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Stability of the Hartree equation with time-dependent coefficients

Boundary Value Problems20172017:129

https://doi.org/10.1186/s13661-017-0854-1

  • Received: 2 March 2017
  • Accepted: 9 August 2017
  • Published:

Abstract

In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients
$$i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{ \vert x \vert }u+\beta(t) \bigl(W \ast \vert u \vert ^{2}\bigr)u=0. $$
We first obtain the Lipschitz continuity of the solution \(u=u(\alpha ,\beta)\) with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients. Our results improve some recent results.

Keywords

  • nonlinear Hartree equation
  • stability
  • time-dependent coefficients
  • Lipschitz continuity

MSC

  • 35Q55
  • 49J20

1 Introduction

Motivated by the nonlinearity management and dispersion management [1, 2] in the experimental work in Bose-Einstein condensates and optics, nonlinear Schrödinger equations have attracted more and more attention in both the physics and the mathematics fields; see [111] and the references therein.

In this paper, we consider the stability for the following nonlinear Schrödinger equation of Hartree type under the perturbation of coefficients:
$$ \textstyle\begin{cases} i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{\vert x\vert }u+\beta(t)(W\ast \vert u\vert ^{2})u=0,& (t,x)\in[0,\infty)\times\mathbb{R}^{3},\\ u(0,x) =u_{0}(x), \end{cases} $$
(1.1)
where \(u(t,x)\) is a complex-valued function in \((t,x)\in[0,\infty )\times\mathbb{R}^{3}\), \(u_{0} \in H^{1}(\mathbb{R}^{3})\), \(\alpha(t)\) and \(\beta(t)\) are two real-valued functions in \(t\in[0,\infty )\).

Equation (1.1) arises as phenomenological models in many different contexts: Hartree-Fock theory, quantum field theory, etc. In particular, when \(W=\vert x\vert ^{-\gamma}\) with \(0<\gamma<N\), this equation describes the mean-field limit of many-body quantum systems and has been extensively studied in [4, 1222]. An essential feature of the Hartree equations is that the convolution kernel \(\vert x\vert ^{-\gamma}\) still retains the fine structure of microscopic two-body interactions of the quantum system. Therefore, it is interesting to extend mathematical methods developed for nonlinear Schrödinger equations with power nonlinearities \(\vert u\vert ^{p}u\) to the study of nonlinear Hartree equations.

When \(\alpha(t)\) and \(\beta(t)\) are two constants, Ma and Zhao in [23] investigated the stability of (1.1) under the perturbation of parameters. More precisely, they proved the following result.

Theorem A

[23], Theorem 1

Assume that \(\alpha_{j}\), \(\beta_{j}\) are positive numbers satisfying \(\alpha_{j}\rightarrow1\), \(\beta_{j}\rightarrow1\). Let \(u_{j}\) be the \(H^{1}\) solution of the following perturbed Hartree equation:
$$ i\partial_{t}u_{j}=\Delta u_{j}+ \frac{\alpha_{j}}{\vert x\vert }u_{j}+\beta_{j} \biggl(\frac{1}{\vert x\vert } \ast \vert u_{j}\vert ^{2} \biggr)u_{j} \quad \textit{on }\mathbb{R}^{3}, $$
(1.2)
with initial data
$$u_{j}\vert_{t=0}=u(0)\in H^{1}. $$
Then there exists a subsequence, still denoted by \(u_{j}\), such that
$$ u_{j}\rightarrow u \quad\textit{weakly in }L^{\infty}\bigl(H^{1}\bigr), $$
(1.3)
where u a weakly continuous solution of (1.1) with \(\alpha (t)\equiv1\), \(\beta(t)\equiv1\) and the initial data \(u(0)\). Moreover, the energy and mass inequalities
$$ E\bigl(u(t)\bigr)\leq E\bigl(u(0)\bigr),\qquad \bigl\Vert u(t) \bigr\Vert _{L^{2}}\leq\bigl\Vert u(0)\bigr\Vert _{L^{2}}, \quad\forall t>0, $$
(1.4)
follow for this weak solution u, where
$$ E\bigl(u(t)\bigr)= \int_{\mathbb{R}^{3}} \biggl[\frac{1}{2}\bigl\vert \nabla u(t,x) \bigr\vert ^{2}\,dx-\frac{1}{4}\bigl(W*\vert u\vert ^{2}\bigr) (x)\bigl\vert u(t,x)\bigr\vert ^{2}\,dx - \frac{1}{2}\frac{\vert u(t,x)\vert ^{2}}{\vert x\vert } \biggr]\,dx. $$
(1.5)
In this paper, we will extend and improve this result in several aspects:
  1. 1.

