Scattering in the Energy Space for Solutions of the Damped Nonlinear Schrödinger Equation on ${\mathbb{R}}^d\times \mathbb{T}$

Abstract

We will show, in any space dimension $d\ge 3$, the decay and scattering in the energy space for the solution to the damped nonlinear Schrödinger equation posed on ${\mathbb{R}}^d\times \mathbb{T}$ and initial data in $H^1({\mathbb{R}}^d\times \mathbb{T})$. We will also derive new bilinear Morawetz identities and corresponding localized Morawetz estimates.

1. Introduction

We will consider the following Cauchy problem for the nonlinear defocusing damped Schrödinger equation posed on ${\mathbb{R}}^d\times \mathbb{T}$, for any space dimensions $d\ge 1$:

$$\left\{\begin{array}{c}i{\partial }_{t}u+{\Delta }_{x,y}u+ib\left(t\right)u-\lambda u{\left|u\right|}^{\beta }=0,(t,x,y)\in [0,\infty )\times {\mathbb{R}}^d\times \mathbb{T},\hfill \\ u(0,x,y)=f(x,y)\in H^1({\mathbb{R}}^d\times \mathbb{T}),\hfill \end{array}\right.$$

(1)

where

$${\Delta }_{x,y}={\Delta }_x+{\partial }_y^2=\sum _{i=1}^d{\partial }_{x_i}^2+{\partial }_y^2,$$

with $\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}$ endowed with the flat metric and $\lambda >0$. Here, $u=u(t,x,y):[0,\infty )\times {\mathbb{R}}^d\times \mathbb{T}\to \mathbb{C}$, $b:[0,\infty )\to \mathbb{C}$ is a measurable function that contains dissipative and oscillatory terms and $0<\beta <4/(d-1)$. We shall also assume $\Re b\left(t\right),\Im b\left(t\right)\in \mathcal{C}\left(\right[0,\infty \left)\right)$ with $\Re b\left(t\right)\ge 0,$

$$\begin{array}{c}\hfill \mathbf{B}\left(t\right)={\int }_0^{t}b\left(s\right)ds,\underset{t>0}{inf}\left({\displaystyle \frac{\Re \mathbf{B}\left(t\right)}{t}}\right)=\mathbf{b}\ge 0,\end{array}$$

(2)

$$\begin{array}{c}\hfill {\parallel e^{-\Re \mathbf{B}\left(t\right)}\parallel }_{L^{\infty }\left([0,\infty )\right)}<\infty .\end{array}$$

(3)

The first two conditions (2) above ensure, in the strict inequality regime, that every global solution of (1) behaves like the solution to the associated free equation, that is, $b\left(t\right)=\lambda =0$ if $t\to +\infty$. The main aim of this paper is to show the decay and scattering of the solutions to (1) in the energy space. More explicitly, we will prove the following result:

Theorem 1.

Let $d\ge 3$, $\lambda >0$ and let $u\in \mathcal{C}([0,\infty );H^1({\mathbb{R}}^d\times \mathbb{T}))$ be a global solution to (1) with initial data $f\in H^1({\mathbb{R}}^d\times \mathbb{T})$ such that (2) and (3) are satisfied. Then, for $2<r<{\displaystyle \frac{2d+2}{d-1}}$, one achieves

$$\underset{t\to \infty }{lim}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x,y)\parallel }_{L^r({\mathbb{R}}^d\times \mathbb{T})}=0.$$

(4)

In addition, the solution to (1) scatters, that is, there exist $f_{\pm }\in H^1({\mathbb{R}}^d\times \mathbb{T})$ such that

$$\underset{t\to \pm \infty }{lim}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x,y)-e^{it{\Delta }_{x,y}}{f}_{\pm }\parallel }_{H^1({\mathbb{R}}^d\times \mathbb{T})}=0,$$

(5)

if the following are true:

(a) 

${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$;

(b) 

$0<\beta \le 4/d$ under the condition that (2) is fulfilled with strict inequality.

The nonlinear Schrödinger equation

$$\left\{\begin{array}{c}i{\partial }_{t}u+\Delta u-{\left|u\right|}^{\tilde{\beta }}u=0,\hfill \\ u\left(0\right)=f\left(x\right)\in H^1\left({\mathbb{R}}^d\right),\hfill \end{array}\right.$$

(6)

represents a cornerstone in mathematical physics, with deep implications across quantum mechanics, nonlinear optics, plasma physics, and fluid mechanics. In quantum mechanics, for example, this equation provides crucial insights into Bose–Einstein condensates by modeling the self-interactions of charged particles. The scattering properties of solutions to (6) represent a classical problem in mathematical physics, thoroughly documented in [1] and references therein. Understanding the asymptotic behavior of these solutions fundamentally depends on analytical tools such as the Morawetz multiplier technique and the resulting estimates. Morawetz estimates were first established in [2] for the Klein–Gordon equation with general nonlinearity. Their significance in scattering theory became evident when they were subsequently employed to prove asymptotic completeness in noteworthy studies, first by the authors of [3] for the cubic NLS in ${\mathbb{R}}^3$ (that is, (6) with $\tilde{\beta }=2$), and later by the authors of [4] for the Schrödinger equation in ${\mathbb{R}}^d$ with pure power nonlinearity as in (6), when $4/d<\tilde{\beta }<4/(d-2)$. A significant methodological advancement has recently simplified scattering proofs through the development of bilinear Morawetz inequalities, also termed interaction or quadratic Morawetz inequalities. Key contributions to this approach can be found in numerous studies, including the following: Refs. [5,6], examining cubic and quintic defocusing NLS in ${\mathbb{R}}^3$; the analysis in [7], demonstrating interaction Morawetz properties and asymptotic completeness for cubic defocusing nonlinear Schrödinger equation in ${\mathbb{R}}^2$; the work [8], presenting interaction Morawetz estimates without Bi-Laplacian involvement for $L^2$-supercritical $H^1$-subcritical nonlinear Schrödinger equation in ${\mathbb{R}}^d$ with $d\ge 1$, including applications to the nonlinear problem in 3d exterior domains; and the comprehensive survey [9], establishing quadratic Morawetz estimates and scattering for the nonlinear Schrödinger equation in an $L^2-H^1$-intercritical case. In our previous work [10] (see also [11,12,13]), we developed a method combining Morawetz inequalities with localization steps and interpolation within a contradiction framework to demonstrate solution decay in energy space, applicable to Equation (6). Motivated by this, we present a generalization of this technique to the damped nonlinear Schrödinger equation defined on the wave-guide spaces ${\mathbb{R}}^d\times \mathbb{T}$ for any space dimensions $d\ge 1$. The nonlinear Schrödinger equation with linear damping is essential across various scientific fields, such as nonlinear optics, plasma physics, and fluid dynamics. It provides key insights into complex phenomena, including optical pulse propagation in nonlinear materials, wave dynamics in plasmas subject to magnetic fields, and specific fluid flow behaviors. For further details and examples, we refer to the studies presented in [14,15]. Our novel contribution simplifies and extends the approaches used in [16,17,18,19]. We also point out that the methods utilized in the aforementioned papers are circumvented here due to the necessity of dealing directly with the complex-valued function $b\left(t\right)$ appearing in Equation (1). To be more precise, here, we establish new Morawetz-type identities and their interaction variants, along with corresponding inequalities, applicable to Equation (1). We focus on localizing the nonlinear terms appearing in these Morawetz inequalities onto suitable space–time regions, where the spatial sets are specifically selected as cubes in ${\mathbb{R}}^d$. Finally, by employing an argument by contradiction, we deduce the decay behavior of the $L^r({\mathbb{R}}^d\times \mathbb{T})$-norms of solutions to (1) when $t\to \infty$, provided that the exponent r satisfies $2<r<2(d+1)(d-1)$ and, as a straightforward effect, the scattering in $H^1({\mathbb{R}}^d\times \mathbb{T}).$

2. Preliminaries

Before outlining our main results, we will unveil some necessary notations and several useful results. For any two positive real numbers $a,b,$ we write $a\lesssim b$ (respectively, $a\gtrsim b$) to denote $a\le Cb$ (respectively, $Ca\ge b$), with $C>0,$ and unravel the constant only when it is necessary. We introduce the Banach spaces $L^r({\mathbb{R}}^d\times \mathbb{T})=L_{x,y}^r$ and $L_t^q{L}_{x,y}^r$, for $1\le r<\infty$, endowed with the norms

$$\begin{array}{c}\hfill {∥f∥}_{L_{x,y}^r}^r={\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\left|f(x,y)\right|}^{r}dxdy<\infty \end{array}$$

and

$${\parallel f\parallel }_{L_t^q{L}_{x,y}^r}={\left({\int }_0^{\infty }{∥f\left(t\right)∥}_{L_{x,y}^r}^{q}dt\right)}^{1/q},$$

respectively, with obvious modification for $r,q=\infty$. We also define

$$H^{1,r}({\mathbb{R}}^d\times \mathbb{T})=H_{x,y}^{1,r}={\left(1-{\Delta }_x-{\partial }_y\right)}^{-{\displaystyle \frac{1}{2}}}{L}^r({\mathbb{R}}^d\times \mathbb{T})$$

and

$$H_x^s{H}_y^{\sigma }={(1-{\Delta }_x)}^{-{\displaystyle \frac{s}{2}}}{(1-{\partial }_y^2)}^{-{\displaystyle \frac{\sigma }{2}}}{L}_{x,y}^2.$$

We adopt the notation $L_{(t_1,t_2)}^{q}X$ when one restricts t in some interval having endpoints $0\le t_1,t_2\le \infty$. We itemize, at this point, a series of achievements available in [13]. We recall the following existence and uniqueness result.

