Scattering in the Energy Space for the Damped Nonlinear Fourth-Order Schrödinger Equation in Higher Dimensions Under Spherical Symmetry

Abstract

We establish global decay and scattering in the energy space $H^2\left({\mathbb{R}}^d\right)$, for $d\ge 5$, of radial solutions to the damped nonlinear biharmonic Schrödinger equation with general complex-valued, time-dependent damping coefficients. Assuming radial data exploits $O\left(d\right)$-symmetry and strengthens Morawetz-type controls through spherical averaging, we introduce new Morawetz-type identities and localized inequalities adapted to the fourth-order dispersive flow and compatible with this symmetry. As a consequence, and under explicit conditions for the damping coefficients that include slowly decaying or oscillatory profiles, we prove that solutions decay in Lebesgue norms and scatter to free biharmonic evolutions.

1. Introduction

We study the following Cauchy problem for the damped nonlinear defocusing fourth-order Schrödinger equation on ${\mathbb{R}}^d$, for any space dimensions $d\ge 5$:

$$\left\{\begin{array}{c}i{\partial }_{t}u+{\Delta }_x^{2}u+ib\left(t\right)u+\lambda u{\left|u\right|}^{\alpha }=0,(t,x)\in [0,\infty )\times {\mathbb{R}}^d,\hfill \\ u(0,x)=f\left(x\right)\in H^2\left({\mathbb{R}}^d\right),\hfill \end{array}\right.$$

(1)

where ${\Delta }_x^2={\Delta }_x{\Delta }_x,$ is the Bi-Laplacian operator. Here, $\lambda >0$, $u=u(t,x):[0,\infty )\times {\mathbb{R}}^d\to \mathbb{C}$, and $b:[0,\infty )\to \mathbb{C}$ are measurable functions that include dissipative and oscillatory terms, and the nonlinearity parameter $\alpha$ is constrained to the following conditions:

$$\begin{array}{cc}\hfill & {\alpha }_{*}\left(d\right)<\alpha <{\alpha }^{*}\left(d\right),{\alpha }_{*}\left(d\right)={\displaystyle \frac{8}{d}},{\alpha }^{*}\left(d\right)={\displaystyle \frac{8}{d-4}}.\hfill \end{array}$$

(2)

We also assume that $\mathfrak{R}b\left(t\right),\mathfrak{I}b\left(t\right)\in \mathcal{C}\left(\right[0,\infty \left)\right)$ with $\mathfrak{R}b\left(t\right)\ge 0,$ and

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

(3)

The two conditions stated above guarantee, in the case of strict inequalities, that every global solution to (1) asymptotically behaves like the solution to the associated free equation (i.e., the case $b\left(t\right)=\lambda =0$) as $t\to +\infty$. The primary objective of this paper is to establish decay and scattering properties for solutions to (1) in the energy space. In view of this, the following theorem constitutes the main analytical step of our study. It asserts that under hypotheses (2) and (3), solutions to the damped biharmonic nonlinear Schrödinger equation not only exist globally in $H^2\left({\mathbb{R}}^d\right)$, but also decay in Lebesgue norms and scatter to free solutions. From a physical perspective, this means that the long-time dynamics of the damped system resemble those of the undamped biharmonic flow, with the damping mechanism ensuring dispersion and preventing concentration. Mathematically, this establishes asymptotic completeness in the energy space for a wide class of time-dependent damping profiles, extending classical results for the damped nonlinear Schrödinger equation to the fourth-order case.

Theorem 1. 

Let $d\ge 5$, $\lambda >0$, and let $u\in \mathcal{C}([0,\infty );H^2\left({\mathbb{R}}^d\right))$ be a global solution to (1) with radial initial data $f\in H^1\left({\mathbb{R}}^d\right)$ such that (2) and (3) are satisfied. Then, for $2<r<\frac{2d}{d-4}$, one achieves

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

(4)

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

$$\underset{t\to \pm \infty }{lim}{\parallel e^{\mathbf{B}\left(t\right)}u(t,x)-e^{it{\Delta }_x^2}{f}_{\pm }\parallel }_{H^2\left({\mathbb{R}}^d\right)}=0.$$

(5)

Remark 1. 

The damping function $b:[0,\infty )\to \mathbb{C}$ plays a central role in the dynamics of Equation (1). Its real part $\mathfrak{R}b\left(t\right)\ge 0$ is responsible for dissipation, ensuring that the mass and energy decay are average, while its imaginary part $\mathfrak{I}b\left(t\right)$ produces only oscillatory modulation without affecting the amplitude. The structural assumptions in (3) guarantee that damping is sufficiently persistent to derive decay estimates and, in turn, to prove scattering in the energy space. These conditions ensure that the damping term does not degenerate too quickly, while still admitting a wide class of coefficients, including time-dependent and oscillatory profiles.

Example 1. 

Several representative cases illustrate the scope of (3):

(i) 

Constant damping: $b\left(t\right)=\beta >0$, yielding $\mathbf{B}\left(t\right)=\beta t$. This corresponds to uniform exponential decay and is classical in nonlinear optics and plasma models.

(ii) 

Polynomially decaying damping: $b\left(t\right)=\beta {(1+t)}^{-\gamma }$, with $\beta >0$ and $0\le \gamma <1$. In this case $\mathbf{B}\left(t\right)\sim \frac{\beta }{1-\gamma }{(1+t)}^{1-\gamma }$, which still grows unboundedly and ensures $\mathfrak{R}\mathbf{B}\left(t\right)/t\to 0$.

(iii) 

Logarithmic damping: $b\left(t\right)=\beta /(1+t)$, with $\beta >0$, so that $\mathbf{B}\left(t\right)=\beta log(1+t)$, with $\beta >0$. Here $\mathfrak{R}\mathbf{B}\left(t\right)/t\to 0$, and the damping is weak but still compatible with the scattering results (see also [1]).

(iv) 

Oscillatory damping: $b\left(t\right)=({\beta }_0+{\beta }_1{e}^{i\omega t})/(1+t)$, with ${\beta }_0\ge 0$, ${\beta }_1\in \mathbb{R}$, such that ${\beta }_0\ge {\beta }_1$, where $\mathfrak{R}b\left(t\right)=\frac{{\beta }_0+{\beta }_{1}cos\left(\omega t\right)}{1+t}$. The average real part remains non-negative, while the oscillations reflect alternating absorption and gain. Such terms arise naturally in optics with time-dependent refractive indices [2].

(v) 

Multi-frequency oscillations: $b\left(t\right)={\sum }_{k=1}^m\frac{{\beta }_{0,k}+{\beta }_{1,k}{e}^{i{\omega }_{k}t}}{{(1+t)}^{{\gamma }_k}}$, with ${\beta }_{0,k}\ge 0$, ${\beta }_{1,k}\in \mathbb{R}$ such that ${\beta }_{0,k}\ge {\beta }_{1,k}$, allowing superposition of several oscillatory modes with possibly different decay rates.

(vi) 

Quasi-periodic oscillations: $b\left(t\right)=\frac{{\beta }_0+{\beta }_1{e}^{i({\omega }_{1}t+{\omega }_2{t}^2)}}{1+t}$, with ${\beta }_0\ge 0$, ${\beta }_1\in \mathbb{R}$ such that ${\beta }_0\ge {\beta }_1$, where the modulation is not purely periodic but still admits a controlled real part in the sense of (3).

(vii) 

Polynomially damped oscillations: $b\left(t\right)=\frac{{\beta }_0+{\beta }_1\beta e^{i\omega t}}{{(1+t)}^{\gamma }}$, with ${\beta }_0\ge 0$, ${\beta }_1\in \mathbb{R}$, such that ${\beta }_0\ge {\beta }_1$ and $0\le \gamma <1$. The amplitude decays slowly, while the oscillatory factor models alternating dissipation and gain.