    Our results hold for more general Hartree nonlinearities \((W\ast \vert u\vert ^{2})u\), where \(W\in L^{p}(\mathbb{R}^{3})+L^{\infty}(\mathbb {R}^{3})\) for some \(p\geq1\).

     
  2. 2.

    We extend this result to the time-dependent coefficients \(\alpha(t)\) and \(\beta(t)\).

     
  3. 3.

    We prove the locally Lipschitz continuity of the solution \(u(\alpha,\beta)\) with respect to the coefficients α and β.

     
  4. 4.

    We prove that \(u_{j}\) strongly converge to u in \(L^{\gamma}((0,T),W^{1,\rho})\) as \(j\rightarrow\infty\), for every admissible pair \((\gamma, \rho)\) and all \(0< T< T^{*}\).

     

More precisely, we will prove the following results.

Theorem 1.1

Let \(W:\mathbb{R}^{3}\rightarrow\mathbb{R}\) be an even, real-valued function and \(W\in L^{p}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})\) for some \(p\geq 1\), \(\alpha_{j}\rightarrow\alpha\) and \(\beta_{j}\rightarrow\beta\) in \(H^{1}(0,T^{*})\). Assume that u is the solution of (1.1) defined on the maximal interval \([0,T^{*})\) with the initial value \(u_{0} \in H^{1}\). Suppose that \(u_{j}\) is the solution to the equation
$$ \textstyle\begin{cases} i\partial_{t}u+\Delta u+ \alpha_{j}(t)\frac{1}{\vert x\vert }u+\beta_{j}(t)(W\ast \vert u\vert ^{2})u=0, \\ u(0,x) =u_{0}(x).\end{cases} $$
(1.6)
Then:
  1. (i)

    Given any \(0< T< T^{*}\), the solution \(u_{j}\) exists on \([0,T]\) if j is sufficiently large.

     
  2. (ii)
    For every admissible pair \((\gamma, \rho)\) and \(0< T< T^{*}\), \(u_{j}\rightarrow u\) in \(L^{\gamma}((0,T),W^{1,\rho})\) as \(j\rightarrow \infty\). In particular, convergence holds in \(C([0,T],H^{1})\) for \(0< T< T^{*}\). In addition, for every admissible pair \((\gamma, \rho)\)
    $$ \Vert u_{j}-u\Vert _{L^{\gamma}((0,T),W^{1,\rho})}\leq C\Vert \alpha_{j}-\alpha \Vert _{H^{1}(0,T)}+C\Vert \beta_{j}- \beta \Vert _{H^{1}(0,T)}, $$
    (1.7)
    where C depends on \(u_{0}\), T, γ, ρ.
     

Remark

For physical reasons, in this paper, we only study the three-dimensional case. In fact, we can investigate more general unbounded potentials in \(\mathbb{R}^{N}\). Our results also follow if the potential \(V:\mathbb{R}^{N}\rightarrow\mathbb{R}\) is a real-valued function, satisfying:
  1. (V1)

    \(V\in L^{q}(\mathbb{R}^{N})+L^{\infty}(\mathbb{R}^{N})\) for some \(q\geq1\), \(q>N/2\);

     
  2. (V2)

    \(V, \nabla V\in L^{q}(\mathbb{R}^{N})+L^{\infty}(\mathbb {R}^{N})\) for some \(q \geq2\), \(q >N/2\).

     
A typical example satisfying these assumptions on V is \(\frac{1}{\vert x\vert ^{\alpha}}\) for some \(\alpha>0\).

