Decay estimates for Schrödinger systems with time-dependent potentials in 2D

1.   Introduction

We consider the following Cauchy problem for the nonlinear Schrödinger system in space dimension two

where $\partial _{t} = \partial/\partial t,$ $\Delta_{V_{j}} = \Delta-V_{j}(t, x), \Delta = \sum_{j = 1}^{2}\partial^{2}/\partial x_j^{2}, V_{j}(t, x)$ is a prescribed $\mathbb{R}$-valued function on $[0, \infty) \times \mathbb{R}^2$, $G_j(v_1, v_2) = K_j(v_1, v_2)+F_j(v_1, v_2)$, $K_j(v_1, v_2) = \lambda_{j}|v_j|^{p-1}v_j$, $F_{1}\left(v_{1}, v_{2}\right) = \overline{v_1}v_2$, $F_{2}\left(v_{1}, v_{2}\right) = v_1^2, \lambda_j \in \mathbb{C}\backslash \{0\}, p\geq2, m_j$ is a mass of a particle, $\phi _{j}(x)$ is a prescribed $\mathbb{C}$-valued function on $\mathbb{R}^2$, and $j = 1, 2$. In this paper, our aim is to prove the time decay estimates of solutions to (1.1)

and

for all $t \geq 0,$ where $C_1, C_2 > 0$, when the small initial data $\phi _{j}(x)$ belongs to ${\dot{H}}^{0, \alpha}(\mathbb{R}^{2})\cap {\dot{H}}^{0, \beta}(\mathbb{R}^{2})$, $2m_1 = m_2$, $\Im \lambda_j < 0$ and $V_j(t, x)$ satisfies

for all $t\geq 0$, where $c_1 > 0, 0 < \alpha < 1 < \beta < 2, j = 1, 2$ and

for all $t\geq 0$, where $U_{\sigma}(t) = \mathcal{F}^{-1}E(t)^{\sigma}\mathcal{F},$ $E(t) = e^{-\frac{i}{2}t|\xi|^{2}}, \sigma \neq 0, \mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier and its inverse transform operators, $c_2 > 0, 0 < \alpha < 1 < 1+2\mu < \beta < 2, 0 < \theta < \mu$, and $j = 1, 2$.

The Cauchy problem for nonlinear Schrödinger equations with time-dependent potentials

appears in physics, where $\Delta_V = \Delta-V(t, x), V (t, x)$ is a prescribed $\mathbb{R}$-valued function on $[0, \infty) \times \mathbb{R}^n, f: \mathbb{C}\rightarrow \mathbb{C}$. If the nonlinear term $f(v) = \lambda |v|^{q-1}v, q > 1$ and $\lambda \in \mathbb{R}$, the Cauchy problem (1.2) was considered from the mathematical point of view in ^[1] and ^[2]. In ^[1], the global existence of solutions to (1.2) with $f(v) = \lambda |v|^{q-1}v$ was studied when the initial data $\phi (x) \in H^{1, 0}(\mathbb{R}^{n})\cap H^{0, 1}(\mathbb{R}^{n})$ and the external potential $V(t, x)$ satisfy some assumptions. The time decay estimates of solutions to (1.2) with $f(v) = \lambda |v|^{q-1}v$ were considered if the initial data $\phi (x) \in H^{1, 0}(\mathbb{R}^{n})\cap H^{0, 1}(\mathbb{R}^{n})$, $\lambda > 0$, and the time-dependent potential $V(t, x)$ meets some conditions. When the time-dependent potential $V (t, x) = \sigma(t)|x|^{2}/2$ and the nonlinear term $f(v) = \nu F_L(v)v+\mu F_S(v)v$ in (1.2), where $F_L: \mathbb{C}\rightarrow \mathbb{R}, F_S: \mathbb{C}\rightarrow \mathbb{R}$, and $\nu, \mu \in \mathbb{R}$, the asymptotic behavior and time decay estimates of solutions to (1.2) for small initial data satisfying $\phi (x) \in H^{\gamma, 0}(\mathbb{R}^{2})\cap H^{0, \gamma}(\mathbb{R}^{2})$ with $\gamma > n/2$ were investigated in ^[3]. In ^[4], a sharp time decay estimate for the global in time solution to (1.2) with cubic nonlinear term and the potential $V(x)$ which satisfies $< \cdot > ^s V \in W^{1, 1}(\mathbb{R})$ was obtained in 1D. The cubic nonlinear Schrödinger equation with potential in 1D also has been studied in ^[5] and ^[6]. If the time-dependent potential $V (t, x)\equiv 0$, then the Cauchy problem (1.2) becomes

