}:=[0,\infty)$ and taking values on a Banach space $X.$ Henceforth we use the notation,

[TABLE]

Then the following property holds: $J_{t}^{\alpha+\gamma}u(t)=J_{t}^{\alpha}J_{t}^{\gamma}u(t),$ for $\alpha,\gamma>0,$ in which $u$ is suitable enough.

We shall consider the following definition of the fractional derivative of order $\alpha\in(0,1).$ Assume that $u\in C(\mathbb{R}_{+};X)$ and that the convolution $g_{1-\alpha}*u$   belongs to   $C^{1}((0,\infty);X).$ Then the Caputo fractional derivative of order $\alpha\in(0,1),$ can be interpreted as

[TABLE]

Furthermore if $u\in AC(\mathbb{R}_{+};X),$ in which $AC(\mathbb{R}_{+};X)$ is the space of absolutely continuous functions on $\mathbb{R}_{+},$ then we can also realize the Caputo derivative as

[TABLE]

see e.g., [13] for further properties and definitions.

Henceforth we shall denote the Caputo derivative by $D_{t}^{\alpha}u(t)$.

Remark 2.1**.**

We recall the Mittag-Leffler function (see e.g., [13]),

[TABLE]

The function $E_{\alpha,\eta}$ is an entire function of $z.$ We denote $E_{\alpha,1}(z)=E_{\alpha}(z),$ for $\alpha>0,z\in\mathbb{C}.$ Next we record the following estimates satisfied by the Mittag-Leffler function [13, Theorem 1.5, page 35] (see also [6, Lemma 2.2]). For $\alpha\in(0,1),\beta>0$ there exists a positive constant $M_{0},$ such that

[TABLE]

2.2 Fractional Sobolev spaces

For $\beta>0$ and $p\geq 1$, we define the fractional Sobolev space,

[TABLE]

endowed with the norm

[TABLE]

in which $\hat{u}:=\mathcal{F}(u)$ stands for the Fourier transform of $u.$ Then $H^{\beta,p}$ is Banach space endowed with the norm $\|\cdot\|_{H^{\beta,p}},$ see [1, 16]. In particular, we shall denote $H^{\beta,2}$ as $H^{\beta}.$ We define

[TABLE]

Therefore we denote, $\|u\|_{H^{\beta,p}}=\displaystyle\|(I-\Delta)^{\beta/2}(u)\|_{L^{p}}.$

Definition 2.1**.**

(see [10]) Let $\beta>0$ be fixed. Then we define the fractional laplacian $(-\Delta)^{\beta/2},$ as follows

[TABLE]

on the domain

[TABLE]

Let $\beta>0.$ Then from the Definition 2.1 follows that

[TABLE]

Let ${\mathcal{S}}_{0}:={\mathcal{S}}_{0}(\mathbb{R}^{n})$ be the space of all those $u$ in the Schwartz space ${\mathcal{S}}:={\mathcal{S}}(\mathbb{R}^{n})$ such that its Fourier transform $\widehat{u}$ vanishes on a neighborhood of the origin. Then we define for $0<\beta<n/2$ and $1\leq p<\infty$ the homogeneous Sobolev space $\dot{H}^{\beta,p}$ as the completion of ${\mathcal{S}}_{0}$ with the norm

[TABLE]

Then, $\dot{H}^{\beta,p}$ is a Banach space contained in the space of tempered distributions ${\mathcal{S}}^{\prime}$; see [1]. In particular, we shall denote $\dot{H}^{\beta,2}$ as $\dot{H}^{\beta}.$

We state the following known facts that will be needed in the forthcoming sections.

Remark 2.2**.**

(Sobolev’s embedding, see [16]).