This paper is organized as follows: in Section 2, we will collect some preliminaries such as local well-posedness, global existence and a Gronwall-type estimate. In Section 3, we firstly obtain the Lipschitz continuity of the solution \(u=u(\alpha,\beta)\) with respect to coefficients α and β, and then prove Theorem 1.1.

Notation

In this paper, we use the following notation. \(C>0\) denotes various positive constants. Because we only consider \(\mathbb{R}^{3}\), we often use the abbreviations \(L^{r}=L^{r}(\mathbb{R}^{3})\), \(H^{1}=H^{1}(\mathbb{R}^{3})\). Given any interval \(I\subset \mathbb{R}\), the norms of mixed spaces \(L^{q}(I,L^{r}(\mathbb{R}^{3}))\) and \(L^{q}(I,H^{s}(\mathbb{R}^{3}))\) are denoted by \(\Vert \cdot \Vert _{L^{q}_{t}L_{x}^{r}(I)}\) and \(\Vert \cdot \Vert _{L^{q}(I,H^{s})}\), respectively. We recall that a pair \((q,r)\) is admissible if \(\frac{2}{q}=3(\frac{1}{2}-\frac{1}{r})\) and \(2 \leq r \leq 6\). For simplicity, we always denote \(V_{1}=\frac{1}{\vert x\vert }\chi_{B_{0}}\) and \(V_{2}=\frac{1}{\vert x\vert }(1-\chi_{B_{0}})\), where \(B_{0}\) is the unit ball in \(\mathbb{R}^{3}\) centered at the original point, \(\chi_{B_{0}}\) is its characteristic function. It is obvious that \(V_{1}\in L^{\frac {3}{1+3\varepsilon}}\), \(\nabla V_{1}\in L^{\frac{3}{2+3\varepsilon}}\) with \(\varepsilon>0\) sufficiently small, \(V_{2}\in L^{\infty}\) and \(\nabla V_{2}\in L^{\infty}\).

2 Preliminaries

Firstly, we investigate the local well-posedness for (1.1). When α and β are two constants, the local well-posedness of (1.1) has been studied in [4]. In our case, when the terms \(\alpha\frac{u}{\vert x\vert }\) and \(\beta W\ast \vert u\vert ^{2}u\) have to be estimated in some norms, due to \(\alpha, \beta\in H^{1}(0,\infty)\hookrightarrow L^{\infty}(0,\infty )\), we only need to take \(L^{\infty}\) norms of α and β. Therefore, by a similar method as that in [4], we can prove the local well-posedness of (1.1).

Lemma 2.1

Assume that \(W:\mathbb{R}^{3}\rightarrow\mathbb{R}\) is an even, real-valued potential and \(W\in L^{p}+L^{\infty}\) for some \(p\geq1\). Given \(A,B,M > 0\), there exists \(\delta= \delta(A,B,M)\) such that, for all \(\alpha, \beta\in H^{1}(0,\infty)\) satisfying \(\Vert \alpha \Vert _{L^{\infty}}\leq A\), \(\Vert \beta \Vert _{L^{\infty}}\leq B\) and all \(u_{0}\in H^{1}\) satisfying \(\Vert u_{0}\Vert _{H^{1}}\leq M\), there exists a unique solution \(u\in C([0,\delta],H^{1})\) of (1.1) and \(\Vert u\Vert _{L^{\infty}((0,\delta),H^{1})}\leq2\Vert u_{0}\Vert _{H^{1}}\). In addition, the solution u of (1.1) satisfies
$$ \bigl\Vert u(t)\bigr\Vert _{L^{2}}=\Vert u_{0}\Vert _{L^{2}} $$
(2.1)
and
$$ \frac{d}{dt}E(t)=-\frac{\alpha'(t)}{2} \int_{\mathbb{R}^{3}}\frac{\vert u(t,x)\vert ^{2}}{\vert x\vert }\,dx -\frac{\beta'(t)}{4} \int_{\mathbb{R}^{3}}\bigl(W*\vert u\vert ^{2}\bigr) (x)\bigl\vert u(t,x)\bigr\vert ^{2}\,dx, $$
(2.2)
for all \(t\in[0,\delta]\), where
$$ E\bigl(u(t)\bigr)= \int_{\mathbb{R}^{3}} \biggl[\frac{1}{2}\bigl\vert \nabla u(t,x) \bigr\vert ^{2}-\frac{\alpha(t)}{2}\frac{\vert u(t,x)\vert ^{2}}{\vert x\vert }-\frac{\beta (t)}{4} \bigl(W*\vert u\vert ^{2}\bigr) (x)\bigl\vert u(t,x)\bigr\vert ^{2} \biggr]\,dx. $$
(2.3)