When $f(v) = \lambda |v|^{q-1}v+ \kappa |v|^{\eta-1}v, \lambda, \kappa \in \mathbb{R}$ and $q-1 = 2/n < \kappa-1$, (1.3) was investigated in ^[7]. It is known that $\eta = 2/n$ is the critical exponent if the scattering problem for (1.3) with $f(v) = \kappa |v|^{\eta-1}v$ and $\kappa > 0$ is considered (see ^[8] and ^[9]). If the small initial data $\phi(x)$ belongs to $H^{\gamma, 0}(\mathbb{R}^{n})\cap H^{0, \gamma}(\mathbb{R}^{n})$ with $n/2 < \gamma \leq q = 1+2/n$, the existence of modified scattering states for (1.3) was studied, and the sharp time decay estimate of solutions was proved in ^[7]. When $n = 1, f(v) = \lambda |v|^{q-1}v$ and $\lambda = \lambda_1+i\lambda_2, \lambda_j \in \mathbb{R}, \lambda_2 < 0, |\lambda_2| > \frac{q-1}{2\sqrt{q}}|\lambda_1|$ and $1 < q\leq 3$, (1.3) with initial data $\phi (x) \in H^{1, 0}(\mathbb{R})\cap H^{0, 1}(\mathbb{R})$ was studied in ^[10]. The time decay estimates and large time asymptotics of the solution for arbitrarily large initial data were presented if $q = 3$ or $q < 3$ and $q$ is close to 3. In ^[11], the same equation was considered. The asymptotic behavior of solutions of the Cauchy problem for initial data $\phi (x) \in H^{0, \gamma}(\mathbb{R})$ with $1/2 < \gamma \leq1$ was studied. In ^[12], the scattering for solutions to (1.3) was shown in the case of $f(v) = \lambda |v|^{q-1}v, 1+\frac{4}{n+2\gamma} < q < 1+\frac{4}{n}, 0 < \gamma\leq \min\{\frac{n}{2}, 1\}$ and $\lambda = \lambda_1+i\lambda_2, \lambda_j \in \mathbb{R}, \lambda_2 < 0, |\lambda_2| > \frac{q-1}{2\sqrt{q}}|\lambda_1|.$ If $f(v) = \sum_{j\neq 0}\lambda_j |v|^{\sigma_j-j} v^{j}, \lambda_j \in \mathbb{C}, \sigma_j > 3$ and $n = 1$ in (1.3), the existence of the scattering operator was considered in ^[13]. If the potential $V (t, x)\equiv V(x)$ and $f(v) = g(|v|^2)v$, then from (1.2) we have