The fourth-order Schrödinger equation plays a pivotal role in various frameworks of mathematical physics. Initially proposed in [3] to model weak dispersion in the transmission of high-intensity laser beams through a Kerr nonlinear medium, these equations have subsequently been applied to describe the dynamics of a vortex filament within an incompressible fluid, as explored in [4,5,6] and further elaborated upon in [7,8]. On the other hand, 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, for example, to [9,10]. Recent progress on damped nonlinear Schrödinger equations provides a useful backdrop for our study. Several works have clarified how dissipation, even when time-dependent, can enforce long-time regularity and scattering: in [1,2] energy scattering as well as classical blow-up criteria are proven for general damping regimes, complementing earlier asymptotic analyses in the constant-damping setting (see also [11,12]). These developments collectively highlight two robust conclusions: (i) time-dependent damping can still yield asymptotic completeness in the energy space, and (ii) the borderline between scattering and blow-up is sensitive to the precise strength and profile of the damping.

Inspired by these applications, our study examines large-data scattering in $H^2\left({\mathbb{R}}^d\right)$ for (1). This investigation builds upon foundational concepts introduced in [13] for the nonlinear fourth-order Schrödinger equations (NL4S) [14], and for the nonlinear Schrödinger equation, and incorporates novel contributions (see also [15,16]). Specifically, we first establish new Morawetz identities along with associated inequalities for (1). In the subsequent step, by localizing the nonlinear components of these Morawetz inequalities over space–time regions, with spatial domains defined as ${\mathbb{R}}^d$-cubes, we employ a contradiction argument to demonstrate the decay of $L^r\left({\mathbb{R}}^d\right)$-norms of solutions to (1) as $t\to \infty$. This decay holds for $2<r<2d/(d-4)$ when $d\ge 5$, aligning with the results in [13,14], while simplifying the demonstration of this behavior in the context of our study. This decay phenomenon, combined with an extension of the framework developed in [17] to NL4S, enables us to establish asymptotic completeness within the energy space $H^2\left({\mathbb{R}}^d\right)$ for solutions to (1). The Morawetz estimates are formulated to focus solely on their nonlinear contributions, yielding a new class of correlation-type inequalities. These nonlinear estimates facilitate overcoming analytical challenges and provide an alternative, streamlined proof of scattering results previously obtained in [18,19,20] which analyze the asymptotic dynamics in subcritical and energy-critical regimes, and in [21], which first addressed a coupled NL4S system. In these prior works, the authors derived a collection of linear Morawetz estimates, ensuring that the wave and scattering operators for (1), with $b\left(t\right)=0$, are well-defined and bijective in the energy space $H^2\left({\mathbb{R}}^d\right)$ for $d\ge 5$.

As mentioned above, energy scattering for the damped nonlinear Schrödinger equation has previously been established primarily via Strichartz techniques combined with blow-up criteria in [1,2,11,12]. Unlike these works, our contribution is twofold. First, we introduce, for the first time, the use of Morawetz-type identities and associated inequalities in the context of damped biharmonic Schrödinger flows. This new approach not only simplifies the analysis but also extends the available techniques to the fourth-order setting, thereby bridging the gap between extinction and scattering phenomena known for the nonlinear Schrödinger equations and the higher-order dynamics addressed here. Second, unlike the aforementioned studies, which often look to indirect arguments to handle variable damping, our method directly incorporates the complex-valued coefficient $b\left(t\right)$ in (1). This allows us to treat both dissipative and oscillatory components of the damping in a unified framework, without additional reductions. Taken together, these innovations establish our results as genuinely new in the literature, providing a streamlined yet robust pathway to long-time asymptotics for damped 4NLS.

Outline of the paper: This article is organized as follows. In Section 2 we describe the main tools and notational conventions, including functional inequalities, Strichartz estimates, and conservation laws, which will be used throughout the paper. Section 3 is devoted to the derivation of new Morawetz-type identities and inequalities adapted to the damped 4NLS; these are crucial for the subsequent analysis and are of independent interest, as they generalize classical Morawetz estimates to the case of fourth-order dispersion with time-dependent damping. The proof of our main result, Theorem 1, is then presented in two stages. In Section 4 we establish the decay of solutions in Lebesgue norms, showing that the renormalized quantity $e^{\mathbf{B}\left(t\right)}u(t,x)$ cannot concentrate mass at large times. This part relies on the localized Morawetz inequalities and contradiction arguments. In Section 5 we combine these decay properties with the Strichartz framework to derive global space–time bounds for $e^{\mathbf{B}\left(t\right)}u(t,x)$, ultimately proving scattering in $H^2\left({\mathbb{R}}^d\right)$. Each of these two sections, that on the decay via Morawetz inequalities and that on scattering via Strichartz estimates, can be read independently and represent contributions of their own. Finally, in Section 6 we discuss the broader implications of our results, highlighting how they extend the current understanding of damped nonlinear Schrödinger flows to the biharmonic regime, while Section 7 outlines open problems and directions for future research.

2. Preliminaries

Before presenting our main achievements, we introduce essential notation and recall several useful tools. For any two positive real numbers $a,b$, we write $a\lesssim b$ (resp. $a\gtrsim b$) to indicate $a\le C_{1}b$ (resp. $a\ge C_{2}b$) for some positive constants $C_1,C_2$, which we specify explicitly only when necessary. We also use the notation $a\sim b$ if $b\lesssim a\lesssim b$. Additionally, we introduce the Banach spaces $L^r\left({\mathbb{R}}^d\right)=L_x^r$ and $L_t^q{L}_x^r$, for $1\le r<\infty$, endowed with the norms

$$\begin{array}{c}\hfill {∥f∥}_{L_x^r}^r={\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{r}dx<\infty \end{array}$$

and

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

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

$$H^{s,r}\left({\mathbb{R}}^d\right)=H_x^{s,r}={\left(1-{\Delta }_x^s\right)}^{-\frac{1}{2}}{L}^r\left({\mathbb{R}}^d\right)$$

and

$$H^s\left({\mathbb{R}}^d\right)=H_x^s={\left(1-{\Delta }_x^s\right)}^{-\frac{1}{2}}{L}^2\left({\mathbb{R}}^d\right),$$

for $s=1,2.$ We adopt the notation $L_{t>t_1}^{q}X$ when one restricts t in some interval with endpoints $0\le t_1<\infty$. We recall the following existence and uniqueness result (see [18]):

Theorem 2. 

Let $d\ge 1$ and $0<\alpha <{\alpha }^{*}\left(d\right)$ be given. Then, the following holds:

1 

For any initial radial condition $f\in H^2\left({\mathbb{R}}^d\right)$, the equation described in (1), with (2), (3) satisfied, admits a unique local-in-time solution

$$u(t,x)\in \mathcal{C}([0,T);H^2\left({\mathbb{R}}^d\right)),$$

where T depends on ${\parallel f\parallel }_{H_x^2}$, i.e., $T=T(\parallel f{\parallel }_{H_x^2})>0$.

2 

The solution $u(t,x)$ can be extended globally with respect to the time variable.

The local existence and uniqueness for (1) follow directly from the previous theorem in connection with (11), since $e^{-\mathbf{B}\left(t\right)}$ remains a bounded function on $[0,\infty )$, and the global well-posedness from inequality (15) below. We have here the Strichartz estimate

Definition 1. 