Theorem 2.

Let $d\ge 1$ and $0<\beta <{\displaystyle \frac{4}{d-1}}$ be given. Then, the following holds:

1. 

For any initial condition $f\in H^1({\mathbb{R}}^d\times \mathbb{T})$, the equation described in (1) admits a uniquely determined local-in-time solution:

$$u(t,x,y)\in \mathcal{C}([0,T);H^1({\mathbb{R}}^d\times \mathbb{T})),$$

where T depends on ${\parallel f\parallel }_{H_{x,y}^1}$, i.e., $T=T(\parallel f{\parallel }_{H_{x,y}^1})>0$.

2. 

The solution $u(t,x,y)$ admits a global extension with respect to the time variable.

We notice that the local existence and uniqueness for (1) follows directly from the theorem above since $e^{-\mathbf{B}\left(t\right)}$ is a bounded function on the set $[0,\infty )$. The global well-posedness is instead a consequence of the inequality (25) below. Likewise, we have the Strichartz estimate.

Proposition 1.

Let be $d\ge 1,$ and $\sigma \in \mathbb{R}.$ Then, the following homogeneous estimates hold:

$$\begin{array}{c}\hfill {\parallel e^{it{\Delta }_{x,y}}f\parallel }_{L_t^q{L}_x^r{H}_y^{\sigma }}\le C{\parallel f\parallel }_{H_{x,y}^{s,\sigma }},\end{array}$$

(7)

when the pair $(q,r)$ satisfies the condition

$$\begin{array}{c}\hfill {\displaystyle \frac{2}{q}}+{\displaystyle \frac{d}{r}}={\displaystyle \frac{d}{2}}-s,\end{array}$$

(8)

with $s<{\displaystyle \frac{d}{2}}$ and

• $4\le q\le \infty$ and $2\le r\le \infty ,$ for $d=1;$
• $2<q\le \infty$ and $2\le r\le \infty ,$ for $d=2;$
• $2\le q\le \infty$ and $2\le r\le \infty ,$ for $d\ge 3.$

Furthermore, the following inhomogeneous inequalities are satisfied:

$$\begin{array}{c}\hfill {∥{\int }_0^t{e}^{-i(t-\tau ){\Delta }_{x,y}}F\left(\tau \right)d\tau ∥}_{L_t^{\ell }{L}_x^p{H}^{\sigma }}\le C{\parallel F\parallel }_{L_t^{{\tilde{\ell }}^{\prime }}{L}_x^{{\tilde{p}}^{\prime }}{H}^{\sigma }},\end{array}$$

(9)

provided that

$${\displaystyle \frac{2}{\ell }}+{\displaystyle \frac{d}{p}}={\displaystyle \frac{2}{\tilde{\ell }}}+{\displaystyle \frac{d}{\tilde{p}}}={\displaystyle \frac{d}{2}},\ell \ge 2,(\ell ,2)\ne (2,2).$$

Moreover, the following extended inhomogeneous estimates are fulfilled:

$$\begin{array}{c}\hfill {∥{\int }_0^t{e}^{i(t-\tau ){\Delta }_{x,y}}F(\tau ,x,y)d\tau ∥}_{L_t^q{L}_x^r{H}_y^{\sigma }}\le C{∥F∥}_{L_t^{{\tilde{q}}^{\prime }}{L}_x^{{\tilde{r}}^{\prime }}{H}_y^{\sigma }},\end{array}$$

(10)

when the Schrödinger-acceptable pairs $(q,r)$ and $(\tilde{q},\tilde{r})$ verify the condition

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{\tilde{q}}}={\displaystyle \frac{d}{2}}\left({\displaystyle \frac{1}{{\tilde{r}}^{\prime }}}-{\displaystyle \frac{1}{r}}\right),\end{array}$$

(11)

with $2\le q,r\le \infty$ and $2\le \tilde{q},\tilde{r}\le \infty$, and satisfy the following:

• For $d=1,$ no additional conditions are needed;
• For $d=2,$ conditions $r<\infty$ and $\tilde{r}<\infty$ are required;
• For $d\ge 3,$ the further conditions

  $$\begin{array}{c}\hfill {\displaystyle \frac{1}{q}}<{\displaystyle \frac{1}{{\tilde{q}}^{\prime }}},{\displaystyle \frac{d-2}{d}}\le {\displaystyle \frac{r}{\tilde{r}}}\le {\displaystyle \frac{d}{d-2}},\end{array}$$

  (12)

  are needed.

We also have the following (see, again, [13]):

Lemma 1.

Assume that $d\ge 1$, ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$, and $\sigma ={\displaystyle \frac{\beta d-4}{2\beta }}$. Then, one can find $\varrho \in (0,1)$ and $(q_{\varrho },r_{\varrho },{\tilde{q}}_{\varrho },{\tilde{r}}_{\varrho }),$ such that

$$0<{\displaystyle \frac{1}{q_{\varrho }}},{\displaystyle \frac{1}{r_{\varrho }}},{\displaystyle \frac{1}{{\tilde{q}}_{\varrho }}},{\displaystyle \frac{1}{{\tilde{r}}_{\varrho }}}<{\displaystyle \frac{1}{2}},$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{q_{\varrho }}}+{\displaystyle \frac{1}{{\tilde{q}}_{\varrho }}}<1,& {\displaystyle \frac{d-2}{d}}<{\displaystyle \frac{r_{\varrho }}{{\tilde{r}}_{\varrho }}}<{\displaystyle \frac{d}{d-2}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{q_{\varrho }}}+{\displaystyle \frac{d}{r_{\varrho }}}<{\displaystyle \frac{d}{2}},& {\displaystyle \frac{1}{{\tilde{q}}_{\varrho }}}+{\displaystyle \frac{d}{{\tilde{r}}_{\varrho }}}<{\displaystyle \frac{d}{2}},\hfill \\ \hfill {\displaystyle \frac{2}{q_{\varrho }}}+{\displaystyle \frac{d}{r_{\varrho }}}={\displaystyle \frac{d}{2}}-s,& {\displaystyle \frac{2}{q_{\varrho }}}+{\displaystyle \frac{d}{r_{\varrho }}}+{\displaystyle \frac{2}{{\tilde{q}}_{\varrho }}}+{\displaystyle \frac{d}{{\tilde{r}}_{\varrho }}}=d,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{(\beta +1){\tilde{q}}_{\varrho }^{\prime }}}={\displaystyle \frac{\varrho }{q_{\varrho }}},& {\displaystyle \frac{1}{(\beta +1){\tilde{r}}_{\varrho }^{\prime }}}={\displaystyle \frac{\varrho }{r_{\varrho }}}+{\displaystyle \frac{2(1-\varrho )}{\beta d}}.\hfill \end{array}$$

(14)

For $d=1,2$, we obtain the same conclusion provided that we drop conditions (13).

Moreover, we can also assume that

$${\displaystyle \frac{\beta }{q_{\varrho }}}+{\displaystyle \frac{\beta d}{2r_{\varrho }}}=1,{\displaystyle \frac{\beta }{r_{\varrho }}}<1.$$

(15)

Lemma 2.

Assume $d\ge 1$ and ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$. Then, one can find $2<\ell \le \infty ,2\le p\le \infty$ such that

$$\begin{array}{c}{\displaystyle \frac{2}{\ell }}+{\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{2}},\hfill \end{array}$$

(16)

$$\begin{array}{c}{\displaystyle \frac{1}{p^{\prime }}}={\displaystyle \frac{1}{p}}+{\displaystyle \frac{\beta }{r_{\varrho }}},\hfill \end{array}$$

(17)

$$\begin{array}{c}{\displaystyle \frac{1}{{\ell }^{\prime }}}={\displaystyle \frac{1}{\ell }}+{\displaystyle \frac{\beta }{q_{\varrho }}},\hfill \end{array}$$

(18)

where $(q_{\varrho },r_{\varrho })$ is a couple given as in Proposition 1.