where $\Delta_V = \Delta-V(x), V (x)$ is a prescribed $\mathbb{R}$-valued function on $\mathbb{R}^n, g: \mathbb{C}\rightarrow \mathbb{R}$. There is some research on the asymptotic behavior of solutions to (1.4) (see ^{[14,15,16,17]}, and references cited therein). In ^[15], when $V(x)$ is a real-valued measurable function defined in $\mathbb{R}^{2}$, $f(v) = \lambda |v|^{2\sigma}v, \sigma > \frac{1}{2}, \lambda \in \mathbb{C} \backslash \{0\}$ and $\Im \lambda\leq 0$, the scattering problem for (1.4) was studied, and by using the equivalence between the operators $(-\Delta_V)^{\frac{s}{2}}$ and $(-\Delta)^{\frac{s}{2}}$ in $L^{2}$ norm sense for $0\leq s < 1$, the time decay estimates of the solution were obained as $O(t^{-1})$ in $L^{\infty}({\mathbb{R}^{2}})$ as $t \rightarrow +\infty$. In the case of $\sigma > \frac{1}{2}$, the solution of a system of the equations in 2D has the same time decay rate under some assumptions. In ^[18] the solution of the nonlinear Schrödinger systems with quadratic nonlinearities in two space dimensions decays like $O(t^{-1} (\log t)^{-1})$ in $L^{\infty}({\mathbb{R}^{2}})$ as $t \rightarrow +\infty$, when $V(x) \equiv 0$. In ^[19], numerical method is considerd by using Fourier spectral method to solve the multidimensional nonlinear fractional-in-space Schrödinger equation involving the fractional Laplacian operator and the numerical method is effective for long time simulation of integer-order Schrödinger equation.

We find that the time decay estimates of the solutions of the Cauchy problem for two dimensional critical NLS system with potentials is an unsolved problem. So we consider the following nonlinear Schrödinger system. Let $W_{j}(t, x) = \frac{1}{2m_{j}}V_{j}(t, x)$ for $j = 1, 2$. From (1.1) we have

We assume that the masses of particles in the Cauchy problem (1.5) and the time-dependent $\mathbb{R}$-valued potential $W_j(t, x)$ satisfies the following hypotheses.

$(\text{H1})$ $2m_1 = m_2$.

$(\text{H2})$ $\|{W_{j}} \Vert_{{{\dot{H}}^{\alpha, 0}(\mathbb{R}^{2})}\cap{{\dot{H}}^{\beta, 0}(\mathbb{R}^{2})}}\leq c_1(1+t)^{-\beta}$ for $t \geq 0$, where $c_1 > 0, 0 < \alpha < 1 < \beta < 2, j = 1, 2$.

$(\text{H3})$ $\|U_{\frac{1}{m_{j}}}(-t){W_{j}} \Vert_{{{\dot{H}}^{0, \alpha}(\mathbb{R}^{2})}\cap{{\dot{H}}^{0, \beta}(\mathbb{R}^{2})}}\leq c_2(1+t)^{-\theta}$ for all $t\geq 0$, where $U_{\sigma}(t) = \mathcal{F}^{-1}E(t)^{\sigma}\mathcal{F},$ $E(t) = e^{-\frac{i}{2}t|\xi|^{2}}, \sigma \neq 0, \mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier and its inverse transform operators, $c_2 > 0, 0 < \alpha < 1 < 1+2\mu < \beta < 2, 0 < \theta < \mu$, and $j = 1, 2$.

We use the assumption of mass resonance $(\text{H1})$ to deal with the nonlinearity $F_j (v_1, v_2)$ in $(1.5)$ for $j = 1, 2.$ The assumptions of the potential $W_j(t, x)$ are $(\text{H2})$ and $(\text{H3})$. The assumption $(\text{H2})$ is used to investigate Lemma 2.3 and Lemma 3.2. The assumption $(\text{H3})$ is applied to the proofs of Lemma 3.2 and Lemma 3.4.

If the mass resonance condition $(\text{H1})$ holds, then the Cauchy problem (1.5) meet the following gauge condition

for any $\theta \in \mathbb{R}$. If $W_j = 0$ for $j = 1, 2$ in (1.5), then the system (1.5) becomes

where $F_{1}\left(v_{1}, v_{2}\right) = \overline{v_1}v_2, F_{2}\left(v_{1}, v_{2}\right) = v_1^2, m_j$ is a mass of a particle, and $\phi _{j}(x)$ is a prescribed $\mathbb{C}$-valued function on $\mathbb{R}^2$ for $j = 1, 2$. The time decay of small solutions to (1.6) with $\Im \lambda_j\leq0$ for $j = 1, 2$ was studied in the situation of small initial data $\phi _{j} \in H^{\gamma, 0}(\mathbb{R}^{2})\cap H^{0, \gamma}(\mathbb{R}^{2})$ with $1 < \gamma < 2$ in ^[18]. If $\lambda_j = 0$ for $j = 1, 2,$ then from (1.6) we have