Next, we consider the global existence for (1.1).

Lemma 2.2

Let \(W:\mathbb{R}^{3}\rightarrow\mathbb{R}\) be an even, real-valued potential and \(W\in L^{\sigma}+L^{\infty}\) for some \(\sigma> 3/2\). If \(u_{0}\in H^{1}\), then the solution \(u(t)\) of (1.1) exists globally.

Proof

We first deduce from Hölder’s and Young’s inequalities that
$$ \int_{\mathbb{R}^{3}}\frac{\vert u(x)\vert ^{2}}{\vert x\vert }\,dx\leq C\Vert V_{1} \Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert u\Vert _{L^{\frac{6}{2-3\varepsilon}}}^{2}+ C\Vert V_{2}\Vert _{L^{\infty}} \Vert u\Vert _{L^{2}}^{2} $$
(2.4)
and
$$ \int_{\mathbb{R}^{3}} \bigl(W*\vert u\vert ^{2}\bigr)\vert u\vert ^{2}\,dx\leq C\Vert W_{1}\Vert _{L^{\sigma}} \Vert u\Vert _{L^{\frac{4\sigma}{2\sigma-1}}}^{4} +C\Vert W_{2}\Vert _{L^{\infty}} \Vert u\Vert _{L^{2}}^{4}, $$
(2.5)
where \(W=W_{1}+W_{2}\) with \(W_{1} \in L^{\sigma}\) and \(W_{2} \in L^{\infty}\).
Thus, by (2.2), (2.4) and (2.5), we derive
$$\begin{aligned} \bigl\Vert E'\bigr\Vert _{L^{2}(0,T)} \leq& C\bigl\Vert \alpha'\bigr\Vert _{L^{2}(0,T)}\bigl(\Vert V_{1} \Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert u\Vert _{L^{\frac{6}{2-3\varepsilon}}}^{2}+ \Vert V_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{2}\bigr) \\ &{}+C\bigl\Vert \beta'\bigr\Vert _{L^{2}(0,T)}\bigl(\Vert W_{1}\Vert _{L^{\sigma}} \Vert u\Vert _{L^{\frac{4\sigma }{2\sigma-1}}}^{4} +\Vert W_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{4}\bigr). \end{aligned}$$
This and (2.3) yield
$$\begin{aligned} E(t) =&E(0)+ \int_{0}^{t} E'(s)\,ds\leq E(0)+ \biggl(t \int_{0}^{t}\bigl(E'(s) \bigr)^{2}\,ds \biggr)^{1/2} \\ \leq & E(0)+Ct^{1/2}\bigl\Vert \alpha'\bigr\Vert _{L^{2}(0,T)}\bigl(\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert u \Vert _{L^{\frac{6}{2-3\varepsilon}}}^{2}+ \Vert V_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{2} \bigr) \\ &{}+Ct^{1/2}\bigl\Vert \beta'\bigr\Vert _{L^{2}(0,T)}\bigl(\Vert W_{1}\Vert _{L^{\sigma}} \Vert u\Vert _{L^{\frac{4\sigma }{2\sigma-1}}}^{4} +\Vert W_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{4} \bigr). \end{aligned}$$
(2.6)
Combining (2.2), (2.4)-(2.6), we obtain
$$\begin{aligned} \int_{\mathbb{R}^{3}}\bigl\vert \nabla u(t,x)\bigr\vert ^{2} \,dx \leq& E\bigl(u(t)\bigr)+C \int_{\mathbb{R}^{3}}\frac{\vert u(t,x)\vert ^{2}}{\vert x\vert }\,dx +C \int_{\mathbb{R}^{3}}\bigl(W*\vert u\vert ^{2}\bigr) (x)\bigl\vert u(t,x)\bigr\vert ^{2}\,dx \\ \leq& E(0)+C\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert u\Vert _{L^{\frac{6}{2-3\varepsilon}}}^{2}+ C\Vert V_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{2} \\ &{}+C\Vert W_{1}\Vert _{L^{\sigma}} \Vert u \Vert _{L^{\frac{4\sigma}{2\sigma-1}}}^{4} +C\Vert W_{2}\Vert _{L^{\infty}} \Vert u_{0}\Vert _{L^{2}}^{4}. \end{aligned}$$
(2.7)
On the other hand, we have the following Gagliardo-Nirenberg’s inequalities:
$$\begin{aligned}& \Vert u\Vert _{L^{\frac{6}{2-3\varepsilon}}}^{2}\leq C\Vert u\Vert _{H^{1}}^{1+3\varepsilon }\Vert u\Vert _{L^{2}}^{1-3\varepsilon}, \\& \Vert u\Vert _{L^{\frac{4\sigma}{2\sigma-1}}}^{4}\leq C\Vert u \Vert _{H^{1}}^{\frac {3}{\sigma} }\Vert u\Vert _{L^{2}}^{\frac{4\sigma-3}{\sigma}}. \end{aligned}$$
(2.8)
Taking ε such that \(1+3\varepsilon<2\), and \(\frac{3}{\sigma}<2\), we infer from (2.7) and Young’s inequality with ε that
$$ \bigl\Vert u(t)\bigr\Vert _{H^{1}} \leq C\bigl(T, \Vert u_{0}\Vert _{H^{1}},\Vert \alpha \Vert _{H^{1}(0,T)},\Vert \beta \Vert _{H^{1}(0,T)}\bigr)\quad \text{for every }t\in[0,T], $$
(2.9)
for every \(0< T<\infty\). This implies that the solution u of (1.1) is global. □