An exponent pair $(q,r)$ is biharmonic-admissible if $2\le q,r\le \infty ,$$(q,r,d)\ne (2,\infty ,4),$ and

$$\begin{array}{c}\hfill {\displaystyle \frac{4}{q}}+{\displaystyle \frac{d}{r}}={\displaystyle \frac{d}{2}}.\end{array}$$

(6)

Proposition 1. 

Let $(q,r)$ and $(\tilde{q},\tilde{r})$ be two biharmonic-admissible pairs. Assume that $\mathcal{D}={\nabla }_x$ and ${\mathcal{D}}^2={\Delta }_x$.Then we have, for $k=0,1,2$ and $\kappa =0,1$, the following estimates:

$$\begin{array}{c}\hfill \parallel {\mathcal{D}}^k{e}^{-it{\Delta }_x^2}{f\parallel }_{L_t^q{L}_x^r}+{∥{\mathcal{D}}^k{\int }_0^t{e}^{-i(t-\tau ){\Delta }_x^2}F\left(\tau \right)d\tau ∥}_{L_t^q{L}_x^r}\le C(\parallel {\mathcal{D}}^k{f\parallel }_{L_x^2}+{\parallel {\mathcal{D}}^{k}F\parallel }_{L_t^{{\tilde{q}}^{\prime }}{L}_x^{{\tilde{r}}^{\prime }}}).\end{array}$$

(7)

Next we recall the radial Sobolev inequality [22].

Lemma 1. 

For any radial $f\in H_x^1$, $d\ge 2$, and $x\ne 0$, we have

$$\left|f\left(x\right)\right|\le {\displaystyle \frac{1}{{\left|x\right|}^{\frac{d-1}{2}}}}{\parallel f\parallel }_{H_x^1}.$$

(8)

We notice that the solutions to (1) entail the following conservation laws:

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

(9)

where

$$\begin{array}{c}\hfill H\left(u\left(t\right)\right)=e^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\int }_{{\mathbb{R}}^d}|{\Delta }_x{u(t,x)|}^{2}dx+{\displaystyle \frac{2\lambda e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{\alpha +2}}{\int }_{{\mathbb{R}}^d}{\left|u(t,x)\right|}^{\alpha +2}dx\\ \hfill +{\displaystyle \frac{2\alpha \lambda }{\alpha +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d}\mathfrak{R}b\left(s\right)e^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\left|u(t,x)\right|}^{\alpha +2}dx.\end{array}$$

(10)

We make use of the change in variable

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

(11)

and see that u satisfies (1) if v solves

$$\left\{\begin{array}{c}i{\partial }_{t}v+{\Delta }_x^{2}v=-\lambda e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v\right|}^{\alpha }v,(t,x)\in [0,\infty )\times {\mathbb{R}}^d,\hfill \\ v(0,x)=f\left(x\right).\hfill \end{array}\right.$$

(12)

Let us multiply the identity equation by $\overline{v}(t,x)$, and integrate by parts with respect to the x-variable twice, accomplishing

$$\begin{array}{c}\hfill i{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^d}{\left|v(t,x)\right|}^{2}dx+{\int }_{{\mathbb{R}}^d}{\Delta }_x\overline{v}(t,x){\Delta }_{x}v(t,x)dx+{\int }_{{\mathbb{R}}^d}\lambda e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}dx.\end{array}$$

By taking the imaginary part, we arrive at

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

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

$${\parallel v\left(t\right)\parallel }_{L_x^2}^2={\parallel f\parallel }_{L_x^2}^2.$$

that is, the first identity in (9). We now multiply the equation in (12) by ${\partial }_t\overline{v}(t,x)$, integrate by parts with respect to the x-variable, and take the real part; then, we have

$$\begin{array}{c}\hfill \mathfrak{R}{\int }_{{\mathbb{R}}^d}\left({\Delta }_{x}v(t,x){\Delta }_x{\partial }_t\overline{v}(t,x)+\lambda e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha }v(t,x){\partial }_t\overline{v}(t,x)\right)dx=0.\end{array}$$

The previous identity leads to the following:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d}\left({\displaystyle \frac{1}{2}}{\partial }_t|{\Delta }_x{v(t,x)|}^2+{\displaystyle \frac{\lambda }{\alpha +2}}{e}^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\partial }_t{\left|v(t,x)\right|}^{\alpha +2}\right)dx=0\end{array}$$

and then to

$$\begin{array}{c}\hfill {\partial }_t{\int }_{{\mathbb{R}}^d}\left({\displaystyle \frac{1}{2}}|{\Delta }_x{v(t,x)|}^2+{\displaystyle \frac{\lambda }{\alpha +2}}{e}^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}\right)dx\\ \hfill =-{\displaystyle \frac{\alpha \lambda }{\alpha +2}}{\int }_{{\mathbb{R}}^d}\mathfrak{R}b\left(t\right)e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}dx.\end{array}$$

(13)

Integrating the above identity with respect to the t-variable (13), we get

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d}\left({\displaystyle \frac{1}{2}}|{\Delta }_x{v(t,x)|}^2+{\displaystyle \frac{\lambda }{\alpha +2}}{e}^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}\right)dx\\ \hfill +{\displaystyle \frac{\alpha \lambda }{\alpha +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d}\mathfrak{R}b\left(t\right)e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}dxds\\ \hfill ={\int }_{{\mathbb{R}}^d}\left({\displaystyle \frac{1}{2}}|{\Delta }_x{v(0,x)|}^2+{\displaystyle \frac{\lambda }{\alpha +2}}{\left|v(0,x)\right|}^{\alpha +2}\right)dx.\end{array}$$

(14)

The above identity (14) suggests that the quantity

$$\begin{array}{c}\hfill \tilde{H}\left(v\left(t\right)\right)={\int }_{{\mathbb{R}}^d}\left({\displaystyle \frac{1}{2}}|{\Delta }_x{v(t,x)|}^2+{\displaystyle \frac{\lambda }{\alpha +2}}{e}^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}\right)dx\\ \hfill +{\displaystyle \frac{\alpha \lambda }{\alpha +2}}{\int }_0^t{\int }_{{\mathbb{R}}^d}\mathfrak{R}b\left(t\right)e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}{\left|v(t,x)\right|}^{\alpha +2}dxds\end{array}$$

is conserved. Then, we get the local conservation of the Hamiltonian in (9) with $H\left(u\right(t\left)\right)$ as in (10). The above conservation laws (9) also infer the bound

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

(15)

Remark 2. 

The substitution (11), with $\mathbf{B}\left(t\right)$ as in (3), is natural in the damped case. Indeed, it eliminates the linear damping term $ib\left(t\right)u$ from the equation and transfers its effect into the nonlinearity, which then appears with a decaying time-dependent coefficient $e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}$. This mirrors classical techniques for the damped Schrödinger equation, where such exponential renormalization reflects the explicit solution to the associated linear ODE ${\partial }_{t}u+b\left(t\right)u=0$. Multiplying by $e^{\mathbf{B}\left(t\right)}$ precisely cancels this decay factor. In this way, the transformation normalizes the amplitude against exponential decay while retaining the higher-order dispersive structure of the biharmonic operator. From an analytical perspective, this change in variables allows the use of dispersive and Strichartz estimates in a form closely resembling the undamped problem, while the damping manifests only as a time-dependent weight in the nonlinear term. Moreover, because $e^{\mathbf{B}\left(t\right)}$ is time-dependent only, this change in variables preserves spatial symmetries and the isotropy of ${\Delta }_x^2$.

3. Morawetz Identities and Inequalities

Our first contribution in this section is the is the set of Morawetz identities associated with (1). Namely, we have the following:

Lemma 2. 