In ^[20], global existence and time decay estimates of solutions to (1.7) for small initial data $v_{j}(0, x)\in H^{2, 0}(\mathbb{R}^{2})\cap H^{0, 2}(\mathbb{R}^{n})$ with $j = 1, 2$ were investigated under the mass resonance condition $2m_1 = m_2$. Large-time asymptotic behavior of solutions to the Cauchy problem for nonlinear Schrödinger equations

was considered in ^[21] and ^[22]. As far as we know, the time decay of the Cauchy problem (1.5) with the time-dependent potentials $V_j(t, x)$ has not been shown, where $j = 1, 2$.

Multiplying the equations of (1.5) by $i\overline{{v}_j}$ respectively, and taking the real parts of the result, we obtain

Since the assumption $W_{j}(t, x)$ is the $\mathbb{R}$-valued potential, and by integrating in space, we find

Under the assumption that $\Im \lambda_j \leq 0$ for $j = 1, 2$, by (1.9) we obtain

Therefore, we prove stability in time of solutions in the neighborhood of solutions to a suitable approximate equation. Our main purpose in this paper is to show time decay estimates of solutions to the Cauchy problem (1.5) with small initial data in $\dot{H}^{0, \alpha}(\mathbb{R}^{2})\cap \dot{H}^{0, \beta}(\mathbb{R}^{2}), 0 < \alpha < 1 < \beta < 2$. Combining the methods of ^[7] and ^[18], we obtain the following result under the assumptions that $\Im \lambda_j < 0$ for $j = 1, 2,$ $(\text{H1})$, $(\text{H2})$ and $(\text{H3})$ hold. Our main idea is to consider $\Delta v_j$ and $W_{j}v_j$ separately. By the assumptions $(\text{H2})$ and $(\text{H3})$ of the potential $W_j$, we can analyze the linear term $W_{j}v_j$ of the system (1.5). The time decay estimates of the solutions to the nonlinear Schrödinger equations are studies in ^[15] by regarding the $\frac {1}{m_j}\Delta-W_j$ as one whole. Our method differs from the approaches. The potential weakness of this paper is that the potentials $W_{j}$ satisfy the decaying condition $(\text{H2})$ and $(\text{H3})$. So if not, whether the time decay estimates of the solutions to the system (1.5) can be obtained is an open problem.

Theorem 1.1. Assume that (1.5) satisfies the mass resonance condition $(\mathit{\text{H1}})$, the time-dependent potential $W_{j}(t, x)$ satisfies $(\mathit{\text{H2}}), (\mathit{\text{H3}})$ and $\Im \lambda_{j} < 0, j = 1, 2$. Then there exist constants $\varepsilon_0 > 0$ and $C_1, C_2 > 0$ such that for any $\varepsilon \in(0, \varepsilon_0)$ and

where $0 < \alpha < 1 < \beta < 2$, there exist a unique global solution $v = (v_{1}, v_{2})$ of (1.5) satisfying

and the time decay estimates

for $t \geq 0$.

Remark 1.1. By Lemma 2.1, we have

where $0 < \alpha < 1 < \beta < 2, j = 1, 2.$ By the assumption (H2) and (1.12), we have

for $t\geq 1$, which is used in the proof of Lemma 3.4, where $0 < \alpha < 1 < \beta < 2, j = 1, 2.$ We also get Lemma 2.3 by the assumption (H2) and some other conditions, which is applied to the proof of Lemma 3.2.