Finally, we recall a Gronwall-type estimate which is vital to obtain the Lipschitz continuity of solution \(u(\alpha,\beta)\) with respect to coefficients α and β.

Lemma 2.3

[24]

Let \(T >0\), \(1 \leq p_{2} < q_{2} \leq\infty\), \(1 \leq p_{1} < q_{1} \leq \infty\), and \(a,b_{1},b_{2} \geq0\). If \(f_{1}\in L^{q_{1}}(0,T)\), \(f_{2}\in L^{q_{2}}(0,T)\) satisfying
$$\Vert f_{1}\Vert _{L^{q_{1}}(0,t)}+\Vert f_{2}\Vert _{L^{q_{2}}(0,t)}\leq a+b_{1}\Vert f_{1}\Vert _{L^{p_{1}}(0,t)}+b_{2}\Vert f_{2}\Vert _{L^{p_{2}}(0,t)}, $$
for all \(0 < t <T\), then there exists \(\Gamma= \Gamma(b_{1},b_{2},p, q,T )\) such that
$$\Vert f_{1}\Vert _{L^{q_{1}}(0,T)}+\Vert f_{2}\Vert _{L^{q_{2}}(0,T)}\leq a \Gamma. $$

3 The proof of main results

In this section, we first prove that the solution of (1.1) depends local Lipschitz continuously on the coefficients α and β, and then show our main results.

Lemma 3.1

Assume that \(W:\mathbb{R}^{3}\rightarrow\mathbb{R}\) is an even, real-valued function and \(W\in L^{p}+L^{\infty}\) for some \(p\geq1\). Given \(u_{0}\in H^{1}\). Let \(u\in L^{\infty}([0,T^{*}),H^{1})\) be the corresponding solution of (1.1) with coefficients \(\alpha,\beta\in H^{1}(0,T)\). There exists \(\varepsilon> 0\) such that if α̃ and β̃ satisfy \(\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}<\varepsilon\), \(\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}<\varepsilon\) and ũ is the corresponding solution of (1.1) with coefficients α̃ and β̃. Then:
  1. (i)

    Given any \(0< T< T^{*}\), the solution ũ exists on \([0,T]\).