• (i)

Let $\gamma_{1}\leq\gamma_{2},$ $1\leq p\leq\infty.$ Then,

[TABLE]

• (ii)

If $1\leq p<\infty$ and $0<\gamma<\frac{n}{p},$ then,

[TABLE]

in particular if $p=2,$  and  $0<\gamma/2<n/2$ then,

[TABLE]

Furthermore, by (i) for $\gamma_{1}=\gamma/2,\gamma_{2}=\beta,p=2$ together with (ii) for $0<\gamma/2<n/2$ we obtain the embedding,

[TABLE]

Remark 2.3**.**

• (i)

Let $0<\beta<1.$ Then the norm $\|\cdot\|_{H^{\beta}}$ of $H^{\beta}$ is equivalent to the graph norm of the fractional Laplacian operator $(-\Delta)^{\beta/2}$ on $L^{2},$ that is,**

[TABLE]

• (ii)

(Hardy inequality, see [15]). Let $0\leq s<n/2,$ $u\in\dot{H}^{s}.$ Then there exists non-negative constant $C$ such that,**

[TABLE]

• (iii)

(Fractional Leibniz rule, see [8]). Let $\sigma>0,$ $1<r<\infty$ and $1<p_{i},q_{i}\leq\infty$ and suppose that $\displaystyle\frac{1}{r}=\frac{1}{p_{i}}+\frac{1}{q_{i}},$ for $i=1,2$. Then there exists a positive constant $C$ such that for each $u,v\in\mathcal{S},$**

[TABLE]

[TABLE]

• (iv)

(Hardy-Littlewood-Sobolev inequality, see [14]). Let $\gamma\in(0,n),$ $1<p<q<\infty$, $u\in L^{p}.$ Then under the assumption that

[TABLE]

there exists a positive constant $C$ such that**

[TABLE]

• (v)

Given $u\in L^{p}$ and assuming that $\gamma,p$ and $q$ satisfy the same conditions as in ($iv$) above. Then we obtain the following direct consequence of (13)**

[TABLE]

where $C$ is a positive constant.

Lemma 2.1**.**

Let $\gamma\in(0,n),$ $u\in\dot{H}^{\gamma/2}.$ Then there exists a positive constant $C$ such that

[TABLE]

Proof.

We denote the operator translation $\tau_{y}$ of $u$ by the vector $y\in\mathbb{R}^{n}$ as $\tau_{y}u(x):=u(x-y).$ Then $\tau_{y}$ is an isometry over space $\dot{H}^{\gamma/2}$, for $\gamma>0,$

[TABLE]

Since the Lebesgue measure is invariant under translations and by Remark 2.3 part (ii) for $s:=\gamma/2$ together with the identity (17) we find that

[TABLE]

for some $C>0.$ ∎

The next theorem is a direct consequence of [1, Theorem 6.3.2, page 148].

Theorem 2.2**.**

Let $\beta>0$, $1<p<\infty,$ $u\in H^{\beta,p}.$ Then there exists a positive constant $C$ such that

[TABLE]

Proof.

We denote $m(\xi):=|\xi|^{\beta}(1+|\xi|^{2})^{-\beta/2},$ then we notice that $m(\xi)$ is an $L^{p}-$Fourier multiplier on $\mathbb{R}^{n}$ for $1<p<\infty,$ see e.g., [7, page 449]. Now we let $u\in H^{\beta,p}$. Then it follows that

[TABLE]

where $C$ is a positive constant. ∎

Proposition 2.1**.**

Let $\sigma>0,$ $1<r<\infty$ and $1<p_{i},q_{i}\leq\infty$ and $\frac{1}{r}=\frac{1}{p_{i}}+\frac{1}{q_{i}},$ for $i=1,2.$ Suppose that $u\in H^{\sigma,p_{1}}\cap L^{p_{2}},v\in H^{\sigma,q_{2}}\cap L^{q_{1}}.$ Then $u\,v\in\dot{H}^{\sigma,r}$ and

[TABLE]

where $C>0.$

Proof.