Let $d\ge 1$, $0<\alpha <{\alpha }^{*}\left(d\right)$ and $u\in \mathcal{C}([0,\infty );H_x^2)$ be a global solution to (1) with radial initial data $f\in H_x^2$ such that (2) is satisfied. Moreover, let $\varphi =\varphi \left(x\right):{\mathbb{R}}^d\to \mathbb{R}$ be a sufficiently regular and decaying function and introduce the action given by

$$\mathcal{M}\left(t\right)=2\mathfrak{I}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}\overline{u}(t,x)·{\nabla }_{x}u(t,x)·{\nabla }_x\varphi \left(x\right)dx.$$

(16)

The following identity holds:

$$\begin{array}{c}\hfill \dot{\mathcal{M}}\left(t\right)={\int }_{{\mathbb{R}}^d}-{\Delta }_x^3\varphi \left(x\right)e^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\left|u(t,x)\right|}^{2}dx+2{\int }_{{\mathbb{R}}^d}{\Delta }_x^2\varphi \left(x\right)e^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\left|{\nabla }_{x}u(t,x)\right|}^{2}dx\\ \hfill +4\mathfrak{R}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_{x}u(t,x)D_x^2{\Delta }_x\varphi \left(x\right)·{\nabla }_x\overline{u}(t,x)dx\\ \hfill -8\mathfrak{R}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{D}_x^{2}u(t,x)D_x^2\varphi \left(x\right)D_x^2\overline{u}(t,x)dx\\ \hfill -{\displaystyle \frac{2\alpha \lambda }{\alpha +2}}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\left|u(t,x)\right|}^{\alpha +2}{\Delta }_x\varphi \left(x\right)dx,\end{array}$$

(17)

where $D_x^2\varphi ,D_x^{2}u\in {\mathcal{M}}_{d\times d}\left({\mathbb{R}}^d\right)$ are the Hessian matrices of ϕ and u, ${\Delta }_x^2\varphi ={\Delta }_x\left({\Delta }_x\varphi \right)$, and ${\Delta }_x^3\varphi ={\Delta }_x\left({\Delta }_x\left({\Delta }_x\varphi \right)\right)$ are the second and the third powers of the Laplacian operator, respectively.

Proof. 

We choose a smooth rapidly decreasing solution $u=u(t,x)$. The general case $e^{\mathbf{B}\left(t\right)}u\in \mathcal{C}([0,\infty );H_x^2)$ can be handled via a classical density argument. The proof of (17) is similar to the one given in [23]; for instance, it can use transformation (11) and then equation (12). Thus, we skip it. □

We continue with the following:

Lemma 3. 

Assume that $d\ge 1$, $0<\alpha <{\alpha }^{*}\left(d\right)$, and let $u(t,x)\in C([0,\infty ),H^2\left({\mathbb{R}}^d\right))$ be a global solution to (1) such that (3) is satisfied. Furthermore, let $\varphi \left(x\right)$ be regular and decaying so that, for any $f\in {\mathbb{C}}^d$,

$$D_x^{2}f{D}_x^2\varphi \left(\right|x\left|\right)D_x^2\overline{f}\ge C_1{\rho }_1\left(\right|x\left|\right){\left|(w·{\nabla }_{x}f){\displaystyle \frac{w}{{\left|w\right|}^2}}\right|}^2$$

(18)

and

$$\nabla f{D}_x^2{\Delta }_x\varphi \left(\right|x\left|\right){\nabla }_x\overline{f}\le -C_2\left({\rho }_2\left(\right|x\left|\right)|{\nabla }_w^{\perp }{f|}^2-{\rho }_1\left(\right|x\left|\right){\left|(w·{\nabla }_{x}f){\displaystyle \frac{w}{{\left|w\right|}^2}}\right|}^2\right),$$

(19)

with $w\in {\mathbb{R}}^d$, ${\rho }_1\left(\right|x\left|\right),{\rho }_2\left(\right|x\left|\right)$, $C_1>C_2>0,$ where ${\nabla }_w^{\perp }f={\nabla }_{x}f-(w·{\nabla }_{x}f)w/{\left|w\right|}^2$. Then the following inequality holds:

$$\begin{array}{c}\hfill \dot{\mathcal{M}}\left(t\right)\le 2{\int }_{{\mathbb{R}}^d}-e^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\Delta }_x^{3}a\left(x\right){\left|u(t,x)\right|}^{2}dx\\ \hfill +4{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\Delta }_x^2\varphi \left(x\right)|{\nabla }_x{u(t,x)|}^{2}dx-{\displaystyle \frac{2\alpha }{\alpha +2}}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\left|u(t,x)\right|}^{\alpha +2}{\Delta }_x\varphi \left(x\right)dx.\end{array}$$

(20)

Proof. 

As above, we choose a smooth, decaying solution to (1). Then, utilizing the assumptions (18) and (19) in combination with Fubini’s Theorem, one drives the bound

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d}{\nabla }_{x}u(t,x)D_x^2{\Delta }_x\varphi \left(x\right){\nabla }_x\overline{u}(t,x)dx-{\int }_{{\mathbb{R}}^{2d}}{D}_x^{2}u(t,x)D_x^2\varphi \left(x\right)D_x^2\overline{u}(t,x)dx\\ \hfill \le -{\int }_{{\mathbb{R}}^d}\left(C_1-C_2\right){\rho }_1\left(\right|x\left|\right){\left|w·{\nabla }_{x}u(t,x){\displaystyle \frac{w}{{\left|w\right|}^2}}\right|}^{2}dx-C_2{\int }_{{\mathbb{R}}^d}{\rho }_1\left(\right|x\left|\right)|{\nabla }_w^{\perp }{u(t,x)|}^{2}dx\le 0.\end{array}$$

(21)

Hence, according to the above inequality, equality (17), and bound (21), one finally obtains identity (20). □

Inequalities of Morawetz-Type Identities

We start with an outcome that is a consequence of Lemma 2 above. The Morawetz functionals here are defined in a way that respects the isotropy of ${\Delta }_x^2$ and the $O\left(d\right)$-symmetry inherent in the radial setting. More precisely, we determine the following:

Lemma 4. 

Assume that $d\ge 5$ and $0<\alpha <{\alpha }^{*}\left(d\right)$, and let $u\in \mathcal{C}([0,\infty );H_x^2)$ be a global solution to (1) with radial initial data $f\in H_x^2$ such that (2) and (3) are satisfied. Then it holds that

$$\begin{array}{c}\hfill -{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\left|x\right|}^5}}{\left|u(t,x)\right|}^{2}dx\gtrsim \mathfrak{I}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_x\overline{u}(t,x){\displaystyle \frac{x}{\left|x\right|}}·{\nabla }_{x}u(t,x)dx,\end{array}$$

(22)

for $d\ge 6,$ and

$$\begin{array}{c}\hfill -{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx\gtrsim \mathfrak{I}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_x\overline{u}(t,x){\displaystyle \frac{x}{\langle x\rangle }}·{\nabla }_{x}u(t,x)dx,\end{array}$$

(23)

for $d=5,$ where $\langle x\rangle :={(1+|x|}^2{)}^{\frac{1}{2}}.$

Proof. 