The rest of this paper is organized as follows. In Section 2, we give some notations and basic lemmas. We prove Theorem 1.1 in Section 3 by using the strategy introduced in ^[18,23].

2.   Preliminaries

In this section, we give some estimates as preliminaries. In what follows, we use the same notations both for the vector function spaces and the scalar ones. For any $p$ with $1\leq p\leq \infty$, $L^{p}(\mathbb{R}^{2})$ denotes the usual Lebesgue space with the norm $\left\| \phi \right\| _{L^P (\mathbb{R}^{2})} = \left(\int_{\mathbb{R}^{2}} |{\phi(x)}|^{p}dx\right)^{\frac{1}{p}}$, if $1\leq p < \infty$ and $\left\| \phi \right\| _{L^ \infty (\mathbb{R}^{2})} =$ ess $\cdot \sup\limits_{x\in \mathbb{R}^2} |\phi(x)|$. For any $m, s\in \mathbb{R}$, weighted Sobolev space $H^{m, s}(\mathbb{R}^{2})$ is defined by

where the Sobolev norm is defined as

for $j = 1, 2$. Also we define the homogeneous Sobolev seminorm $\| f_{j} \Vert _{\dot{H}^{m, s}(\mathbb{R}^{2})}$ as

for $j = 1, 2$.

We define the dilation operator by

and

Let $U_{\alpha }\left(t\right) = \mathcal{F}^{-1}E(t)^{\alpha}\mathcal{F}$ with $\alpha \neq 0$, where the Fourier transform of $f$ is

and the inverse Fourier transform of $g$ is

The evolution operator $U_{\alpha }\left(t\right)$ and inverse evolution operator $U_{\alpha }\left(-t\right)$ for $t \neq 0$, are written as

and

respectively.

The operator $|J_{\frac{1}{m}}|^{s}(t)$ is given by

which is represented as

for $t \neq 0$. Let $[E, F] = EF-FE$. We have the commutator relations

for $s > 0.$ These formulas are essential tools for studying the asymptotic behavior of solutions to (1.1) (see ^[24]). And in what follows, we denote several positive constants by the same letter C, which may vary from one line to another.

We start with the following lemma.

Lemma 2.1. Let $0 < s_1 < 1 < s_2$. Then we have

By the Cauchy-Schwarz inequality, we have Lemma 2.1. We shall not give the proof here.

We next recall the well known results (see ^[7]).

Lemma 2.2. Let $0\leq s < 2, \rho\geq 2, m > 0$. Then we have

and

Using the factorization formula $U_{\frac{1}{m_{j}}}\left(-t\right) = -M^{m_{j}}{\mathcal{F}}^{-1}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}$ and the assumption (H2), we have

Lemma 2.3. Let $\mathcal{M}_{m_{j}}^{-1} = \mathcal{F}M^{-m_{j}}\mathcal{F}^{-1}$, $m_j > 0$ and $j = 1, 2$. If $W_{j}$ satisfies the assumption (H2), then there exists a constant $C_3 > 0$ such that

for $t\geq 1$, where $0 < \alpha < 1 < \beta < 2$.

Proof. By using the factorization formula $U_{\frac{1}{m_{j}}}\left(-t\right) = -M^{m_{j}}{\mathcal{F}}^{-1}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}},$ we have

for $t \geq 1$, where $s = \alpha$ or $\beta.$ Then we obtain that

for $t\geq 1$, where $0 < \alpha < 1 < \beta < 2$. By the assumption (H2) and (2.5), we get

for $t \geq 1$, where $C_3 = \frac{c_1}{\min\{m_1^s, m_2^s\}}$, and $0 < \alpha < 1 < \beta < 2$. □

3.   Proof of Theorem 1.1

3.1.   A priori estimates of solutions

We define the function space $X_T$ as follows

where $T > 0$ and $U_{\frac{1}{m}}(t)f_ = \left(U_{\frac{1}{m_{1}}}(t)f_{1}, U_{\frac{1}{m_{2}}}(t)f_{2}\right)$. We can obtain the local existence of solutions to (1.1) by the standard contraction mapping principle (see ^[23]).