Consider $D^{\sigma}:\mathcal{S}\times\mathcal{S}\to L^{r}$ the bilinear form defined as

[TABLE]

We claim that $D^{\sigma}$ is continuous. In fact, we endow $\mathcal{S}\times\mathcal{S}$ with the norm of $(H^{\sigma,p_{1}}\cap L^{p_{2}})\times(H^{\sigma,q_{2}}\cap L^{q_{1}})$. Assume that $(u_{n},v_{n})\in\mathcal{S}\times\mathcal{S}$ and $u_{n}\rightarrow 0$ in the norm of $H^{\sigma,p_{1}}\cap L^{p_{2}},$ $v_{n}\rightarrow 0$ in the norm of $H^{\sigma,q_{2}}\cap L^{q_{1}}$ that is

[TABLE]

Hence by Remark 2.3 part (iii) and Theorem 2.2 we have that

[TABLE]

Thus $D^{\sigma}$ is a continuous bilinear map on $\mathcal{S}\times\mathcal{S}.$ On the other hand, since $\overline{\mathcal{S}}=H^{\sigma,p_{1}}\cap L^{p_{2}}$ and $\overline{\mathcal{S}}=H^{\sigma,q_{2}}\cap L^{q_{1}}$ (see [1]). Thus there exists a unique continuous extension of $D^{\sigma}$ from $H^{\sigma,p_{1}}\cap L^{p_{2}}\times H^{\sigma,q_{2}}\cap L^{q_{1}}\to L^{r},$ in which we denote the extended map as $(-\Delta)^{\sigma/2}.$ Moreover by density it follows that the unique extension satisfy (19), i.e,

[TABLE]

for each $u\in H^{\sigma,p_{1}}\cap L^{p_{2}},$ and $v\in H^{\sigma,q_{2}}\cap L^{q_{1}}.$ ∎

Proposition 2.2**.**

Let $\sigma>0,$ $1<r<\infty$ and $1<p_{i},q_{i}\leq\infty$ and $\frac{1}{r}=\frac{1}{p_{i}}+\frac{1}{q_{i}},$ for $i=1,2.$ Suppose that $u\in H^{\sigma,p_{1}}\cap L^{p_{2}},$ and  $v\in H^{\sigma,q_{2}}\cap L^{q_{1}}.$ Then $u\,v\in H^{\sigma,r}.$ Furthermore,

[TABLE]

for some $C>0.$

Proof.

The proof is a direct consequence of the fractional Leibniz rule given by Remark 2.3 (iii). Then, as we argue in the proof of the Proposition 2.1 the proof now follows. ∎

Next we state some properties for the operator $K_{\gamma}$ defined in (2).

Lemma 2.3**.**

Let $\gamma\in(0,n),$ $u\in\dot{H}^{\gamma/2}.$ Then there exists a positive constant $C$ such that

[TABLE]

for $\psi\in L^{\infty}.$

Proof.

We will consider $\gamma\in(0,n),$ $u\in\dot{H}^{\gamma/2}.$ Then by Lemma 2.1 inequality (16) there exists $C>0$ such that

[TABLE]

where $\displaystyle\psi_{\gamma}=\frac{\psi}{|\cdot|^{\gamma}},\psi\in L^{\infty}.$ ∎

Lemma 2.4**.**

Let $\gamma\in(0,n),$ $\beta\geq\gamma/2,$ $u\in H^{\beta}.$ Then there exists a positive constant $C$ such that

[TABLE]

in which we assume that $\psi\in L^{\infty}.$

Proof.

Let us take $u\in H^{\beta}.$ Then by inclusions (10), (11), and Hölder s inequality it follows that $|u|^{2}\in L^{2n/2n-\gamma}.$ Furthermore, from (2) it follows that

[TABLE]

Thus by Remark 2.3 (v) for $q=2n/\gamma$, $p=2n/2n-\gamma$ and $|u|^{2}\in L^{2n/2n-\gamma}(\mathbb{R}^{n})$ together with $H^{\beta}\hookrightarrow L^{2}$, there exists a constant $C>0$ such that,

[TABLE]

∎

Lemma 2.5**.**

Let $\gamma\in(0,n)$, $\beta\geq\gamma/2$ be fixed. Then for each $u\in H^{\beta},$ there exists a positive constant $C$ such that

[TABLE]

for an arbitrary fixed $\psi\in L^{\infty}$ such that $|\psi(x)|\leq Me^{-\mu|x|},$ where $M,\mu\geq 0,x\in\mathbb{R}^{n}.$

Proof.