     
  2. (ii)
    For every admissible pair \((\gamma, \rho)\)
    $$ \Vert \tilde{u}-u\Vert _{L^{\gamma}((0,T),W^{1,\rho})}\leq C\Vert \tilde{ \alpha}-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
    (3.1)
    where C depends on \(u_{0}\), T, γ, ρ. In particular,
    $$\Vert \tilde{u}-u\Vert _{L^{\infty}((0,T),H^{1})}\leq C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}. $$
     

Proof

Firstly, we assume that the solution ũ exists on \([0,T]\). Note that the following Duhamel’s formulation:
$$ u(t)=U(t)u_{0}+i \int_{0}^{t}U(t-s) \biggl( \alpha(s) \frac{1}{\vert x\vert }u(s)+ \bigl(\beta (s)W\ast\bigl\vert u(s)\bigr\vert ^{2} \bigr)u(s) \biggr)\,ds, $$
(3.2)
where \(U(t):=e^{it\triangle}\) denotes the free Schrödinger propagator, which is isometric on \(H^{s}\) for every \(s\geq0\); see [4]. This yields
$$ \tilde{u}(t)-u(t)= i \int_{0}^{t} U(t-r) \biggl( \frac{1}{\vert x\vert }( \tilde{\alpha}\tilde{u}-\alpha u)+\bigl(\tilde{\beta}\bigl(W*\vert \tilde {u} \vert ^{2}\bigr)\tilde{u}-\beta\bigl(W*\vert u\vert ^{2} \bigr)u\bigr)\biggr) (r)\,dr. $$
(3.3)
In the following, we set \(\rho_{1}=\frac{4p}{2p-1}\), taking \(\gamma_{1}\) such that \((\gamma_{1},\rho_{1})\) is an admissible pair. Applying Strichartz’s estimate to (3.3), we deduce from Hölder’s inequality that for \(0< t\leq T\)
$$\begin{aligned}& \Vert \tilde{u}-u\Vert _{L_{t}^{\gamma} L^{\rho}_{x}(0,t)} \\& \quad \leq C\bigl\Vert \tilde{\beta}\bigl(W*\vert \tilde{u}\vert ^{2}\bigr)\tilde{u}-\tilde{\beta}\bigl(W*\vert u\vert ^{2}\bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}} L^{\rho_{1}^{\prime}}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\beta}-\beta) \bigl(W*\vert u\vert ^{2}\bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L^{\rho_{1}^{\prime}}_{x}(0,t)} \\& \qquad{}+ C\bigl\Vert V_{1}(\tilde{\alpha}\tilde{u}-\alpha u)\bigr\Vert _{L^{2}_{t}L^{\frac{6}{5}}_{x}(0,t)} +C\bigl\Vert V_{2}(\tilde{\alpha}\tilde{u}- \alpha u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \quad\leq C\Vert \tilde{\beta} \Vert _{L^{\infty}(0,t)}\bigl(\Vert \tilde{u} \Vert _{{L_{t}^{\infty}L^{\rho_{1}}_{x}(0,t)}}^{2}+ \Vert u\Vert _{L_{t}^{\infty}L^{\rho_{1}}_{x}(0,t)}^{2} \bigr)\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho _{1}}(0,t)} \\& \qquad{}+C\Vert \tilde{\beta}-\beta \Vert _{L^{\infty}(0,t)}\Vert \tilde {u} \Vert _{L_{t}^{\infty}L_{x}^{\rho_{1}}(0,t)}^{3} +C\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert \tilde{\alpha} \Vert _{L^{\infty}(0,t)} \Vert \tilde{u}-u \Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)} \\& \qquad{}+\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert \tilde{\alpha}- \alpha \Vert _{L^{\infty}(0,t)}\Vert u\Vert _{L_{t}^{2}L_{x}^{\frac {2}{1-2\varepsilon}}(0,t)} \\& \qquad{}+C\Vert V_{2}\Vert _{L^{\infty}} \Vert \tilde{\alpha} \Vert _{L^{\infty}(0,t)}\Vert \tilde{u}-u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} +\Vert V_{2}\Vert _{L^{\infty}} \Vert \tilde{\alpha}-\alpha \Vert _{L^{\infty}(0,t)}\Vert u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} \\& \quad\leq C\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho_{1}}(0,t)}+C\Vert \tilde {u}-u \Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)} +C\Vert \tilde{u}-u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} \\& \qquad{}+C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,t)} +C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,t)}, \end{aligned}$$