We now choose $\varphi \left(x\right)=\left|x\right|$. Elementary computations yield

$${\Delta }_x\left|x\right|={\displaystyle \frac{(d-1)}{\left|x\right|}},$$

(24)

for $d>1$,

$${\Delta }_x^2\left|x\right|=\left\{\begin{array}{c}-\frac{(d-1)(d-3)}{{\left|x\right|}^3}\mathrm{if}d\ge 4,\hfill \\ \\ -4\pi {\delta }_{x=0}\mathrm{if}d=3\hfill \end{array}\right.$$

(25)

and

$${\Delta }_x^3\left|x\right|=\left\{\begin{array}{cc}C\delta \left(x\right),\hfill & d=5,\hfill \\ {3(d-1)(d-3)(d-5)\left|x\right|}^{-5},\hfill & d\ge 6.\hfill \end{array}\right.$$

(26)

Additionally, for $d\ge 2$ and with $w=x$, they satisfy

$$D_x^{2}u\left(x\right)D_x^2\left|x\right|D_x^2\overline{u}\left(x\right)\ge {\displaystyle \frac{(d-1)}{{\left|x\right|}^3}}{\left|{\nabla }_{w}u\left(x\right)\right|}^2,$$

(27)

(that is, bound (18)) (see [24]) and

$${\nabla }_{x}u\left(x\right)D_x^2{\Delta }_x^2\left|x\right|{\nabla }_x\overline{u}\left(x\right)=-{\displaystyle \frac{(d-1)}{{\left|x\right|}^3}}\left(\right|{\nabla }_w^{\perp }{u\left(x\right)|}^2-2|{\nabla }_{x}u\left(x\right)-{\nabla }_w^{\perp }u\left(x\right){|}^2),$$

(28)

(that is, bound (19)) (here we refer to [23]). Similarly, one can get, for $\varphi \left(x\right)=\langle x\rangle$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\Delta }_x\langle x\rangle ={\displaystyle \frac{d-1}{\langle x\rangle }}+{\displaystyle \frac{1}{{\langle x\rangle }^3}},\hfill \\ \hfill & {\Delta }_x^2\langle x\rangle =-{\displaystyle \frac{(d-1)(d-3)}{{\langle x\rangle }^3}}-{\displaystyle \frac{6(d-3)}{{\langle x\rangle }^5}}-{\displaystyle \frac{15}{{\langle x\rangle }^7}},\hfill \\ \hfill & {\Delta }_x^3\langle x\rangle ={\displaystyle \frac{3(d-1)(d-3)(d-5)}{{\langle x\rangle }^5}}+{\displaystyle \frac{45(d-3)(d-5)}{{\langle x\rangle }^7}}+{\displaystyle \frac{315(d-5)}{{\langle x\rangle }^9}}+{\displaystyle \frac{945}{{\langle x\rangle }^{11}}},\hfill \end{array}\end{array}$$

(29)

as well as

$$\begin{array}{cc}\hfill {\Delta }_x^2\langle x\rangle {\left|{\nabla }_{x}u\right|}^2& =\left(-{\displaystyle \frac{(d-1)(d-3)}{{\langle x\rangle }^3}}-{\displaystyle \frac{6(d-3)}{{\langle x\rangle }^5}}-{\displaystyle \frac{15}{{\langle x\rangle }^7}}\right){\left|{\nabla }_{x}u\right|}^2\hfill \\ \hfill & =\left(-{\displaystyle \frac{(d+5)(d-3)}{{\langle x\rangle }^5}}-{\displaystyle \frac{{(d-1)(d-3)\left|x\right|}^2}{{\langle x\rangle }^5}}-{\displaystyle \frac{15}{{\langle x\rangle }^7}}\right){\left|{\nabla }_{x}u\right|}^2\hfill \\ \hfill & \le -{\displaystyle \frac{(d+5)(d-3)|{\nabla }_{w}u{|}^2}{{\langle x\rangle }^5}},\hfill \end{array}$$

(30)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -{\partial }_{jk}^x\langle x\rangle {\partial }_{ij}^x\overline{u}{\partial }_{ik}^{x}u=-\left({\displaystyle \frac{{\delta }_{ij}}{\langle x\rangle }}-{\displaystyle \frac{{\left(x\right)}_i{\left(x\right)}_j}{{\langle x\rangle }^3}}\right){\partial }_{ij}^x\overline{u}{\partial }_{ik}^{x}u=-{\displaystyle \frac{|{\nabla }_x{\partial }_i^{x}u|}{\langle x\rangle }}+{\displaystyle \frac{{\left|x\right|}^2{\left|{\nabla }_w{\partial }_i^{x}u\right|}^2}{{\langle x\rangle }^3}}\\ \hfill \le -{\displaystyle \frac{|{\nabla }_x{\partial }_i^{x}u{|}^2-{\left|{\nabla }_w{\partial }_i^{x}u\right|}^2}{\langle x\rangle }}\le -{\displaystyle \frac{(d-1)|{\nabla }_{w}u{|}^2}{{\langle x\rangle }^3}}\end{array}\end{array}$$

(31)

and

$$\begin{array}{c}\hfill {\partial }_{jk}^x{\Delta }_{x}a{\partial }_j^x\overline{u}{\partial }_k^{x}u\end{array}$$

(32)

$$\begin{array}{c}\hfill =\left(-{\displaystyle \frac{(d-1){\delta }_{ij}}{{\langle x\rangle }^3}}+{\displaystyle \frac{3(d-1)x_i{x}_j}{{\langle x\rangle }^5}}-{\displaystyle \frac{3{\delta }_{ij}}{{\langle x\rangle }^5}}+{\displaystyle \frac{15x_i{x}_j}{{\langle x\rangle }^7}}\right){\partial }_j^x\overline{u}{\partial }_k^{x}u\end{array}$$

(33)

$$\begin{array}{c}\hfill \le -{\displaystyle \frac{(d-1)|{\nabla }_{x}u{|}^2}{{\langle x\rangle }^3}}+{\displaystyle \frac{{3(d-1)\left|x\right|}^2{\left|{\nabla }_{w}u\right|}^2}{{\langle x\rangle }^5}}-{\displaystyle \frac{3|{\nabla }_{x}u{|}^2}{{\langle x\rangle }^5}}+{\displaystyle \frac{{15\left|x\right|}^2{\left|{\nabla }_{w}u\right|}^2}{{\langle x\rangle }^7}}\end{array}$$

(34)

$$\begin{array}{c}\hfill \le {\displaystyle \frac{2(d-1)|{\nabla }_{w}u{|}^2}{{\langle x\rangle }^3}}-{\displaystyle \frac{3(d-5)|{\nabla }_{w}u{|}^2}{{\langle x\rangle }^5}},\end{array}$$

(35)

for $i,j,k=1,\dots ,d$, where we utilized in (31) the inequality

$$\begin{array}{c}\hfill \sum _i\left(\right|{\nabla }_x{\partial }_i^{x}u{|}^2-|{\nabla }_w{\partial }_i^{x}u{|}^2)\ge {\displaystyle \frac{d-1}{{\left|x\right|}^2}}{\left|{\nabla }_{w}u\right|}^2,\end{array}$$

which can be found in [24], for instance. We will now take into account the cases $d\ge 6$ and $d=5$, separately. For the former, one can directly use (24)–(26) in (20), considering (27) and (28). This gives the inequality (22). For $d=5$, we will argue in a similar way as in the proof of (20), with minor changes. Namely, by plugging (29)–(32) into (17), we have the chain of inequalities

$$\begin{array}{c}\hfill \dot{\mathcal{M}}\left(t\right)\lesssim -{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx-2{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\displaystyle \frac{(d+5)(d-3)}{{\langle x\rangle }^5}}{\left|{\nabla }_{x}u(t,x)\right|}^2,dx\\ \hfill +4\mathfrak{R}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\displaystyle \frac{2(d-1)}{{\langle x\rangle }^3}}{\left|{\nabla }_{x}u(t,x)\right|}^{2}dx\\ \hfill -8\mathfrak{R}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\displaystyle \frac{(d-1)}{{\langle x\rangle }^3}}{\left|{\nabla }_{x}u(t,x)\right|}^{2}dx\\ \hfill -{\displaystyle \frac{2\alpha \lambda }{\alpha +2}}{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^3}}{\left|u(t,x)\right|}^{\alpha +2}dx\lesssim -{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx,\end{array}$$

(36)

which gives the desired estimate (23). □

We obtain the following proposition, which is a consequence of the previous lemma:

Proposition 2. 