We recall that $(I-\Delta)^{\beta/2}u={\mathcal{F}}^{-1}[(1+|\xi|^{2})^{\beta/2}\widehat{u}(\xi)],$ belongs to $\mathcal{S}$  for $u\in{\mathcal{S}}.$ We claim that,

[TABLE]

Since, $|\psi(x)|\leq Me^{-\mu|x|},$ decreases faster than any power of $1/|x|^{r}$ for each integers $r>0,$ by hypothesis. Then $\psi_{\gamma}=\displaystyle\frac{\psi(\cdot)}{|\cdot|^{\gamma}}$ belongs to $\mathcal{O}_{C}^{\prime}\subseteq\mathcal{S}^{\prime},$ see e.g. [17, Definition 30.1, page 315] for definition and properties of $\mathcal{O}_{C}^{\prime}$. Moreover, if $u\in{\mathcal{S}}$ is a given function, then $|u|^{2}=u\overline{u}$ also defines an element of the Schwartz space. Therefore the convolution product

[TABLE]

exists as a tempered distribution. Hence,

[TABLE]

Furthermore, $(1+|\xi|^{2})^{\beta/2}{\mathcal{F}}(|u|^{2})\in\mathcal{S}$. Thus,

[TABLE]

Since $\overline{\mathcal{S}}^{\|\cdot\|_{\beta}}=H^{\beta},$ $\beta>0.$ Then (29) holds on $H^{\beta}.$ But then (29) together with Remark 2.3 (v) in which we choose $q=2n/\gamma,$ $p=2n/2n-\gamma$ implies that,

[TABLE]

Next, we estimate the right hand side of (30) by applying Fractional Leibniz rule (20). First we prove the following inequality

[TABLE]

In fact, from Proposition 2.2, in which we now choose the parameters as follows: $\sigma=\beta,$ $r=2n/2n-\gamma,$ $p_{1}=q_{2}=2,$ $q_{1}=p_{2}=2n/n-\gamma,$ and $u=v=|u|,$ where $u\in H^{\beta}\cap L^{2n/n-\gamma}.$ Then $|u|^{2}\in H^{\beta,2n/2n-\gamma}$ and

[TABLE]

Therefore from (30) and (32) the proof follows. ∎

Lemma 2.6**.**

Let $\gamma\in(0,n),$ $\beta\geq\gamma/2$. Then for each $u,v\in H^{\beta},$ there exists a positive constant $C$ such that

[TABLE]

Proof.

Let $u,$ $v$ be in $H^{\beta}.$ Then, $u,v\in L^{2n/n-\gamma},$ since $\beta\geq\gamma/2$. Moreover, $(|u|+|v|)|u-v|$ belongs to $L^{2n/2n-\gamma}.$ Thus, by Hölder’s inequality and Sobolev inclusion, it follows that

[TABLE]

where $C>0.$ ∎

Lemma 2.7**.**

For $\gamma\in(0,n),$ $\beta\geq\gamma/2,$ $n\geq 1$ and $\psi\in L^{\infty}.$ Then there exists a positive constant $C$ such that the map $u\mapsto K_{\gamma}(|u|^{2})u$ from $L^{2}$ to $L^{2}$ satisfies

[TABLE]

for $u,v\in H^{\beta}$.

Proof.