(3.4)
which, together with Lemma 2.3, implies that for every admissible pair \((\gamma,\rho)\)
$$ \Vert \tilde{u}-u\Vert _{L_{t}^{\gamma}L_{x}^{\rho} (0,T)}\leq C\Vert \tilde{ \alpha }-\alpha \Vert _{H^{1}(0,T)} +C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
(3.5)
where C depends on \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\), \(\Vert u\Vert _{L^{\infty}((0,T),H^{1})}\), T, γ. In addition, by a similar argument to that of (3.4), the embedding \(W^{1,\frac {2}{1-2\varepsilon}}\hookrightarrow L^{\frac{6}{1-6\varepsilon}}\), we obtain
$$\begin{aligned}& \Vert \nabla\tilde{u}-\nabla u\Vert _{L_{t}^{\gamma}L_{x}^{\rho}(0,t)} \\& \quad \leq C\bigl\Vert \tilde{\beta}\bigl(W*\vert \tilde{u}\vert ^{2}\bigr)\nabla\tilde{u}-\beta \bigl(W*\vert u\vert ^{2} \bigr)\nabla u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L_{x}^{\rho_{1}^{\prime}}(0,t)} \\& \qquad{}+ C\bigl\Vert \tilde{\beta}\bigl(W*\nabla \vert \tilde{u}\vert ^{2}\bigr)\tilde{u}-\beta\bigl(W*\nabla \vert u\vert ^{2} \bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L_{x}^{\rho_{1}^{\prime}}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha)\nabla V_{1}\tilde{u} \bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)}+C\bigl\Vert \alpha\nabla V_{1}( \tilde{u}-u)\bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)} \\& \qquad {}+C\bigl\Vert (\tilde{\alpha}-\alpha)\nabla V_{2}\tilde{u} \bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}+C\bigl\Vert \alpha\nabla V_{2}( \tilde{u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha) V_{1}\nabla\tilde{u} \bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)}+C\bigl\Vert \alpha V_{1}\nabla( \tilde{u}-u)\bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha) V_{2}\nabla\tilde{u} \bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}+C\bigl\Vert \alpha V_{2}\nabla( \tilde{u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \quad\leq C\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}} L_{x}^{\rho_{1}}(0,t)}+C\bigl\Vert \nabla( \tilde {u}- u)\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho_{1}}(0,t)} \\& \qquad{}+C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,t)}+C\Vert \tilde{ \beta }-\beta \Vert _{H^{1}(0,t)} +C\Vert \tilde{u}-u\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert \nabla(\tilde {u}-u)\bigr\Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)}+C\bigl\Vert \nabla(\tilde {u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}. \end{aligned}$$
(3.6)
Hence, it follows from Lemma 2.3 and (3.5) that for every admissible pair \((\gamma,\rho)\)
$$ \Vert \tilde{u}-u\Vert _{L^{\gamma}((0,T),W^{1,\rho})}\leq C\Vert \tilde{ \alpha }-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
(3.7)
where C depends on \(\Vert u_{j}\Vert _{L^{\infty}((0,T),H^{1})}\), \(\Vert u\Vert _{L^{\infty}((0,T),H^{1})}\), T, γ.

Therefore, in order to prove this lemma, we only need to show that there exist \(\varepsilon> 0\) and \(M>0\) such that if α̃ and β̃ satisfy \(\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}<\varepsilon\) and \(\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}<\varepsilon\), then the corresponding solution ũ exists on \([0,T]\) and \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\leq M\).