Let $d\ge 5$, $0<\alpha <{\alpha }^{*}\left(d\right)$ and $u\in \mathcal{C}([0,\infty );H_x^2)$ be a global solution to (1) with radial initial data $f\in H_x^1$ such that (2) and (3) are satisfied. Then, for 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

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_{{\mathcal{Q}}_{\tilde{x}}^d\left(r\right)}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\left|x\right|}^5}}{\left|u(t,x)\right|}^{2}dx<\infty ,\end{array}$$

(37)

for $d\ge 6$ and

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_{{\mathcal{Q}}_{\tilde{x}}^d\left(r\right)}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx<\infty ,\end{array}$$

(38)

for $d=5.$

Proof. 

By integrating (22) with respect to the time variable on the interval $J=[t_1,t_2]$, where $t_1,t_2\in [0,\infty )$, one obtains

$$\begin{array}{c}\hfill \left|{\left[\mathfrak{I}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_x\overline{u}(t,x){\displaystyle \frac{x}{\left|x\right|}}·{\nabla }_{x}u(t,x)dx\right]}_{t=t_1}^{t=t_2}\right|\\ \hfill \gtrsim {\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\left|x\right|}^5}}{\left|u(t,x)\right|}^{2}dx\gtrsim {\int }_0^{\infty }{\int }_{{\mathcal{Q}}_{\tilde{x}}^d\left(r\right)}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\left|x\right|}^5}}{\left|u(t,x)\right|}^{2}dx.\end{array}$$

Applying the Cauchy–Schwartz inequality again and according to the conservation laws (9), one also infers

$$\begin{array}{c}\hfill {\left[\mathfrak{I}{\int }_{{\mathbb{R}}^{2d}}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_x\overline{u}(t,x){\displaystyle \frac{x}{\left|x\right|}}·{\nabla }_{x}u(t,x)|dx\right]}_{t=t_1}^{t=t_2}\lesssim {\parallel f\parallel }_{H_x^2}^2<\infty .\end{array}$$

(39)

Finally, we get (37) upon letting $t_1\to 0,t_2\to \infty$. Similarly, integrating (23) with respect to the time variable on J, we arrive at

$$\begin{array}{c}\hfill \infty >\left|{\left[\mathfrak{I}{\int }_{{\mathbb{R}}^d}{e}^{2\mathfrak{R}\mathbf{B}\left(t\right)}{\nabla }_x\overline{u}(t,x){\displaystyle \frac{x}{\langle x\rangle }}·{\nabla }_{x}u(t,x)dx\right]}_{t=t_1}^{t=t_2}\right|\\ \hfill \gtrsim {\int }_{{\mathbb{R}}^d}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx\gtrsim {\int }_0^{\infty }{\int }_{{\mathcal{Q}}_{\tilde{x}}^d\left(r\right)}{\displaystyle \frac{e^{2\mathfrak{R}\mathbf{B}\left(t\right)}}{{\langle x\rangle }^{11}}}{\left|u(t,x)\right|}^{2}dx.\end{array}$$

The proof of the proposition is now completed. □

4. The Decay of Solutions

This section is devoted to proving the first part of Theorem 1. Specifically, we present the following:

Proof of 

(4). Firstly, we consider dimensions $d\ge 6$, and subsequently focus specifically on the case $d=5$. Following [25], it suffices to demonstrate property (4) for some suitable $2<q<\frac{2d}{d-4}$ when $d\ge 5$. The result for the general case then follows by employing mass conservation and kinetic energy conservation in (9), and interpolation arguments. Setting $v(t,x)=e^{\mathbf{B}\left(t\right)}u(t,x)$, our goal is to establish that

$$\underset{t\to \pm \infty }{lim}{\parallel v\left(t\right)\parallel }_{L_x^{\frac{2d+4}{d}}}=0.$$

(40)

To prove (40), we follow [14], assuming, for the sake of contradiction, the existence of a sequence $\left\{t_n\right\}$ satisfying $t_n\to +\infty$ and

$$\underset{n}{inf}{\parallel v(t_n,x)\parallel }_{L_x^{{\displaystyle \frac{2d+4}{d}}}}={ϵ}_0>0.$$

(41)

Next, we recall the localized Gagliardo–Nirenberg inequality from [13] (see also [26,27]):

$${\parallel \phi \parallel }_{L_x^{\frac{2d+4}{d}}}^{\frac{2d+4}{d}}\le C{\left(\underset{x\in {\mathbb{R}}^d}{sup}{\parallel \phi \parallel }_{L^2\left(Q_x\right)}\right)}^{\frac{4}{d}}{\parallel \phi \parallel }_{H_x^2}^2,$$

(42)

where $Q_x+{[-1,1]}^d$. Using (41) and (42), and choosing $\phi =v(t_n,x)$, which satisfies $\parallel v(t_n,x){\parallel }_{H_x^2}<+\infty$, we infer the existence of $x_n\in {\mathbb{R}}^d$ such that

$$\parallel v(t_n,x){\parallel }_{L^2\left(Q_{x_n}\right)}={\delta }_0>0.$$

(43)

We now assert the existence of some $\overline{t}>0$ for which

$${\parallel v(t,x)\parallel }_{L^2\left({\tilde{Q}}_{x_n}\right)}\ge {\delta }_0/2,$$

(44)

for all $t\in (t_n,t_n+\overline{t})$, with ${\tilde{Q}}_x=x+{[-2,2]}^d$ indicating the cube of side length 2 centered at x. To verify (44), we choose a radially symmetric cutoff function $\tilde{\chi }\left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^d\right)$, such that $\tilde{\chi }\left(x\right)=1$ for $x\in Q_x$ and $\tilde{\chi }\left(x\right)=0$ for $x\notin {\tilde{Q}}_x$. Thus, we obtain

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^d}\tilde{\chi }(x-x_n){\left|v(t,x)\right|}^{2}dx\right|& \lesssim \left|{\int }_{{\mathbb{R}}^d}{\Delta }_x\tilde{\chi }(x-x_n)\mathfrak{I}\left({\Delta }_{x}v(t,x)\overline{v}(t,x)\right)dx\right|\hfill \\ \hfill & +\left|{\int }_{{\mathbb{R}}^d}{\nabla }_x\tilde{\chi }(x-x_n)\mathfrak{I}\left({\Delta }_{x}v(t,x){\nabla }_x\overline{v}(t,x)\right)dx\right|\hfill \\ \hfill & \lesssim \underset{t}{sup}{\parallel v(t,x)\parallel }_{H_x^2}^2.\hfill \end{array}$$

By taking advantage of estimate (15) and applying the fundamental theorem of calculus, we obtain

$$\left|{\int }_{{\mathbb{R}}^d}\tilde{\chi }(x-x_n)\left(\right|v(s,x){|}^2-|v(t,x){|}^2)dx\right|\le C|t-s|,$$

(45)

for some constant $C>0$ independent of n. Setting $t=t_n$, it follows immediately that

$${\int }_{{\tilde{Q}}_{x_n}}{\left|v(s,x)\right|}^{2}dx\ge {\int }_{Q_{x_n}}|v(t_n,x){|}^{2}dx-C|t_n-s|.$$

(46)

Therefore, choosing a sufficiently small value of $\overline{t}>0$, specifically $3{\delta }_0^2>4C\overline{t}$, we achieve (44). Observe also that, from (43) and (8) of Lemma 1, one sees that $|x_n|\lesssim 1$, because $\left|x\right|\sim |x_n|$ if $x\in {\tilde{Q}}_{x_n}$. Thus, for $d\ge 6$, one concludes that