From the definition of $K_{\gamma}(|u|^{2})$ given in (2) we have that,

[TABLE]

Next, applying Lemma 2.1 for $\gamma\in(0,n)$ together with Theorem 2.2 for $\beta=\gamma/2,p=2$ and $u\in H^{\gamma/2},$ we have that

[TABLE]

Thus from (11) we get for $\beta\geq\gamma/2$, that

[TABLE]

On the other hand, since $K_{\gamma}(|u|^{2}-|v|^{2})\in L^{2n/\gamma},$ for $u,v\in H^{\beta}\hookrightarrow L^{2n/n-\gamma}$ by Lemma 2.4 and the embedding (11). Moreover, we have that $|u|^{2}-|v|^{2}\in L^{2n/2n-\gamma}$ by Lemma 2.6. Thus the second summand on right hand side of (2.2) satisfy

[TABLE]

Thus, combining the estimate (2.2), (35), and (36) we get that

[TABLE]

and the proof of Lemma 2.7 is now complete. ∎

We next show that the nonlinear function $u\mapsto K_{\gamma}(|u|^{2})u$ is Lipschitz continuous from the closed ball in $H^{\beta}$ into itself. To show this we state the following lemma.

Lemma 2.8**.**

For $\gamma/2<\beta<1,n\geq 2$ and $\psi\in L^{\infty}.$ Then there exists a positive constant $C$ such that the map $u\mapsto K_{\gamma}(|u|^{2})u,$ satisfies the following estimate on $H^{\beta},$

[TABLE]

for each $u,v\in H^{\beta}$.

Proof.

First we show that there exists a positive constant $C$ such that,

[TABLE]

for each $u,v\in H^{\beta}.$ Indeed, it follows from the definition of the convolution operator (2) that

[TABLE]

Then proceed to estimate $K_{\gamma}(|u|^{2})u-K_{\gamma}(|v|^{2})v$ on the space $\dot{H}^{\beta}$ for $u,v\in H^{\beta}.$ In fact, from identity (38) we have that,

[TABLE]

Thus, it is sufficient to obtain bounds for the following two quantities

[TABLE]

For this purpose let us consider first the expression $I$. We notice that $K_{\gamma}(|u|^{2})$ belongs to $H^{\beta,2n/\gamma}\cap L^{\infty},$ for $u\in H^{\beta},$ because of Theorem 2.2, Lemma 2.3, and Lemma 2.5. Moreover, if $u,v\in H^{\beta},$ then by the Proposition 2.1 when $\sigma=\beta,r=2,p_{1}=2n/\gamma,q_{1}=2n/n-\gamma,p_{2}=\infty,q_{2}=2,$ we obtain that

[TABLE]

Now using the embedding (11) together with the Lemma 2.5 we can estimate the first term of right side of (40), that is,

[TABLE]

Next, by the Lemma 2.3, Theorem 2.2 for $p=2,$ $\beta=\gamma/2$ and the embedding (11), we can estimate the second term of (40), that is,

[TABLE]

Next, it remains to obtain estimates for $J=\|(-\Delta)^{\beta/2}K_{\gamma}(|u|^{2}-|v|^{2})v\|_{L^{2}},$ for $u,v\in H^{\beta}.$ Once again we appeal to Proposition 2.1 in the case that $\sigma=\beta,r=2,p_{1}=2n/\gamma,q_{1}=2n/n-\gamma,p_{2}=\infty,q_{2}=2.$ Hence we have that

[TABLE]

Thus by (11) and Remark 2.3 (v) for $q=2n/\gamma,$ $p=2n/2n-\gamma$ together with Proposition 2.2 for $\sigma=\beta,$ $r=2n/2n-\gamma,$ $p_{1}=2,q_{1}=2n/n-\gamma$ $p_{2}=2n/n-\gamma,q_{2}=2.$ Since $u,v,\in H^{\beta}\cap L^{2n/n-\gamma},$ it follows that

[TABLE]

Since $H^{\beta}\hookrightarrow L^{2n/n-\gamma}$ we have

[TABLE]

and

[TABLE]

Therefore, thanks to (2.2)-(46) we obtain

[TABLE]

Now, applying the same reasoning above we estimate $J_{2}.$ Thus,

[TABLE]

Hence, the proof of assertion (37) follows from the estimates (40)-(42), (43), (47)-(48) together with the inequality (39).