Lemma 3.

For every $0<s<1,\beta >0$, there exist two positive constants $C_1=C_1(\beta ,s)$ and $C_2=C_2(\beta ,s)$, such that

$${\parallel u\left|u\right|}^{\beta }{\parallel }_{{\dot{H}}_y^s}\le C_1{\parallel u\parallel }_{{\dot{H}}_y^s}{\parallel u\parallel }_{L^{\infty }}^{\beta }\le C_2{\parallel u\parallel }_{{\dot{H}}_y^s}^{\beta +1}.$$

We observe that the solutions to (1) enjoy the following conservation laws:

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_{L_{x,y}^2}=e^{-\Re \mathbf{B}\left(t\right)}{\parallel f\parallel }_{L_{x,y}^2},\mathcal{H}\left(u\left(t\right)\right)=\mathcal{H}\left(f\right),\end{array}$$

(19)

where

$$\begin{array}{c}\hfill H\left(u\left(t\right)\right)=e^{2\Re \mathbf{B}\left(t\right)}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}|{\nabla }_{x,y}{u(t,x,y)|}^{2}dxdy+{\displaystyle \frac{2\lambda e^{2\Re \mathbf{B}\left(t\right)}}{\beta +2}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\left|u(t,x,y)\right|}^{\beta +2}dxdy\\ \hfill +{\displaystyle \frac{2\beta \lambda }{\beta +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\Re b\left(s\right)e^{2\Re \mathbf{B}\left(t\right)}{\left|u(t,x,y)\right|}^{\beta +2}dxdy.\end{array}$$

(20)

We utilize the change in variable

$$v(t,x,y):=e^{\mathbf{B}\left(t\right)}u(t,x,y)$$

(21)

and see that u satisfies (1) if v solves

$$\left\{\begin{array}{c}i{\partial }_{t}v+{\Delta }_{x,y}v=\lambda e^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v\right|}^{\beta }v,(t,x,y)\in [0,\infty )\times {\mathbb{R}}^d\times \mathbb{T},\hfill \\ v(0,x,y)=f(x,y).\hfill \end{array}\right.$$

(22)

We multiply the above equation by $\overline{v}(t,x,y)$ and integrate by parts with respect to the x-variable, achieving

$$\begin{array}{c}\hfill i{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\left|v(t,x,y)\right|}^{2}dxdy-{\int }_{{\mathbb{R}}^d}{\nabla }_{x,y}\overline{v}(t,x,y){\nabla }_{x,y}v(t,x,y)dxdy\\ \hfill -{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\lambda e^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta +2}dxdy.\end{array}$$

Then, by taking the imaginary part, we arrive at

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\left|v(t,x,y)\right|}^{2}dxdy=0.\end{array}$$

Thus, solutions that are local in time satisfy the conservation of mass

$${\parallel v\left(t\right)\parallel }_{L_{x,y}^2}^2={\parallel f\parallel }_{L_{x,y}^2}^2,$$

that is, the first identity in (19). We multiply the equation in (22) by ${\partial }_t\overline{v}(t,x,y)$, then integrate by parts with respect to the $x,y$-variable and take the imaginary part. We thus have

$$\begin{array}{c}\hfill \Re {\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\nabla }_{x,y}v(t,x,y){\nabla }_{x,y}{\partial }_t\overline{v}(t,x)+\lambda e^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta }v(t,x){\partial }_t\overline{v}(t,x)\right)dxdy=0.\end{array}$$

The previous identity leads to the following:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\displaystyle \frac{1}{2}}{\partial }_t|{\nabla }_{x,y}{v(t,x,y)|}^2+{\displaystyle \frac{\lambda }{\beta +2}}{e}^{-\beta \Re \mathbf{B}\left(t\right)}{\partial }_t{\left|v(t,x,y)\right|}^{\beta +2}\right)dxdy=0\end{array}$$

and then to

$$\begin{array}{c}\hfill {\partial }_t{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\displaystyle \frac{1}{2}}|{\nabla }_{x,y}{v(t,x,y)|}^2+{\displaystyle \frac{\lambda }{\beta +2}}{e}^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta +2}\right)dxdy\\ \hfill =-{\displaystyle \frac{\beta \lambda }{\beta +2}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\Re b\left(t\right)e^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta +2}dxdy.\end{array}$$

(23)

Integrating with respect to the t-variable in the above identity (23), we get

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\displaystyle \frac{1}{2}}|{\nabla }_{x,y}{v(t,x,y)|}^2+{\displaystyle \frac{\lambda }{\beta +2}}{e}^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta +2}\right)dxdy\\ \hfill +{\displaystyle \frac{\beta \lambda }{\beta +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\Re b\left(s\right)e^{-\beta \Re \mathbf{B}\left(s\right)}{\left|v(s,x,y)\right|}^{\beta +2}dxdyds\\ \hfill ={\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\displaystyle \frac{1}{2}}|{\nabla }_{x,y}{v(0,x,y)|}^2+{\displaystyle \frac{\lambda }{\beta +2}}{\left|v(0,x,y)\right|}^{\beta +2}\right)dxdy.\end{array}$$

(24)

The above identity (24) indicates that the quantity

$$\begin{array}{c}\hfill \tilde{H}\left(v\left(t\right)\right)={\int }_{{\mathbb{R}}^d\times \mathbb{T}}\left({\displaystyle \frac{1}{2}}|{\nabla }_{x,y}{v(t,x,y)|}^2+{\displaystyle \frac{\lambda }{\beta +2}}{e}^{-\beta \Re \mathbf{B}\left(t\right)}{\left|v(t,x,y)\right|}^{\beta +2}\right)dxdy\\ \hfill +{\displaystyle \frac{\beta \lambda }{\beta +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\Re b\left(s\right)e^{-\beta \Re \mathbf{B}\left(s\right)}{\left|v(s,x,y)\right|}^{\beta +2}dxdyds\end{array}$$

is conserved. Hence, we obtain the local conservation of the Hamiltonian in (19) with $H\left(u\right(t\left)\right)$ as in (20). The above conservation laws (19) also infer the bound

$${∥e^{\mathbf{B}\left(t\right)}u∥}_{H_{x,y}^1}\lesssim {∥e^{\mathbf{B}\left(t\right)}u∥}_{L_{x,y}^2}+{∥{\nabla }_{x,y}{e}^{\mathbf{B}\left(t\right)}u∥}_{L_x^2}\lesssim H\left(u\left(0\right)\right)+{∥f∥}_{L_{x,y}^2}.$$

(25)

3. Morawetz Identities and Inequalities

Our first contribution here is the Morawetz equalities associated with (1). We start with the following:

Lemma 4

(Morawetz identities). Let $d\ge 1$ and $u\in \mathcal{C}([0,\infty );H_{x,y}^1)$ be a global solution to (1) with initial data $f\in H_{x,y}^1$ such that (2) and (3) are satisfied. Moreover, let $\psi =\psi \left(x\right):{\mathbb{R}}^d\to \mathbb{R}$ be a sufficiently regular and decaying function, and denote the virial by

$$\mathcal{V}\left(t\right):={\int }_{{\mathbb{R}}^d\times \mathbb{T}}\psi \left(x\right){\left|e^{\mathbf{B}\left(t\right)}u(t,x,y)\right|}^{2}dxdy.$$

Then, the following identities hold:

$$\begin{array}{c}\hfill \dot{\mathcal{V}}\left(t\right)=2\Im {\int }_{{\mathbb{R}}^d\times \mathbb{T}}{e}^{2\Re \mathbf{B}\left(t\right)}\overline{u}(t,x,y){\nabla }_x\psi \left(x\right)·{\nabla }_{x}u(t,x,y)dxdy\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \ddot{\mathcal{V}}\left(t\right)=-{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\Delta }_x^2\psi \left(x\right){\left|e^{\mathbf{B}\left(t\right)}u(t,x,y)\right|}^{2}dxdy\\ \hfill +4{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{e}^{2\Re \mathbf{B}\left(t\right)}{\nabla }_{x}u(t,x,y)D_x^2\psi \left(x\right)·\overline{{\nabla }_{x}u(t,x,y)}dxdy\\ \hfill -4\Im {\int }_{{\mathbb{R}}^d\times \mathbb{T}}{e}^{2\Re \mathbf{B}\left(t\right)}u(t,x,y){\nabla }_x\psi \left(x\right)·B\left(x\right)\overline{{\nabla }_{x}u(t,x,y)}dxdy\\ \hfill +{\displaystyle \frac{2\lambda \beta }{\beta +2}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}{\Delta }_x{e}^{2\Re \mathbf{B}\left(t\right)}\psi \left(x\right){\left|u(t,x,y)\right|}^{\beta +2}dxdy,\end{array}$$

(27)

where $D_x^2\psi \in {\mathcal{M}}_{d\times d}\left({\mathbb{R}}^d\right)$ is the Hessian matrix of ψ and ${\Delta }_x^2\psi ={\Delta }_x\left({\Delta }_x\psi \right)$ is the Bi-Laplacian operator.