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{{\left|x\right|}^5}}dxdt\gtrsim {\int }_0^{\infty }\left(\underset{x\in {\mathbb{R}}^d}{sup}{\int }_{Q_x}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{{\left|x\right|}^5}}dx\right)dt\\ \hfill \gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\int }_{{\tilde{Q}}_{x_n}}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{|x_n{|}^5}}dxdt\gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\int }_{{\tilde{Q}}_{x_n}}{\left|v(t,x)\right|}^{2}dxdt\\ \hfill \gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\delta }_0^{2}dt=\infty ,\end{array}$$

(47)

where in the last two lines, we applied (44) in conjunction with the fact that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a bounded sequence. This generates a contradiction to (37). For $d=5$ we can proceed as in the previous case, arriving at

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{{\langle x\rangle }^{11}}}dxdt\gtrsim {\int }_0^{\infty }\left(\underset{x\in {\mathbb{R}}^d}{sup}{\int }_{Q_x}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{{\langle x\rangle }^{11}}}dx\right)dt\\ \hfill \gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\int }_{{\tilde{Q}}_{x_n}}{\displaystyle \frac{{\left|v(t,x)\right|}^2}{{\langle x_n\rangle }^{11}}}dxdt\gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\int }_{{\tilde{Q}}_{x_n}}{\left|v(t,x)\right|}^{2}dxdt\\ \hfill \gtrsim \sum _n{\int }_{t_n}^{t_n+\overline{t}}{\delta }_0^{4}dt=\infty ,\end{array}$$

(48)

Again, the above inequality gives rise to a contradiction with the interaction estimate (38). The proof is now accomplished. □

5. Scattering

Here, we detail the proof of the second part of Theorem 1. To achieve this, the following result is employed to obtain the space–time summability required for the scattering; that is,

Lemma 5. 

Let us assume that α as in (2). Then, for any $u\in \mathcal{C}([0,\infty ),H_x^2)$ global solution to (1) satisfying (3), we have

$$\begin{array}{c}\hfill e^{\mathbf{B}\left(t\right)}u\in L^q([0,\infty ),H_x^{2,r}),\end{array}$$

(49)

for every biharmonic-admissible pair $(q,r)$.

Proof. 

We take into account, for any $f\in H_x^2$, the integral operator associated with (1), defined as

$$\begin{array}{c}\hfill e^{it{\Delta }_x^2}f+\lambda {\int }_0^t{e}^{i(t-\tau ){\Delta }_x^2}g(u\left(\tau \right),\alpha )d\tau :={\mathcal{T}}_f\left(e^{\mathbf{B}\left(t\right)}u\right)\\ \hfill =e^{it{\Delta }_x^2}f+\lambda {\int }_0^t{e}^{-\alpha \mathfrak{R}\mathbf{B}\left(\tau \right)}{e}^{i(t-\tau ){\Delta }_x^2}\left(|e^{\mathbf{B}\left(\tau \right)}{u\left(\tau \right)|}^{\alpha }{e}^{\mathbf{B}\left(\tau \right)}u\left(\tau \right)\right)d\tau .\end{array}$$

(50)

We obtained the thesis using the Strichartz estimates (7) (see again [18,19], for instance). Moreover, it will be enough to handle the inhomogeneous part in (50). Select $(q^{\prime },r^{\prime })$ such that

$$(q,r):=\left({\displaystyle \frac{8(\alpha +2)}{d\alpha }},\alpha +2\right).$$

(51)

The Hölder inequality, the Leibniz rule, and (3) give

$$\begin{array}{cc}\hfill {\parallel g(u,\alpha )\parallel }_{L_{t>T}^{q^{\prime }}{H}_x^{2,r^{\prime }}}\lesssim \parallel e^{-\alpha \mathfrak{R}\mathbf{B}\left(\tau \right)}{\parallel }_{L_t^{\infty }\left((0,\infty )\right)}\parallel \parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{H_x^{2,r}}\parallel e^{\mathbf{B}\left(\tau \right)}u{{|}^{\alpha }\parallel }_{L_x^{{\displaystyle \frac{r}{p}}}}{\parallel }_{L_{t>T}^{q^{\prime }}}& \\ \hfill \lesssim \parallel \parallel e^{\mathbf{B}\left(\tau \right)}u{{\parallel }_{H_x^{2,r}}\left(\parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_x^r}^{\alpha (1-\theta )}{\parallel e^{\mathbf{B}\left(\tau \right)}u\parallel }_{L_x^r}^{\alpha \theta }\right)\parallel }_{L_{t>T}^{q^{\prime }}}.\end{array}$$

(52)

We now select $\theta \in (0,1)$ so that $\theta =(q-q^{\prime })/\alpha q^{\prime }$; this implies that the term in (52) is controlled by

$$\begin{array}{c}\hfill {∥\parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{H_x^{2,r}}\parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_x^r}^{\alpha (1-\theta )}{\parallel e^{\mathbf{B}\left(\tau \right)}u\parallel }_{L_x^r}^{\alpha \theta }∥}_{L_{t>T}^{q^{\prime }}}\\ \hfill \lesssim \parallel \parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{H_x^{2,r}}\parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_x^r}^{\frac{q}{q^{\prime }}-1}{\parallel e^{\mathbf{B}\left(\tau \right)}u\parallel }_{L_x^r}^{\alpha +1-\frac{q}{q^{\prime }}}{\parallel }_{L_{t>T}^{q^{\prime }}}& \\ \hfill \lesssim \parallel \parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{H_x^{2,r}}^{\frac{q}{q^{\prime }}}\parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_x^{\alpha +2}}^{\alpha +1-\frac{q}{q^{\prime }}}{\parallel }_{L_{t>T}^{q^{\prime }}}\lesssim \parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_{t>T}^{\infty }{L}_x^r}^{\alpha +1-\frac{q}{q^{\prime }}}{\parallel e^{\mathbf{B}\left(\tau \right)}u\parallel }_{L_{t>T}^q{H}_x^{2,r}}^{q-1}& ,\hfill \end{array}$$

(53)

with all the constants independent of $t,T$. These conclusions, in association with (50) and (4) and the use of the inhomogeneous Strichartz estimates in (7), lead to

$$\begin{array}{c}\hfill \parallel e^{\mathbf{B}\left(\tau \right)}{u\parallel }_{L_{t>T}^q{H}_x^{2,r}}\le {C\parallel f\parallel }_{H_x^2}+ϵ\left(T\right){\parallel e^{\mathbf{B}\left(\tau \right)}u\parallel }_{L_{t>T}^q{H}_x^{2,r}}^{q-1},\end{array}$$

(54)

where $ϵ\left(T\right)\to 0$ as $T\to \infty .$ Then for a sufficiently large T, we arrive at

$${\parallel u\parallel }_{L^q((T,t),H_x^{2,r})}\le \overline{C},$$

with the constant $\overline{C}$ independent of t. In this way we determine that $e^{\mathbf{B}\left(t\right)}u\in L^q((T,\infty ),H_x^{2,r}).$ We conclude, using a continuity, that $e^{\mathbf{B}\left(t\right)}u\in L^q([0,\infty ),H_x^{2,r})$. □

A consequence of the above lemma is the following:

Proof of 

(5). The proof of (5) is now a straight consequence of Lemma 5 above: we present it here in brief for the sake of completeness. We write $\overline{w}\left(t\right)=e^{\mathbf{B}\left(\tau \right)}{e}^{-it{\Delta }_x^2}u\left(t\right)$, getting

$$\begin{array}{c}\hfill \overline{w}\left(t\right)=f+i{\int }_0^t{e}^{-is{\Delta }_x^2}g(u,\alpha )ds,\end{array}$$