On the other hand, due to the equivalence of norm provided by the Remark 2.3 part (i) applied to $K_{\gamma}(|u|^{2})u-K_{\gamma}(|v|^{2})v$ under the condition $\beta\in(0,1),$ we have that

[TABLE]

Thus, using the Lemma 2.7 together with the estimate (37) and estimate (49) we get that

[TABLE]

for some positive constant $C$, and the proof of Lemma 2.8 is now complete. ∎

3 Non linear fractional Schrödinger equation

In this section, we establish local existence in time for the fractional evolution problem

[TABLE]

where $\alpha\in(0,1)$, $\beta>0$, $\lambda\in\mathbb{R}\backslash\{0\}.$ We consider in (50), a Hartree type non-linearity, given by

[TABLE]

where $\gamma\in(0,n),$ $\psi\in L^{\infty}.$ In this section we prove the existence and uniqueness of the solution for (50). For this purpose, our main tool will be Banach’s fixed point theorem and the results of the previous sections.

3.1 Existence and local uniqueness

In this section, we will prove the existence and uniqueness of solutions on $C([0,T];H^{\beta})$ for equation (50).

Hereafter we consider the $L^{\infty}$ norm on the space $C([0,T];H^{\beta}),$ that is,

[TABLE]

We denote $u(t,\cdot)=u$ unless otherwise is specified. Furthermore, if $X$ is any of the function spaces under consideration, we simply write $u\in X$ whenever $u(t,\cdot)\in X$ for each $t\in[0,T].$

Definition 3.1** (mild solution).**

Let $\alpha\in(0,1),$ $\beta>0$ be fixed. Assume that $u_{0}\in H^{\beta}.$ A function $u\in C([0,T];H^{\beta})$ is called a mild solution of (50) if $u$ satisfies the integral equation

[TABLE]

for each $t>0,x\in\mathbb{R}^{n}.$

We are now ready to prove the main result of this paper.

Theorem 3.1**.**

Let $\alpha\in(0,1)$, $\gamma/2\leq\beta<1$, $n\geq 2.$ Suppose that $u_{0}\in H^{\beta},$ $\psi\in L^{\infty}$ with $|\psi(x)|\leq M\,e^{-\mu|x|},$ $M,\mu\geq 0,x\in\mathbb{R}^{n}.$ Then, there exists $T>0,$ such that the nonlinear equation (50) has unique mild solution $u\in C([0,T];H^{\beta}),$ such that

[TABLE]

for some positive constant $C.$ Moreover, the map

[TABLE]

is continuous.

Proof.

Let us fix $T>0,$ and choose $r<2\sqrt{2}M_{0}\|u_{0}\|_{H^{\beta}},$ in which the constant $M_{0}>0$ is taken from the Remark 2.1. We recall our notation (51), that is, for a given $u\in C([0,T];H^{\beta})$ we have that $\|u(t)\|_{H^{\beta}}\leq\displaystyle\sup_{t\in[0,T]}\|u(t)\|_{H^{\beta}}=\|u\|_{\infty},$ for all $t$. Furthermore, we denote the closed ball of radius $r$ on $C([0,T];H^{\beta})$ as

[TABLE]

Next, under these considerations we define the nonlinear operator $\Phi_{u_{0}}:B_{r}\to B_{r}$ by

[TABLE]

First we claim that $\Phi_{u_{0}}$ is well defined and $\Phi_{u_{0}}$ maps $B_{r}$ to $B_{r}.$ We notice that $t\mapsto\Phi_{u_{0}}u(t)$ is continuous. Moreover, since the mapping $\xi\mapsto E_{\alpha}((-it)^{\alpha}|\xi|^{\beta})$ is bounded by Remark 2.1. Then for $u_{0}\in H^{\beta}\subseteq L^{2}$, and $0<\gamma\leq 2\beta.$ It follows by Hölder’s inequality, that