Proof. 

We choose a smooth rapidly decreasing solution $u=u(t,x)$. The general case $e^{\mathbf{B}\left(t\right)}u\in \mathcal{C}([0,\infty );H_{x,y}^1)$ can be recovered via a classical density argument. The proof of (26) and (27) is similar to the one given in [10], for instance, since one can use the transformation (21) and then Equation (22). Thus, we skip this. □

We continue with the following.

Lemma 5

(Tensor Morawetz). Assume $d\ge 1$ and let $u(t,x,y)\in C([0,\infty ),H^1\left({\mathbb{R}}^d\right))$ be a global solution to (1) such that (2) and (3) are satisfied. Furthermore, let us denote by $z(t,x_1,x_2,y)=e^{2\mathbf{B}\left(t\right)}{u}_1(t,x_1,y_1)u_2(t,x_2,y_2)$ and $a(x_1,x_2)=\psi (x_1-x_2)$ and set the tensor action

$$\mathcal{M}\left(t\right)=2\Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}\overline{z}(t,x_1,x_2,y_1,y_2){\nabla }_{x_1,x_2}z(t,x_1,x_2,y_1,y_2)·{\nabla }_{x_1,x_2}a(x_1,x_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2,$$

(28)

where ${\nabla }_{x_1,x_2}=({\nabla }_{x_1},{\nabla }_{x_2})$ and ${({\mathbb{R}}^d\times \mathbb{T})}^2=({\mathbb{R}}^d\times \mathbb{T})\times ({\mathbb{R}}^d\times \mathbb{T})$. Then, the following identity holds:

$$\begin{array}{c}\hfill \dot{\mathcal{M}}\left(t\right)=-2{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{D}_{x_1{x}_2}^{2}a(x_1,x_2){\nabla }_{x_1}|e^{\mathbf{B}\left(t\right)}{u}_1(t,x_1,y_1){|}^2{\nabla }_{x_2}{\left|e^{\mathbf{B}\left(t\right)}{u}_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +{\displaystyle \frac{4\beta }{\beta +2}}{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u_1(t,x_1,y_1){|}^{\beta +2}{\left|u_2(t,x_2,y_2)\right|}^2{\Delta }_{x}a(x_1,x_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}u(t,x_1,y_1)D_{x_1{x}_1}^{2}a(x_1,x_2){\nabla }_{x_1}\overline{u}(t,x_1,y_1){\left|u(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_2}u(t,x_2,y_2)D_{x_2{x}_2}^{2}a(x_1,x_2){\nabla }_{x_2}\overline{u}(t,x_2,y_2){\left|u(t,x_1,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(29)

Proof. 

As above, we choose a smooth, decaying solution to (1). From now on, we hide the variables t, $y_1$, and $y_2$ to simplify the calculations. Note that

$$\begin{array}{c}\hfill i{\partial }_{t}z(x,y)+{\Delta }_{x_1,x_2}z(x_1,x_2)=e^{2\mathbf{B}\left(t\right)}{u}_1\left(x_1\right)|u_1\left(x_1\right){|}^{\beta }{u}_2\left(x_2\right)+e^{2\mathbf{B}\left(t\right)}{u}_1\left(x_1\right){\left|u_2\left(x_2\right)\right|}^{\beta }{u}_2\left(x_2\right),\end{array}$$

(30)

with ${\Delta }_{x_1,x_2}={\Delta }_{x_1,y_1}+{\Delta }_{x_2,y_2}$. Then, we differentiate the action (28) with respect to the time variable, obtaining

$$\begin{array}{c}\hfill \dot{\mathcal{M}}\left(t\right)=2\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1,x_2}z(x_1,x_2)[\left({\Delta }_{x_1,x_2}a(x_1,x_2)\overline{z}(x_1,x_2)\right]d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1,x_2}z(x_1,x_2)[({\nabla }_{x_1},{\nabla }_{x_2})a(x_1,x_2)·({\nabla }_{x_1},{\nabla }_{x_2})\overline{z}(x_1,x_2)]d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2+{\mathcal{N}}^{\beta }\left(t\right),\end{array}$$

(31)

where by the identity (30) and after exploiting the symmetry of $a(x_1,x_2)$ in combination with Fubini’s Theorem, we have

$$\begin{array}{c}\hfill {\mathcal{N}}^{\beta }\left(t\right)={\displaystyle \frac{2\beta }{\beta +1}}{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u_1\left(x_1\right){|}^{\beta +2}{\left|u_2\left(x_2\right)\right|}^2{\Delta }_{x_1}a(x_1,x_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +{\displaystyle \frac{2\beta }{\beta +1}}{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u_2\left(x_2\right){|}^{\beta +2}{\left|u_1\left(x_1\right)\right|}^2{\Delta }_{x_2}a(x_1,x_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill ={\displaystyle \frac{4\beta }{\beta +1}}{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u_1\left(x_1\right){|}^{\beta +2}{\left|u_2\left(x_2\right)\right|}^2{\Delta }_{x_1}a(x_1,x_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(32)

We will consider now the linear terms that are associated with ${\Delta }_{x,y}$. The approach displayed in [7,11] brings us to

$$\begin{array}{c}\hfill 2\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1,x_2}z(x_1,x_2)[\left({\Delta }_{x_1,x_2}a(x_1,x_2)\overline{z}(x_1,x_2)\right]d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1,x_2}z(x_1,x_2)[({\nabla }_{x_1},{\nabla }_{x_2})a(x_1,x_2)·({\nabla }_{x_1},{\nabla }_{x_2})\overline{z}(x_1,x_2)]d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill =-2{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1}^{2}a(x_1,x_2)|e^{\mathbf{B}\left(t\right)}u\left(x_1\right){|}^2{\left|e^{\mathbf{B}\left(t\right)}u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4e^{4\Re \mathbf{B}\left(t\right)}\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\nabla }_{x_1}{u}_1\left(x_1\right)D_{x_1{x}_1}^{2}a(x_1,x_2){\nabla }_{x_1}{\overline{u}}_1\left(x_1\right){\left|u_2\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4e^{4\Re \mathbf{B}\left(t\right)}\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\nabla }_{x_2}{u}_2\left(x_2\right)D_{x_2{x}_2}^{2}a(x_1,x_2){\nabla }_{x_2}{\overline{u}}_2\left(x_2\right){\left|u_1\left(x_1\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(33)

Notice that, by using again the symmetry of $a(x_1,x_2)$ and integration by parts, we get

$$\begin{array}{c}\hfill {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1}^{2}a(x_1,x_2)|e^{\mathbf{B}\left(t\right)}u\left(x_1\right){|}^2{\left|e^{\mathbf{B}\left(t\right)}u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill ={\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\Delta }_{x_1}{\Delta }_{x_2}a(x_1,x_2)|e^{\mathbf{B}\left(t\right)}u\left(x_1\right){|}^2{\left|e^{\mathbf{B}\left(t\right)}u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill ={\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{D}_{x_1{x}_2}^{2}a(x_1,x_2){\nabla }_{x_1}|e^{\mathbf{B}\left(t\right)}u\left(x_1\right){|}^2{\nabla }_{x_2}{\left|e^{\mathbf{B}\left(t\right)}u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(34)

By merging (32) and (33), one finally earns the identity (29). □

A Localized Morawetz Inequality

We start this section with an outcome that is a consequence of Lemma 4 above. More precisely, we introduce Lemma 6.

Lemma 6.