Multiplying both sides of (1.5) by $D_{\frac{1}{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t)$, $j = 1, 2$, we use the factorization formula $\mathcal{F}U_{\frac{1}{m_{j}}}(-t) = -\mathcal{M}_{m_{j}}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}$ with $\mathcal{M}_{m_{j}} = \mathcal{F}M^{m_{j}}\mathcal{F}^{-1}$ to get

Using the identity operator $I = -D_{\frac{t}{m_{j}}}D_{\frac{m_{j}}{t}}, j = 1, 2$, we have

We now use the identity ${D_a}E^{-b}f(t, x) = {\frac{1}{ia}}e^{\frac{ibt{|\xi|^2}}{2a^2}}f\left(t, {\frac{x}{a}}\right) = E^{\frac{b}{a^2}}{D_a}f(t, x)$ for $a\neq0$ to get

Let $\theta_j = e^{-\frac{t{|\xi|^2}}{2{m_{j}}^2}}$, then we have

Let $\widetilde{{{v}}}_{k} = E^{\frac{1}{m_{k}}}D_{\frac{m_{k}}{t}}{v_{k}}, k = 1, 2$. By the factorization formula $\mathcal{F}U_{\frac{1}{m_{k}}}(-t) = -\mathcal{M}_{m_{k}}E^{\frac{1}{m_{k}}}D_{\frac{m_{k}}{t}}$, we have $\widetilde{{v}}_{k} = -{\mathcal{M}^{-1}_{m_{k}}}\mathcal{F}U_{\frac{1}{m_{k}}}(-t){v_{k}}$. By the mass resonance condition $(\text {H1})$, we get

Next we consider $\mathcal{F}U_{\frac{1}{m_{j}}}(-t)\left(W_{j}{v_{j}}\right)$ similarly, we have

Let $D_{\frac{m_{j}}{t}}{W_{j}} = E^{-\frac{1}{m_{j}}}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}{W_{j}},$ $D_{\frac{m_{j}}{t}}{v_{j}} = E^{-\frac{1}{m_{j}}}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}{v_{j}},$ and $\widetilde{{{W}}}_{j} = E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}{W_{j}}, \widetilde{{{v}}}_{j} = E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}{v_{j}}, j = 1, 2$. By the factorization formula $\mathcal{F}U_{\frac{1}{m_{j}}}(-t) = -\mathcal{M}_{m_{j}}E^{\frac{1}{m_{j}}}D_{\frac{m_{j}}{t}}$, we have $\widetilde{{W}}_{j} = -{\mathcal{M}^{-1}_{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t){W_{j}},$ $\widetilde{{v}}_{j} = -{\mathcal{M}^{-1}_{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t){v_{j}}$. By the definition of the operator $E(t)$, we get

We set

and $u_j = D_{\frac{1}{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t){v_{j}}, S_j = D_{\frac{1}{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t){W_{j}}$, then we have

Multiplying both sides of the above identity by $D_{\frac{1}{m_{j}}}$, we have

Therefore, from (1.5) we have

for $j = 1, 2$. Multiplying both sides of (3.1) by $\overline{{u}_j}$, taking the imaginary parts, we obtain

By the definitions of the nonlinear terms $F_{1}\left(u_{1}, u_{2}\right) = \overline{u_1}u_2$, $F_{2}\left(u_{1}, u_{2}\right) = u_1^2,$ and $\Im \lambda_j < 0$ for $j = 1, 2$, we obtain

We integrate the inequality above in time and use $u_j = D_{\frac{1}{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t)v_j$ to obtain that

Next we estimate $\sum_{j = 1}^{2}\|\left({E^{-{m_j}}}S_j\right) u_j\Vert_{L^{\infty}(\mathbb{R}^{2})}$ and $\sum_{j = 1}^{2}\sum_{l = 1}^{2} \|R_{lj}\Vert_{L^{\infty}(\mathbb{R}^{2})}$.