[TABLE]

Thus, from (3.1) we obtain that

[TABLE]

for every $u_{0}\in H^{\beta}.$ Then, it follows from (55) that

[TABLE]

Therefore, it suffices to estimate $\|K_{\gamma}(|u|^{2})u\|_{H^{\beta}}$ to ensure that $\Phi_{u_{0}}(u)\in B_{r}$. Thus from the Remark 2.3 part (i) for $\beta\in(0,1),$ we have that

[TABLE]

In order to estimate both quantities of the right hand side of (57). We notice that $K_{\gamma}(|u|^{2})\in L^{\infty}$ by Lemma 2.3. But then $K_{\gamma}(|u|^{2})u\in L^{2}$ and

[TABLE]

On the other hand, since $u\in\dot{H}^{\beta}\cap L^{2n/n-\gamma}$ then $K_{\gamma}(|u|^{2})\in\dot{H}^{\beta,2n/\gamma}\cap L^{\infty}$, by Lemma 2.3 together with Lemma 2.5. Furthermore let us assume in Proposition 2.1 that $\sigma=\beta,$ $r=2,p_{1}=2n/\gamma,q_{1}=2n/n-\gamma,$ $p_{2}=\infty,$ $q_{2}=2.$ But then follows that $K_{\gamma}(|u|^{2})u\in\dot{H}^{\beta}.$

Hence,

[TABLE]

Thus, by (57) together with (58),(59) we obtain that

[TABLE]

for some constant $C.$

Therefore, because of (11), (60), Lemma 2.3, and Lemma 2.5 we obtain that

[TABLE]

Hence, from (56) and (61) we obtain,

[TABLE]

if $T>0$ is small enough, we can conclude that the operator $\Phi_{u_{0}}$ leaves the closed ball $B_{r}$ invariant.

Next, we show that $\Phi_{u_{0}}$ is an operator Lipschitz for $T$ sufficiently small. In what follows we assume that $u,v$ belongs to $B_{r}\subseteq C([0,T];H^{\beta}),$ then we have that

[TABLE]

in which we denote

[TABLE]

Then, according to Remark 2.1, Hölder inequality and the last equality we get that

[TABLE]

that is, $\|\Phi_{u_{0}}(u)-\Phi_{u_{0}}(v)\|_{\infty}\leq\lambda TM_{0}\|K_{\gamma}(|u|^{2})u-K_{\gamma}(|v|^{2})v\|_{\infty},$ for $\lambda\neq 0,T>0,M_{0}$ positive constant. In fact, by the Lemma 2.8 we find the following estimate

[TABLE]

for some constant $C$. In this way we have that

[TABLE]

then, if we assume $(3r^{2}+r)T\lambda CM_{0}<1$, we get that $\Phi_{u_{0}}$ defines a contraction on closed ball $B_{r}$.

It remains to prove the continuous dependence of $\Phi(u(t))=\Phi_{u_{0}}(u(t))$ with respect to $u_{0},$ we notice that if $u,v$ are the corresponding mild solutions of (50) with initial data $u_{0},v_{0}$, respectively. Thus, we have that

[TABLE]

From (3.1) we have that

[TABLE]

Therefore from (3.1) together with the fact that

[TABLE]

by Lemma 2.8. Next we denote by,

[TABLE]

and

[TABLE]

Then we obtain that

[TABLE]

Since,

[TABLE]

and

[TABLE]

it then follows that

[TABLE]

for each $u,v\in B_{r}\subseteq H^{\beta}.$

Hence by (67) it follows

[TABLE]

Hence, by taking the supremum in $t$ on the left hand side of (68) we obtain that

[TABLE]

in which $T$ is appropriate.

∎

Acknowledgements:

This work has been partially supported by FONDECYT grant # 1170571.