Assume $d\ge 1$ and let $u\in \mathcal{C}([0,\infty );H_{x,y}^1)$ be a global solution to (1) with initial data $f\in H_{x,y}^1$ such that (2) and (3) are satisfied. Then, it holds that

$$\begin{array}{c}\hfill {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\displaystyle \frac{e^{4\Re \mathbf{B}\left(t\right)}}{|x_1-x_2{|}^3}}|u_1(t,x_1,y_1){|}^2{\left|u_2(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \lesssim \Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1(t,x_1,y_1){\displaystyle \frac{(x_1-x_2)}{|x_1-x_2|}}·{\nabla }_{x_1}{u}_1(t,x_1,y_1){\left|u_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2,\end{array}$$

(35)

for $d\ge 3,$

$$\begin{array}{c}\hfill {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\displaystyle \frac{e^{4\Re \mathbf{B}\left(t\right)}}{|x_1-x_2|}}|u_1(t,x_1,y_1){|}^{\beta +2}{\left|u_2(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \lesssim \Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1(t,x_1,y_1){\displaystyle \frac{(x_1-x_2)}{|x_1-x_2|}}·{\nabla }_{x_1}{u}_1(t,x_1,y_1){\left|u_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2,\end{array}$$

(36)

for $d=2$ and

$$\begin{array}{c}\hfill {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\displaystyle \frac{e^{4\Re \mathbf{B}\left(t\right)}}{{\langle x_1-x_2\rangle }^3}}|u_1(t,x_1,y_1){|}^{\beta +2}{\left|u_2(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \lesssim \Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1(t,x_1,y_1){\displaystyle \frac{(x_1-x_2)}{\langle x_1-x_2\rangle }}·{\nabla }_{x_1}{u}_1(t,x_1,y_2){\left|u_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2,\end{array}$$

(37)

for $d=1$, where $\langle x_1-x_2\rangle :=(1+|x_1-x_2{{|}^2)}^{{\displaystyle \frac{1}{2}}}.$

Proof. 

We first notice, by Fubini’s Theorem, that

$$\begin{array}{c}\hfill \mathcal{M}\left(t\right)=2\Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1\left(x_1\right){\nabla }_{x_1}a(x_1,x_2)·{\nabla }_{x_1}{u}_1\left(x_1\right){\left|u_2\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(38)

We choose now $\psi =\psi (x_1,x_2)=|x_1-x_2|$ if $d\ge 2$ and $\psi =\langle x_1-x_2\rangle$ for $d=1$. Elementary computations then bring us to

$$\begin{array}{c}\hfill {\nabla }_{x_1}|x_1-x_2|={\displaystyle \frac{x_1-x_2}{|x_1-x_2|}},{\Delta }_{x_1}|x_1-x_2|={\displaystyle \frac{d-1}{|x_1-x_2|}},\\ \hfill {\Delta }_{x_1}^2|x_1-x_2|=-{\displaystyle \frac{(d-1)(d-3)}{|x_1-x_2{|}^3}},\\ \hfill D_{x_1{x}_1}^2|x_1-x_2|=D_{x_2{x}_2}^2|x_1-x_2|=-D_{x_1{x}_2}^2|x_1-x_2|\\ \hfill ={\displaystyle \frac{1}{|x_1-x_2|}}\left[{\delta }_{\ell k}-{\displaystyle \frac{{(x_1-x_2)}_{\ell }}{|x_1-x_2|}}{\displaystyle \frac{{(x_1-x_2)}_k}{|x_1-x_2|}}\right],\end{array}$$

(39)

for $\ell ,k=1,\dots ,d$, and

$$\begin{array}{c}\hfill {\nabla }_{x_1}\langle x_1-x_2\rangle ={\displaystyle \frac{x_1-x_2}{\langle x_1-x_2\rangle }},{\Delta }_{x_1}\langle x_1-x_2,\rangle ={\displaystyle \frac{1}{{\langle x_1-x_2\rangle }^3}},\\ \hfill D_{x_1{x}_1}^2\langle x_1-x_2\rangle =D_{x_2{x}_2}^2\langle x_1-x_2\rangle =-D_{x_1{x}_2}^2\langle x_1-x_2\rangle \\ \hfill ={\displaystyle \frac{1}{\langle x_1-x_2\rangle }}\left[{\delta }_{\ell k}-{\displaystyle \frac{{(x_1-x_2)}_{\ell }}{\langle x_1-x_2\rangle }}{\displaystyle \frac{{(x_1-x_2)}_k}{\langle x_1-x_2\rangle }}\right].\end{array}$$

(40)

We will look now at the identity (29) in the case $d\ge 2$ only, since the case $d=1$ can be handled in a similar way. We then have

$$\begin{array}{c}\hfill \Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}u\left(x_1\right)D_{x_1{x}_1}^{2}a(x_1,x_2){\nabla }_{x_1}\overline{u}\left(x_1\right){\left|u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill =\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\displaystyle \frac{|{\tilde{\nabla }}_{x_2}u\left(x_1\right){|}^2}{|x_1-x_2|}}{\left|u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\ge 0,\end{array}$$

(41)

with

$${\tilde{\nabla }}_{x_i}u\left(x_j\right)=\nabla u\left(x_j\right)-\nabla u\left(x_j\right)·{\displaystyle \frac{x_i-x_j}{|x_i-x_j|}}{\displaystyle \frac{x_i-x_j}{|x_i-x_j|}},$$

for $i,j=1,2,i\ne j$. Analogously, we have

$$\begin{array}{c}\hfill \Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_2}u\left(x_2\right)D_{x_2{x}_2}^{2}a(x_1,x_2){\nabla }_{x_2}\overline{u}\left(x_2\right){\left|u\left(x_1\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill =\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\displaystyle \frac{|{\tilde{\nabla }}_{x_1}u\left(x_2\right){|}^2}{|x_1-x_2|}}{\left|u\left(x_1\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\ge 0.\end{array}$$

(42)

Thus, (29), (34), and the inequalities (41) and (42) give (35). Moreover, by the Cauchy–Schwartz inequality, one also achieves

$$\begin{array}{c}\hfill 2{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{D}_{x_1{x}_2}^{2}a(x_1,x_2){\nabla }_{x_1}|e^{\mathbf{B}\left(t\right)}u\left(x_1\right){|}^2{\nabla }_{x_2}{\left|e^{\mathbf{B}\left(t\right)}u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill =8{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{D}_{x_1{x}_2}^{2}a(x_1,x_2)\Re \left(\overline{u}\left(x_1\right){\nabla }_{x_1}u\left(x_1\right)\right)\Re \left(\overline{u}\left(x_2\right){\nabla }_{x_2}u\left(x_2\right)\right)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill =8{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\displaystyle \frac{e^{4\Re \mathbf{B}\left(t\right)}}{|x_1-x_2|}}\Re \left(\overline{u}\left(x_1\right){\tilde{\nabla }}_{x_2}u\left(x_1\right)\right)·\Re \left(\overline{u}\left(x_2\right){\tilde{\nabla }}_{x_1}u\left(x_2\right)\right)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \le 4{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}\left({\displaystyle \frac{|{\tilde{\nabla }}_{x_2}u\left(x_1\right){|}^2}{|x_1-x_2|}}|u\left(x_2\right){|}^2+{\displaystyle \frac{|{\tilde{\nabla }}_{x_1}u\left(x_2\right){|}^2}{|x_1-x_2|}}{\left|u\left(x_1\right)\right|}^2\right)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(43)

The above inequality in combination with (41) and (42) yields

$$\begin{array}{c}\hfill -2{\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{D}_{x_1{x}_2}^{2}a(x_1,x_2){\nabla }_{x_1}|e^{\mathbf{B}\left(t\right)}{u}_1(t,x_1){|}^2{\nabla }_{x_2}{\left|e^{\mathbf{B}\left(t\right)}{u}_2(t,x_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}u\left(x_1\right)D_{x_1{x}_1}^{2}a(x_1,x_2){\nabla }_{x_1}\overline{u}\left(x_1\right){\left|u\left(x_2\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill +4\Re {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_2}u\left(x_2\right)D_{x_2{x}_2}^{2}a(x_1,x_2){\nabla }_{x_2}\overline{u}\left(x_2\right){\left|u\left(x_1\right)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\ge 0.\end{array}$$

By the use of the previous bound in the equality (29), one arrives at (36). □

We obtain the following corollary, which is a direct outcome of the previous lemma.

Corollary 1.

Let $u\in \mathcal{C}([0,\infty );H_{x,y}^1)$ be a global solution to (1) with initial data $f\in H_{x,y}^1$ such that (2) and (3) are satisfied. Then, far any ${\mathcal{Q}}_{\tilde{x}}^d\left(r\right)=\tilde{x}+{[-r,r]}^d$, with $r>0$ and $\tilde{x}\in {\mathbb{R}}^d$, one has

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_{{({\mathcal{Q}}_{\tilde{x}}^d\left(r\right)\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u(t,x_1,y_1){|}^2{\left|u(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2<\infty .\end{array}$$

(44)

Proof. 