To this aim, we set
$$ M=2\sup_{0\leq t\leq T} \bigl\Vert u(t)\bigr\Vert _{H^{1}}+1. $$
(3.8)
With the notation of Lemma 2.1, let \(\delta= \delta(A,B,M)\), where \(A=\Vert \tilde{\alpha} \Vert _{L^{\infty}}\), \(B=\Vert \tilde{\beta} \Vert _{L^{\infty}}\) and M is given by (3.8). We infer from Lemma 2.1 that ũ exists on \([0,\delta]\) and that
$$ \Vert \tilde{u}\Vert _{L^{\infty}((0,\delta),H^{1})}\leq2\Vert u_{0}\Vert _{H^{1}}. $$
(3.9)
On the other hand, by (3.7), we have
$$ \bigl\Vert \tilde{u}(\delta)-u(\delta)\bigr\Vert _{H^{1}}\leq C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}+C \Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}. $$
(3.10)
Taking ε such that \(C\varepsilon< M/4\), it follows that \(\Vert \tilde{u}(\delta)\Vert _{H^{1}}< M\). Hence, we can repeat the argument to continue the solution also in the time interval \([\delta,2\delta]\), and so on. Since the solution \(u(t)\) exists on \([0,T^{*})\), and for any \(0< T< T^{*}\), we consider \([0,T]\subset[0,\delta]\cup\cdots\cup[(N-1)\delta ,N\delta]\), \(N=[\frac{T}{\delta}]+1\), where \([\cdot]\) denotes the integer part of the number. Thus, the solution ũ exists on \([0,T]\) and \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\leq M\). This completes the proof. □

Proof of Theorem 1.1

Given \(T\in(0,T^{\ast})\). Let \(A=\Vert \alpha \Vert _{L^{\infty}(0,T)}\), \(B=\Vert \beta \Vert _{L^{\infty}(0,T)}\) and \(M=2 {\sup}_{0\leq t\leq T}\Vert u(t)\Vert _{H^{1}}+1\). With the notation of Lemma 2.1, setting \(\delta=\delta(A,B,M)\). We infer from Lemma 2.1 that \(u_{j}\) exists on \([0,\delta]\) and satisfies
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,\delta),H^{1})}< 2\Vert u_{0}\Vert _{H^{1}}. $$
(3.11)
Applying Lemma 3.1, we see that the conclusion holds on the interval \([0,\delta]\).
Let \(0< l\leq T\) be such that \(u_{j}\) exists on \([0,l]\) for j sufficiently large and
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,l),H^{1})}< \infty. $$
(3.12)
Then we infer from Lemma 3.1 that
$$ u_{j}\rightarrow u\quad \text{in }L^{\gamma} \bigl((0,l),W^{1,\rho }\bigr)\text{ as }j \rightarrow\infty $$
(3.13)
for every admissible pair \((\gamma,\rho)\). In particular, \(u_{j}(l)\rightarrow u(l)\) in \(H^{1}\) as \(j \rightarrow\infty\), which, together with the definition of M yields that \(\Vert u_{j }(l)\Vert _{H^{1}}< M\). Applying Lemma 2.1 with the initial value \(u_{j }(l)\), it follows that \(u_{j }\) exists on \([0,l+\delta]\) and
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,l+\delta),H^{1})}\leq C. $$
(3.14)
It follows from Lemma 3.1 that the estimate (3.13) holds with l replaced by \(l+\delta\) provided \(l+\delta\leq T\). Iterating this argument, we see that
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,T),H^{1})}\leq C. $$
(3.15)
This estimate with Lemma 3.1 yields the desired results. □

4 Conclusions

In this paper, we consider the stability for the nonlinear Schrödinger equation (1.1) of Hartree type under the perturbation of coefficients. We first obtain the Lipschitz continuity of the solution \(u=u(\alpha,\beta)\) with respect to coefficients α and β by using Strichartz’s estimates, and then prove that this equation is stable under the perturbation of coefficients by a bootstrap argument. Our results improve some recent results. In particular, the proof of the locally Lipschitz continuity contains a very general method that may be useful for other related problems.

Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China
(2)
College of Humanities and Sciences, Northeast Normal University, Changchun, 130117, P.R. China

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