Lemma 3.1. We have

for $t\geq1,$ where $0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$, ${\|U_{\frac{1}{m}}(-t)v\Vert}_{{{\dot{H}}^{0, \alpha}(\mathbb{R}^{2})}\cap{{\dot{H}}^{0, \beta}(\mathbb{R}^{2})}}: = \sum_{j = 1}^{2}{\|U_{\frac{1}{m_j}}(-t)v_j\Vert}_{{{\dot{H}}^{0, \alpha}(\mathbb{R}^{2})}\cap{{\dot{H}}^{0, \beta}(\mathbb{R}^{2})}}$.

Proof. By the definitions of $S_{j}$ and $u_{j}$, Lemma 2.1 and the assumption $(\text{H3})$, we obtain

where $0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$. Thus we have the desired result. □

Lemma 3.2. We have

for $t\geq1,$ where $p\geq 2, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$, and

Proof. Let $h_{k, j} = -D_{\frac{m_{j}}{m_k}}{\mathcal{M}^{-1}_{m_{k}}}\mathcal{F}U_{\frac{1}{m_{k}}}(-t){v_{k}}$. By the definition of $R_{1j}$, the Cauchy-Schwarz inequality, Lemmas 2.1–2.3 and the assumption $(\text{H3})$, we have

for $t \geq 1$, where $p\geq 2, 0 < s_1 < 1 < s_2, s_1 = \alpha, s_2+2\mu = \beta < 2$, $0 < \theta < \mu$.

We next consider the estimate $\|R_{2j}\Vert_{L^{\infty}(\mathbb{R}^{2})}$. By the definition of $R_{2j}$, Lemma 2.1 and the assumption $(\text{H3})$, we get

where $p\geq 2, 0 < \mu < 1$, $0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$.

Therefore we obtain

where $p\geq 2, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$. □

By Lemmas 3.1 and 3.2, we have from (3.2)

By the Lemma 2.1 and the existence of local solutions of (1.5), we have

where $T > 1$. Then we obtain

where $0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$.

Lemma 3.3. Let $f\in{{{\dot{H}}^{0, \alpha}(\mathbb{R}^{2})}\cap{{\dot{H}}^{0, \beta}(\mathbb{R}^{2})}}.$ Then we get

for $t\geq1$, where $0 < \alpha < 1 < 1+2\mu < \beta$.

Similar to the proof of Lemma 3.2, we obtain Lemma 3.3. We omit the proof of Lemma 3.3 here. By Lemmas 2.2 and 3.3, the assumptions $(\text{H2})$ and $(\text{H3})$, we have the following lemma.

Lemma 3.4. We have

for any $t\in[1, T]$, where $p\geq 2, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$.

Proof. Let us consider the integral equation of (1.5) which is written as

Multiplying both sides of (3.4) by $|J_{\frac{1}{m_j}}|^s(t) = U_{\frac{1}{m_j}}(t)|x|^sU_{\frac{1}{m_j}}(-t), s = \alpha$ or $\beta$, by Lemma 2.2 we have

for $j = 1, 2$.

By Lemma 3.3, (1.13) in Remark 1.1, the assumption $(\text{H3})$, we obtain

where $s = \alpha$ or $\beta$, $p\geq 2, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$. This completes the proof of the lemma. □

Lemma 3.5. There exists a small $\delta > 0$ such that

for any $t\in[1, T]$, where $p\geq2,$ ${\varepsilon}^{\frac{1}{2}} < \delta < \min\{\theta, \frac{\mu}{p}\}$, $0 < \alpha < 1 < 1+2\mu < \beta < 2,$ $0 < \theta < \mu$.