By integrating (35) with $a(x_1,x_2)$ as in (39) with respect to the time variable on the interval $J=[t_1,t_2]$, with $t_1,t_2\in [0,\infty )$, one obtains

$$\begin{array}{c}\hfill {\left[\Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1(t,x_1,y_1){\displaystyle \frac{(x_1-x_2)}{|x_1-x_2|}}·{\nabla }_{x_1}{u}_1(t,x_1,y_1){\left|u_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\right]}_{t=t_1}^{t=t_2}\\ \hfill \gtrsim {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{\displaystyle \frac{e^{4\Re \mathbf{B}\left(t\right)}}{|x_1-x_2{|}^3}}|u_1(t,x_1,y_1){|}^2{\left|u_2(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \gtrsim {\int }_0^{\infty }{\int }_{{({\mathcal{Q}}_{\tilde{x}}^d\left(r\right)\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u(t,x_1,y_1){|}^2{\left|u(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2,\end{array}$$

where in the last line of the previous chain of inequalities we used, for any $\tilde{x}\in {\mathbb{R}}^d$,

$$\underset{x_1,x_2\in {\mathcal{Q}}_{\tilde{x}}^d\left(r\right)}{inf}{\displaystyle \frac{1}{|x_1-x_2|}}=\underset{x_1,x_2\in {\mathcal{Q}}_0^d\left(r\right)}{inf}{\displaystyle \frac{1}{|x_1-x_2|}}>0.$$

Applying again the Cauchy–Schwartz inequality and by the conservation laws (19), one also infers

$$\begin{array}{c}\hfill {\left[\Im {\int }_{{({\mathbb{R}}^d\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}{\nabla }_{x_1}{\overline{u}}_1(t,x_1,y_1){\displaystyle \frac{(x_1-x_2)}{|x_1-x_2|}}·{\nabla }_{x_1}{u}_1(t,x_1,y_1){\left|u_2(t,x_2,y_2)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\right]}_{t=t_1}^{t=t_2}\\ \hfill \lesssim {\parallel f\parallel }_{H_{x,y}^1}^4<\infty .\end{array}$$

(45)

Finally, we get (44) when $t_1\to 0,t_2\to \infty$. □

4. The Decay of Solutions

This section focuses on the demonstration of the first part of Theorem 1. Namely, one has the following.

Proof of Equation (4).

It is enough to prove the decay in (4) for a suitable $2<q<{\displaystyle \frac{2d+2}{d-1}}$, because the general case follows the conservation laws of (19) and interpolation. More precisely, it is enough to show that

$$\underset{t\to \pm \infty }{lim}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x,y)\parallel }_{L_{x,y}^{d+1}}=0.$$

(46)

Then, the property (4) follows for all $2<q<{\displaystyle \frac{2d+2}{d-1}}$ by (46) and the fact that

$$\underset{t\in \mathbb{R}}{sup}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x,y)\parallel }_{H_{x,y}^1}<\infty .$$

(47)

We recall the following localized Gagliardo–Nirenberg inequality (see [12,13]):

$${\parallel \chi \parallel }_{L_{x,y}^{\frac{2d+6}{d+1}}}\le C\underset{\tilde{x}\in {\mathbb{R}}^d}{sup}{\left({\parallel \chi \parallel }_{L_{({\mathcal{Q}}_{\tilde{x}}^d\left(1\right)\times \mathbb{T})}^2}\right)}^{\frac{2}{d+3}}{\parallel \chi \parallel }_{H_{x,y}^1}^{\frac{d+1}{d+3}},$$

(48)

where ${\mathcal{Q}}_x^d\left(1\right)=\tilde{x}+{[-1,1]}^d$. Now, assume by contradiction that (46) is not fulfilled, then by (47) and by (48), we deduce the existence of a sequence $(t_n,x_n)\in \mathbb{R}\times {\mathbb{R}}^d$ with $t_n\to \infty$ and ${ϵ}_0>0$, such that

$$\underset{n}{inf}{\parallel e^{\mathbf{B}\left(t_n\right)}u(t_n,x)\parallel }_{L^2({\mathcal{Q}}_{x_n}^d\left(1\right)\times \mathbb{T})}^2={ϵ}_0^2.$$

(49)

Note that by (26) and (47), one attains

$$\underset{n,t}{sup}\left|{\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\xi (x-x_n){\left|e^{\mathbf{B}\left(t\right)}u(t,x,y)\right|}^{2}dxdy\right|<\infty ,$$

where $\xi \left(x\right)$ is a smooth, non-negative cut-off function, so that $\xi \left(x\right)=1$ for $x\in {\mathcal{Q}}_0^d\left(1\right)={[-1,1]}^d$ and $\xi \left(x\right)=0$ for $x\notin {\mathcal{Q}}_0^d\left(2\right)={[-2,2]}^d$. Therefore, by the application of the fundamental theorem of calculus, we establish the following inequality:

$$\left|{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\xi (x-x_n)|e^{\mathbf{B}\left(\sigma \right)}u(\sigma ,x,y){|}^{2}dxdy-{\int }_{{\mathbb{R}}^d\times \mathbb{T}}\xi (x-x_n){\left|e^{\mathbf{B}\left(t\right)}u(t,x,y)\right|}^{2}dxdy\right|\le \tilde{C}|t-\sigma |,$$

(50)

for $\tilde{C}>0$, which does not depend on n. We choose $t=t_n$ and have

$${\int }_{{\mathbb{R}}^d\times \mathbb{T}}\xi (x-x_n)|e^{\mathbf{B}\left(\sigma \right)}u(\sigma ,x,y){|}^{2}dxdy\ge {\int }_{{\mathbb{R}}^d\times \mathbb{T}}\xi (x-x_n)|e^{\mathbf{B}\left(t_n\right)}u(t_n,x,y){|}^{2}dxdy-\tilde{C}|t_n-\sigma |,$$

(51)

which—keeping in mind the support property of the function $\xi \left(x\right)$—results in

$${\int }_{{\mathcal{Q}}_{x_n}^d\left(2\right)\times \mathbb{T}}|e^{\mathbf{B}\left(\sigma \right)}{u(\sigma ,x,y)|}^{2}dxdy\ge {\int }_{{\mathcal{Q}}_{x_n}^d\left(1\right)\times \mathbb{T}}|e^{\mathbf{B}\left(t_n\right)}u(t_n,x,y){|}^{2}dxdy-\tilde{C}|t_n-\sigma |.$$

(52)

By combining this result with (49), it follows that there exists $T>0$ so that

$$\underset{n}{inf}\left(\underset{t\in (t_n,t_n+T)}{inf}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x)\parallel }_{L^2({\mathcal{Q}}_{x_n}\left(2\right)\times \mathbb{T})}^2\right)\gtrsim {ϵ}_1^2,$$

(53)

for some ${ϵ}_1>0$. Observe also that, because $t_n\to \infty$, it is possible to assume, up to a subsequence, that the intervals $(t_n,t_n+T)$ are mutually disjointed. In particular, we arrive at

$$\begin{array}{c}\hfill \sum _{n}T{ϵ}_1^4\lesssim \sum _n{\int }_{t_n}^{t_n+T}{\int }_{{({\mathcal{Q}}_{x_n}^d\left(2\right)\times \mathbb{T})}^2}|e^{\mathbf{B}\left(t\right)}u(t,x_1,y_1){|}^2{\left|e^{\mathbf{B}\left(t\right)}u(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \lesssim {\int }_0^{\infty }\underset{\tilde{x}\in {\mathbb{R}}^d}{sup}{\int }_{{({\mathcal{Q}}_{\tilde{x}}^d\left(2\right)\times \mathbb{T})}^2}|e^{\mathbf{B}\left(t\right)}u(t,x_1,y_1){|}^2{\left|e^{\mathbf{B}\left(t\right)}u(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2\\ \hfill \lesssim {\int }_0^{\infty }\underset{\tilde{x}\in {\mathbb{R}}^d}{sup}{\int }_{{({\mathcal{Q}}_{\tilde{x}}^d\left(2\right)\times \mathbb{T})}^2}{e}^{4\Re \mathbf{B}\left(t\right)}|u(t,x_1,y_1){|}^2{\left|u(t,x_2,y_1)\right|}^{2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.\end{array}$$

(54)

Thus, we obtain a contradiction since the right-hand side of (54) is bounded, as can be seen form (44). □

Remark 1.