(55)

Moreover, for $0<t_1<t_2$, one has

$$\begin{array}{c}\hfill \overline{w}\left(t_2\right)-\overline{w}\left(t_1\right)=i{\int }_{t_1}^{t_2}{e}^{-is{\Delta }_x^2}g(u,\alpha )ds.\end{array}$$

(56)

By applying the Strichartz estimates (7), we infer that

$$\begin{array}{c}\hfill \parallel \overline{w}\left(t_1\right)-\overline{w}\left(t_2\right){\parallel }_{H_x^2}\lesssim {∥{\int }_{t^{\prime }}^t{e}^{-is{\Delta }_x^2}\left(e^{-\alpha \mathfrak{R}\mathbf{B}(·)}{\left|e^{\mathbf{B}(·)}u\right|}^{\alpha }{e}^{\mathbf{B}(·)}u\right)\left(s\right)ds∥}_{H_x^2}\\ \hfill \lesssim \parallel e^{-\alpha \mathfrak{R}\mathbf{B}(·)}|e^{\mathbf{B}(·)}u{{|}^{\alpha }{e}^{\mathbf{B}(·)}u\parallel }_{L^{q^{\prime }}((t_1,t_2),H_x^{2,r^{\prime }})},\end{array}$$

(57)

with $(q,r)$ is a Schrödinger-admissible biharmonic pair, as in (51). Via the steps we followed in the proof of Lemma 5, one attains

$$\begin{array}{c}\hfill \underset{t_1,t_2\to \infty }{lim}{\parallel \overline{w}\left(t_1\right)-\overline{w}\left(t_2\right)\parallel }_{H_x^2}=0.\end{array}$$

Then we can see that there exists $f^{\pm }\in H_x^2$ and, thus, the map $f\to f^{\pm }$ in $H_x^2$, as $t\to \infty .$

□

Remark 3. 

The proof of Theorem 1 combines the new Morawetz identities with localized decay estimates to rule out the possibility of persistent concentration. The key step is showing that the $L_x^r$ norms of $e^{\mathbf{B}\left(t\right)}u(t,x)$ vanish asymptotically, which, together with the Strichartz framework, yields scattering. This confirms that damping dominates any potential nonlinear growth, leading to dispersion, similarly to the free biharmonic Schrödinger equation. Thus, Theorem 1 provides a rigorous characterization of the long-time behavior of damped fourth-order Schrödinger flows.

Remark 4. 

We also emphasize that one may derive exponential decay simply by interpolating the mass-conservation law (9) with the Sobolev-embedding bounds obtained in (15). Indeed, this yields the uniform bound

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

Nonetheless, this estimate by itself falls short of the sharper decay rate asserted in (4) of Theorem 1, which reflects a strictly stronger dispersive property of solutions to (1). Finally, we note that the above argument breaks down in the degenerate case $\mathfrak{R}\mathbf{B}\left(t\right)=0$, i.e., when $ib\left(t\right)$ reduces to a purely real coefficient.

Remark 5. 

We stress once again that our theorems cover a wide array of damped nonlinearities obeying the general bound (3). In particular, they include the logarithmic decay

$$\mathfrak{R}b\left(t\right)\sim {\displaystyle \frac{\tilde{\beta }}{1+t}},\tilde{\beta }>0,t\ge 0,$$

as well as the slowly decaying regimes studied in [2], namely,

$$\mathfrak{R}b\left(t\right)\sim {\displaystyle \frac{\tilde{\beta }}{{(1+t)}^{\gamma }}},0\le \gamma <1,\tilde{\beta }>0,t\ge 0.$$

Additionally, our approach readily adapts to even richer time-dependent damping profiles. For example, one may consider equations of the form

$$i{\partial }_{t}u+{\Delta }_x^{2}u+i{\displaystyle \frac{\tilde{\beta }{t}^{{\gamma }_1}lnt}{{(1+t)}^{{\gamma }_2}}}u+\tilde{b}\left(t\right)u+\lambda u{\left|u\right|}^{\alpha }=0,t\ge \delta \ge 1,$$

where $\tilde{b}\left(t\right)$ is any real-valued continuous function, and the constants $\tilde{\beta },{\gamma }_1,{\gamma }_2$ define the precise damping rate. Equations of this type have appeared in earlier works (see [1]), illustrating the flexibility and broad reach of our method.

6. Conclusions

We generalize the results of [1,2,11,12] to the case of the damped 4NLS. Our hypotheses on the coefficient function $b\left(t\right)$ reflect the broad framework set forth in [2], yet we also address decay rates of the form

$$e^{-\alpha \mathfrak{R}\mathbf{B}\left(t\right)}\lesssim {(1+t)}^{-\rho },\rho \ge 0.$$

A novel feature of our analysis is the inclusion of an oscillatory term in the perturbed propagator $e^{\mathbf{B}\left(t\right)+it{\Delta }_x^2},$ a scenario that is not addressed in the previously mentioned works. Furthermore, when $\frac{8}{d}<\alpha <\frac{8}{d-4}$, our argument yields a simpler proof of energy-space scattering for the damped 4NLS. By deploying localized Morawetz estimates, we demonstrate that solutions to (1) decay faster in $L_x^r$ than in $e^{-\mathfrak{R}\mathbf{B}\left(t\right)}$ whenever the Hamiltonian (10) remains positive. This stronger decay criterion enlarges the admissible class of perturbations, even including the borderline case $\mathbf{b}\equiv 0$. We have purposely imposed slow-growth restrictions on the nonlinearity in (1), confident that our technique readily extends to any $b\left(t\right)$ satisfying

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

with $\mathbf{B}\left(t\right)$ as in (3), and even to the case of $\mathfrak{R}\mathbf{B}\left(t\right)<0$. These generalizations will be pursued in future investigations. Thus, the present paper advances the second-order theory in a fourth-order (biharmonic) setting with complex, time-dependent damping in the following ways: we (a) introduce Morawetz-type identities adapted to the biharmonic flow and to the renormalized variable $v(t,x)=e^{\mathbf{B}\left(t\right)}u(t,x)$, and (b) achieve decay and scattering in $H_x^2$ under explicit hypotheses on $b\left(t\right)$ that include slowly decaying and oscillatory behaviors (the conditions in (3)). This furnishes a self-contained higher-order, damped analogue of the second-order theory in [1,2,11,12].

7. Open Problems and Further Developments

The abstract framework developed herein is sufficiently versatile and powerful to yield direct insights into the energy-dissipation characteristics of damped Schrödinger flows with local nonlinearities. In particular, one can consider the following unresolved challenges:

• Establishing scattering results in the energy-critical and supercritical regimes for the focusing case $(\lambda <0)$ in (1), thereby clarifying the delicate interplay between damping mechanisms and the nonlinear effects of focusing.
• Determining precise decay estimates and asymptotic completeness for fourth-order nonlinear Schrödinger equations on product domains, for instance,

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

  with ${\left({\Delta }_{x,y}\right)}^2={\Delta }_{x,y}\left({\Delta }_{x,y}\right),$ $(x,y)\in {\mathbb{R}}^d\times \mathbb{T}$, to better understand the long-time behavior and stability in higher-order dispersive models with time-dependent damping.
• Investigating decay and scattering properties for nonlinear beam equations in mixed periodic settings, for example,

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

  which would shed light on the interaction between biharmonic dispersion and variable damping.
• Analyzing global scattering dynamics for damped nonlinear Klein–Gordon models,

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

  including the impact of partial periodicity on long-time dispersion and energy decay.