Proof. Let

By (3.3), Lemma 3.4, we get

Then we have

Since

then from (3.6) we obtain

Thus we get

If we assume that there exists a time $t \in [1, T]$ such that $K(t)+\widetilde{H}(t)\leq{\varepsilon}^{\frac{1}{2}},$ then by (3.7), we have

where $\varepsilon^{\frac{1}{2}} < \delta < \mu, p \geq 2.$ By Gronwall's inequality, we have

where $C_{4} = C e^{C\int_{1}^{t}{\tau}^{-1-\theta}d\tau}.$ Therefore if we choose $\varepsilon > 0$ small enough, from (3.5) we get

where $p\geq2, 0 < \delta < \min \{\frac{\mu}{p}, \theta\}$, $0 < \theta < \mu$.

Thus we have

This contradicts the assumption that there exists a time $t \in [1, T]$ such that $K(t)+\widetilde{H}(t)\leq{\varepsilon}^{\frac{1}{2}}.$ This completes the proof of the lemma. □

3.2.   Time decay estimates of the global solutions

By Lemma 3.5, we have global existence of solutions to the Cauchy problem (1.5). By Lemmas 3.3 and 3.5, we have

Therefore we get the time decay estimate (1.10)

for $t\geq1.$

We are now in the position of proving the delicate decay estimates of solutions to (1.5). From (3.1), we have

where $u_j = D_{\frac{1}{m_{j}}}\mathcal{F}U_{\frac{1}{m_{j}}}(-t){v_{j}}$ for $j = 1, 2$. Let $|u| = \left(\sum_{j = 1}^{2}|u_j|^2\right)^{\frac{1}{2}}$. We can get positive constants $\lambda_{*}, \lambda^{*}$ such that

By Lemmas 3.1, 3.2, 3.5, (3.8) and (3.9), we obtain

for $t \geq 1$, where ${\varepsilon}^{\frac{1}{2}} < \delta < \min\{\theta, \frac{\mu}{p}, \mu-\theta, \frac{p+\mu-2-\theta}{p-1}\}$, $p\geq 2, 0 < \alpha < 1 < 1+2\mu < \beta < 2, 0 < \theta < \mu$.

Let us consider the case of $p = 2$. Multiplying both sides of (3.10) by $(\log t)^{3},$ we have

for $t\geq 1$. By Young's inequality, we obtain:

for $t \geq 1$. Thus we have

for $t \geq 1$, where ${\varepsilon}^{\frac{1}{2}} < \delta < \min\{\theta, \frac{\mu}{2}, \mu-\theta \}, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$.

Integrating the above inequality in time, we have

for $t \geq 2.$ By Lemmas 3.3, 3.5 and (3.11), we have

for $t\geq 2$, where ${\varepsilon}^{\frac{1}{2}} < \delta < \min\{\theta, \frac{\mu}{2}, \mu-\theta \}, 0 < \alpha < 1 < 1+2\mu < \beta < 2$, $0 < \theta < \mu$. Therefore, we get the desired time decay estimate (1.11).

From (3.4), we have

Let ${v_{j+}} = U_{\frac{1}{m_j}}(-1) \phi_j-i\int_{1}^{{\infty}}U_{\frac{1}{m_j}}(-\tau) G_j(v_1, v_2)+U_{\frac{1}{m_j}}(-\tau)\left(W_j v_j\right)d\tau$. Then we have

We can also obtain the scattering $\lim\limits_{t\to+\infty} \Vert U_{\frac{1}{m_j}}(-t)v(t)- v_{+}\Vert_{{\dot{H}}^{0, \alpha}(\mathbb{R}^{2})\cap {\dot{H}}^{0, \beta}(\mathbb{R}^{2})} = 0$ from the time decay estimate (1.10), where $0 < \alpha < 1 < \beta < 2$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The work is supported by the Education Department of Jilin Province of China under Grant Number JJKH20220527KJ and the Program for Young and Middle-aged Leading Talents in Scientific and Technological Innovation of Jilin Province (20200301053RQ).

Conflict of interest

All authors declare no conflict of interest in this paper.