The condition (3) is required to ensure not only an effective use of the Morawetz technique, as one can see from Lemmas 4 and 5, but also that we do not need to impose further lower bounds to $e^{\Re \mathbf{B}\left(t\right)}$ when $4/d<\beta <4/(d-1)$. Notice also that, for $0<\beta \le 4/d$, one has $e^{-\beta \Re \mathbf{B}\left(t\right)}<{(1+t)}^{-1}$; this bound guarantees that $e^{{\Delta }_{x,y}+\mathbf{B}\left(t\right)}u$ has a strong limit in $L_{x,y}^2$ when $\Re b\left(t\right)\ge 0$, as underlined in [19], avoiding the nonexistence of scattering solutions.

5. Analysis of the Solutions in the Strichartz Spaces and Scattering

Here, we present some results associated to the control of the solution of (1) in the Strichartz norms. They are pivotal for the proof of the second part of Theorem 1.

Proposition 2.

Let $d\ge 3$, $\lambda >0$, and let $u\in \mathcal{C}([0,\infty );H^1\left({\mathbb{R}}^d\right))$ be a unique global solution to (1) with initial data $f\in H^1({\mathbb{R}}^d\times \mathbb{T})$ such that (2) is satisfied. One thus has the following:

(a) 

If ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$, and assuming (3) is also fulfilled, then

$$e^{\mathbf{B}\left(t\right)}u(t,x,y)\in L_t^{q_{\varrho }}{L}_x^{r_{\varrho }}{H}_y^{\tilde{s}},$$

(55)

with $(q_{\varrho },r_{\varrho })$ as in Lemma 1.

(b) 

If $0<\beta \le {\displaystyle \frac{4}{d}}$, and assuming (2) is fulfilled with strict inequality, then

$$e^{\mathbf{B}\left(t\right)}u(t,x,y)\in L_t^q{L}_x^r{H}_y^{\tilde{s}},$$

(56)

with $(q,r)=\left({\displaystyle \frac{4(\beta +2)}{d\beta }},\beta +2\right)$.

Proof. 

Case 1: ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$. Bear in mind that the integral operator associated to (22) is defined, for any $f\in H_{x,y}^1$ as

$$\begin{array}{c}\hfill {\mathcal{T}}_f\left(e^{\mathbf{B}\left(t\right)}u\right)=e^{it{\Delta }_{x,y}}f+k{\int }_0^t{e}^{-\beta \Re \mathbf{B}\left(\tau \right)}{e}^{i(t-\tau ){\Delta }_{x,y}}\left(|e^{\mathbf{B}\left(\tau \right)}{u\left(\tau \right)|}^{\beta }{e}^{\mathbf{B}\left(\tau \right)}u\left(\tau \right)\right)d\tau .\end{array}$$

(57)

Then, in view of (2), one obtains via a combination of Proposition 1, Lemmas 1 and 3, and the Hölder inequality

$$\begin{array}{c}\hfill \parallel e^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{q_{\varrho }}{L}_x^{r_{\varrho }}{H}_y^{\tilde{s}}}\lesssim \parallel u\left(t_0\right){\parallel }_{H_x^{\sigma }{H}_y^{\tilde{s}}}+\parallel e^{-\beta \Re \mathbf{B}(·)}|e^{\mathbf{B}(·)}u{{|}^{\beta }{e}^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{{\widetilde{q_{\varrho }}}^{\prime }}{L}_x^{{\tilde{r}}_{\varrho }^{\prime }}{H}_y^{\tilde{s}}}\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel |e^{\mathbf{B}(·)}{u|}^{\beta }{e}^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{{\widetilde{q_{\varrho }}}^{\prime }}{L}_x^{{\tilde{r}}_{\varrho }^{\prime }}{H}_y^{\tilde{s}}}\lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{(1+\beta ){\widetilde{q_{\varrho }}}^{\prime }}{L}_x^{(1+\beta ){\tilde{r}}_{\varrho }^{\prime }}{H}_y^{\tilde{s}}}^{1+\beta }\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{\infty }{L}_x^{\beta d/2}{H}_y^{\tilde{s}}}^{(1-\varrho )(1+\beta )}{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{q_{\varrho }}{L}_x^{r_{\varrho }}{H}_y^{\tilde{s}}}^{\varrho (1+\beta )}.\end{array}$$

(58)

An application of (4) and Lemma 2.5 in [13] gives, for a sufficiently large $t_0>0$, that $e^{\mathbf{B}\left(t\right)}u(t,x,y)\in L_{(t_0,\infty )}^{q_{\varrho }}{L}_x^{r_{\varrho }}{H}_y^{\tilde{s}}.$ Finally, by a continuity argument, we deduce (55).

Case 2: $0<\beta \le d$. Within this framework, the previous approach can be followed with slight adjustments. Keeping in mind the Sobolev embedding $H_y^{\tilde{s}}\subset L_y^{\infty }$, which is valid for $\tilde{s}>{\displaystyle \frac{1}{2}}$, one can acquire the following chain of inequalities:

$$\begin{array}{c}\hfill \parallel e^{\mathbf{B}\left(t\right)}{u\parallel }_{L_{(t_0,\infty )}^q{L}_x^r{H}_y^{\tilde{s}}}\lesssim \parallel u\left(t_0\right){\parallel }_{H_x^{\sigma }{H}_y^{\tilde{s}}}+\parallel e^{-\beta \Re \mathbf{B}(·)}|e^{\mathbf{B}(·)}u{{|}^{\beta }{e}^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{q^{\prime }}{L}_x^{r^{\prime }}{H}_y^{\tilde{s}}}\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{-\mathbf{B}(·)}{e}^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{\eta }{L}_x^r{H}_y^{\tilde{s}}}^{\beta }{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^q{L}_x^r{H}_y^{\tilde{s}}}\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{-\mathbf{B}(·)}{\parallel }_{L_{(t_0,\infty )}^{\eta }}^{\beta }\parallel e^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{\infty }{L}_x^r{H}_y^{\tilde{s}}}^{\beta }{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^q{L}_x^r{H}_y^{\tilde{s}}}\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{\infty }{L}_x^r{H}_y^{\tilde{s}}}^{\beta }{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^q{L}_x^r{H}_y^{\tilde{s}}},\end{array}$$

(59)

with

$$\eta ={\displaystyle \frac{2\beta (\beta +2)}{4-(d-2)\beta }},$$

due to the strict inequality in (2). Once again, by (4), Lemma 2.5 in [13], and a continuity argument, we conclude that (56) holds true. □

Corollary 2.

Assume that $(\ell ,p)$ is given as in Lemma 2 and let $u(t,x,y)$ be the unique solution to (1) with $0<\beta <{\displaystyle \frac{4}{d-1}}$, such that (2) is satisfied. Then, one gets

$$\parallel e^{\mathbf{B}\left(t\right)}{u(t,x,y)\parallel }_{L_t^{\ell }{L}_x^p{L}_y^2}+{\parallel e^{\mathbf{B}\left(t\right)}u(t,x,y){\nabla }_{x,y}u(t,x,y)\parallel }_{L_t^{\ell }{L}_x^p{L}_y^2}<\infty ,$$

(60)

(a) 

if ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$, and assuming (3) is also fulfilled;

(b) 

if $0<\beta \le {\displaystyle \frac{4}{d}}$, and assuming (2) holds with strict inequality.

Proof. 

We will concentrate first on the case ${\displaystyle \frac{4}{d}}<\beta <{\displaystyle \frac{4}{d-1}}$; the proof of the case $0<\beta \le {\displaystyle \frac{4}{d}}$ follows from (59). We display $\parallel e^{\mathbf{B}\left(t\right)}{u(t,x,y)\parallel }_{L_t^{\ell }{L}_x^p{L}_y^2}<\infty$, and the other estimate can be handled in a similar way. By (9), Lemma 2, and Hölder inequality, one achieves

$$\begin{array}{c}\hfill \parallel e^{\mathbf{B}\left(t\right)}{u(t,x,y)\parallel }_{L_{(t_0,\infty )}^{\ell }{L}_x^p{L}_y^2}\lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{-\beta \Re \mathbf{B}\left(t\right)}|e^{\mathbf{B}(·)}u{{|}^{\beta }{e}^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{{\ell }^{\prime }}{L}_x^{p^{\prime }}{L}_y^2}\\ \hfill \lesssim \parallel u\left(t_0\right){\parallel }_{H_{x,y}^1}+\parallel e^{\mathbf{B}(·)}{u\parallel }_{L_{(t_0,\infty )}^{q_{\varrho }}{L}_x^{r_{\varrho }}{H}_y^{\tilde{s}}}^{\beta }{\parallel e^{\mathbf{B}(·)}u\parallel }_{L_{(t_0,\infty )}^{\ell }{L}_x^p{L}_y^2}.\end{array}$$

We conclude by choosing a suitably large $t_0>0$ and utilizing the previous Proposition 2. □

We now return to the proof of the scattering for the solution to (1).

Proof of Equation (5).

By using the integral operator (57) and the Strichartz estimates (7) and (9), we obtain

$$\begin{array}{c}\hfill {∥e^{\mathbf{B}\left(t\right)}{e}^{-it{\Delta }_{x,y}}u\left(t\right)-e^{\mathbf{B}\left(t^{\prime }\right)}{e}^{-it{\Delta }_{x,y}}u\left(t^{\prime }\right)∥}_{H_{x,y}^1}={∥{\int }_{t^{\prime }}^t{e}^{-is{\Delta }_{x,y}}\left(e^{-\beta \Re \mathbf{B}(·)}{\left|e^{\mathbf{B}(·)}u\right|}^{\beta }{e}^{\mathbf{B}(·)}u\right)\left(s\right)ds∥}_{H_{x,y}^1}\\ \hfill \lesssim \parallel e^{-\beta \Re \mathbf{B}(·)}|e^{\mathbf{B}(·)}{u|}^{\beta }{e}^{\mathbf{B}(·)}{u\parallel }_{L_{(t,t^{\prime })}^{\ell }{L}_x^p{L}_y^2}+{\parallel e^{-\beta \Re \mathbf{B}(·)}{\nabla }_{x,y}\left(|e^{\mathbf{B}(·)}{u|}^{\beta }{e}^{\mathbf{B}(·)}u\right)\parallel }_{L_{(t,t^{\prime })}^{\ell }{L}_x^p{L}_y^2}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{t_1,t_2\to \infty }{lim}{∥e^{\mathbf{B}\left(t\right)}{e}^{-it{\Delta }_{x,y}}u\left(t\right)-e^{\mathbf{B}\left(t^{\prime }\right)}{e}^{-it{\Delta }_{x,y}}u\left(t^{\prime }\right)∥}_{H_{x,y}^1}=0,\end{array}$$

(61)

if

$$\underset{t,t^{\prime }\to \infty }{lim}\left(\parallel e^{-\beta \Re \mathbf{B}(·)}|e^{\mathbf{B}(·)}{u|}^{\beta }{e}^{\mathbf{B}(·)}{u\parallel }_{L_{(t,t^{\prime })}^{\ell }{L}_x^p{L}_y^2}+{\parallel e^{-\beta \Re \mathbf{B}(·)}{\nabla }_{x,y}\left(|e^{\mathbf{B}(·)}{u|}^{\beta }{e}^{\mathbf{B}(·)}u\right)\parallel }_{L_{(t,t^{\prime })}^{\ell }{L}_x^p{L}_y^2}\right)=0.$$

This limit can be provided as in the proof of Proposition 2, by exploiting (55), (56) and (60), together with Lemma 2. So, (5) follows from (61) above. □

Remark 2.

We also emphasize that exponential decay can be obtained directly through interpolation between the conservation of mass stated in (19) and the estimates derived from the Sobolev embedding combined with (25). More explicitly, one obtains

$$\parallel e^{\mathbf{B}\left(t\right)}{u\left(t\right)\parallel }_{L_{x,y}^r}\lesssim 1.$$

Nevertheless, this inequality is insufficient to establish a decay behavior as strong as the one described in (4) of Theorem 1, which characterizes a more restrictive property of solutions to (1). Additionally, the analysis above does not take into account the scenario where $\Re \mathbf{B}\left(t\right)=0$, that is, when $ib\left(t\right)$ is a real-valued function.

Remark 3.

It is important to emphasize that our results apply broadly to a wide class of damped nonlinearities satisfying the condition (2). In particular, they cover situations where the damping term behaves as

$$\Re \mathbf{b}\left(t\right)={\displaystyle \frac{\tilde{a}}{(1+t)}},\alpha >0,\tilde{a}>0,t\ge 0,$$

as well as cases of the form

$$\Re \mathbf{b}\left(t\right)={\displaystyle \frac{\tilde{a}}{{(1+t)}^{\alpha }}},0\le \alpha <1,\tilde{a}>0,t\ge 0,$$

which were examined in [19]. Moreover, the techniques and results developed herein extend naturally to even more general forms of damping terms, thereby enabling the analysis of nonlinear Schrödinger equations with complex time-dependent damping structures, such as

$$i{\partial }_{t}u+{\Delta }_{x,y}u+{\displaystyle \frac{i\tilde{a}{t}^{{\alpha }_1}lnt}{{(1+t)}^{{\alpha }_2}}}u+\tilde{b}\left(t\right)u-\lambda u{\left|u\right|}^{\beta }=0,t\ge \delta ,$$

with $\delta \ge 1,$ where $\tilde{b}\left(t\right)$ is a real-valued continuous function, and the parameters $\tilde{a}$, ${\alpha }_1$, and ${\alpha }_2$ characterize the precise rate and structure of the damping effect. These types of equations were previously considered in the literature, for example, in [17], highlighting the flexibility and applicability of our approach.

6. Conclusions

We extend the outcomes obtained in [16,17,18,19] to the partially periodic framework. In our approach, the assumptions on the function $ib\left(t\right)$ closely resemble the more general conditions presented in [19], with the added advantage that our approach can easily handle the case when $e^{-\beta \Re \mathbf{B}\left(t\right)}\lesssim {(1+t)}^{-\delta }$ for $\delta \ge 0$. Moreover, we include an oscillatory component within the perturbed propagator $e^{\mathbf{B}\left(t\right)+it{\Delta }_{x,y}},$ a feature that has not been addressed in the aforementioned works. In addition, our strategy significantly simplifies the proof of scattering in the energy space for the damped Schrödinger equation, even when considered on the flat Euclidean geometry ${\mathbb{R}}^d$ for $0<\beta <{\displaystyle \frac{4}{d-2}}$. The implementation of bilinear Morawetz inequalities provides insight into the decay behavior of the $L_{x,y}^p$-norm of the solutions of (1): specifically, we show that the decay rate is faster than $e^{-\Re \mathbf{B}\left(t\right)}$ when the Hamiltonian (20) is positive. This enhanced decay property broadens the class of admissible perturbations, even allowing the limit case $\mathbf{b}=0$. It is important to emphasize that we have opted for a conservative set of assumptions regarding the nonlinear terms in (1). Indeed, we are convinced that our methodology extends naturally to perturbations $b\left(t\right)$ satisfying

$$e^{-\beta \Re \mathbf{B}\left(t\right)}\le {\displaystyle \frac{\tilde{a}{t}^{{\alpha }_1}}{{(1+t)}^{{\alpha }_2-{\alpha }_1}}},0\le {\alpha }_1<1,{\alpha }_1<{\alpha }_2,\tilde{a}>0,t\ge 0,$$

with $\mathbf{B}\left(t\right)$ defined as in (2), as well as to the case $\Re \mathbf{B}\left(t\right)<0$. These extensions form the basis of our future investigations.

7. Open Problems and Further Developments

The theoretical framework developed in this paper is general and robust, allowing for a direct analysis of energy decay in solutions to damped Schrödinger equations with partial periodic local nonlinearities. The versatility of our approach suggests its applicability to several significant open problems, particularly the following:

• A detailed analysis of scattering phenomena in energy spaces for solutions to Equation (1) within the focusing regime, characterized by $\lambda <0$. Such an investigation would deepen the understanding of the interplay between damping and focusing nonlinearities.
• An exploration of decay and scattering behavior of solutions to fourth-order nonlinear Schrödinger equations, such as

  $${\partial }_{t}u-{\left({\Delta }_{x,y}\right)}^{2}u+ib\left(t\right)u-\lambda u{\left|u\right|}^{\beta }=0,$$

  where ${\left({\Delta }_{x,y}\right)}^2={\Delta }_{x,y}\left({\Delta }_{x,y}\right)$ is the Bi-Laplacian operator. This would help clarify the long-term dynamics and stability of higher-order dispersive models under nonlinear damping effects.
• A thorough investigation into the decay rates and scattering properties of solutions to other related nonlinear dispersive equations, including the nonlinear Beam equation such as

  $${\partial }_{tt}u+{\Delta }_x^{2}u+b\left(t\right)u+u+\lambda u{\left|u\right|}^{\beta }=0,$$

  with ${\left({\Delta }_x\right)}^2={\Delta }_x\left({\Delta }_x\right)$. This can also be carried out in a partially periodic setting.
• A comprehensive study of the scattering dynamics for nonlinear Klein–Gordon equations of the form

  $${\partial }_{tt}u-{\Delta }_{x,y}u+b\left(t\right)u+u+\lambda u{\left|u\right|}^{\beta }=0,$$

  including the the partially periodic case.