Inhomogeneous NLS with partial harmonic confinement

1.   Introduction

We consider the nonlinear, focusing, inhomogeneous Schrödinger equation with a partial harmonic confinement

Hereafter, the space dimension is $N\geq2$ and the wave function is denoted by $u: = u(t, x):\mathbb{R}\times \mathbb{R}^N\to \mathbb{C}$. The set of partial confinement components is $\emptyset\neq J: = \{i_1, \dots, i_k\}$, where $1\leq k < N$ and $1\leq i_1 < \dots < i_k < N$. Finally, the exponent of the source term is $p > 1$, and the singular inhomogeneity satisfies $0 < \varrho < \tilde{2} : = \frac{N}{3}\chi_{[2, 3]} + 2\chi_{[4, \infty)}$.

The cubic nonlinear Schrödinger equation, commonly referred to as the Gross–Pitaevskii equation (GPE), plays a crucial role in physics. Specifically, (1.1) with $p = 3, \, \varrho = 0$, and an external trapping potential provides an effective description of Bose–Einstein condensates (BEC). A Bose–Einstein condensate is a macroscopic collection of bosons that, at extremely low temperatures, occupy the same quantum state. This phenomenon was experimentally observed only in the last two decades ^[1,2], which has spurred extensive theoretical and numerical research. In experiments, BEC is observed in the presence of a confining potential trap, and its macroscopic behavior is highly dependent on the shape of this trapping potential. When the trap potential is confined along partial directions in the space, ^[3,4] showed the same properties as the whole confinement in the space under some assumptions. At low enough temperature, neglecting the thermal and quantum fluctuations, a Bose condensate can be represented by (1.1). Specifically, if we consider a condensate of particles of mass and negative effective scattering length in a partial confining potential using variables rescaled by the natural quantum harmonic oscillator units of time, we get (1.1). Equation (1.1) arises also in the propagation of mutually incoherent wave packets in nonlinear optics. For more details we refer to ^[5].

Several historic works have addressed the non-linear Schrödinger equation with partial confinement. The existence of orbitally stable ground states was investigated in ^[4], and the strong instability of standing waves was studied in ^[6]. A mixed source term was considered in ^[7]. The energy scattering of global solutions in the focusing intercritical regime was treated in ^[3,8,9,10], while the finite-time blow-up of energy solutions was examined in ^[11,12]. Thresholds for global existence versus energy concentration were established in ^[13,14]. See also ^[15,16] for the mass-critical NLS concentration without any potential. All these works focus on the homogeneous regime, corresponding to (1.1) with $\varrho = 0$. The only paper addressing the inhomogeneous case appears to be ^[17], which investigated the existence and stability of standing waves. This work aims to extend the existing literature to the inhomogeneous regime, specifically treating the case $\varrho \neq 0$ in (1.1). The challenge lies in addressing the partial confinement, which breaks the scaling invariance, as well as the singular inhomogeneous term $|\cdot|^{-\varrho}$. The method used here does not cover the energy-critical regime, which is investigated in a work in progress. A solid theoretical understanding of the problem motivates us to explore its numerical and practical aspects in future work.

Let us outline the plan of the manuscript. Section 2 proves a global existence result. Section 3 establishes a threshold of global existence versus blowup of mass-critical solutions. Sections 4-5 investigate the finite-time blow-up of mass-critical non-global solutions. In the appendix, some variance-type identities are established.

For simplicity, let us denote the Lebesgue space $L^p: = L^p(\mathbb{R}^N)$ and the classical norms $\|\cdot\|_{p}: = \|\cdot\|_{L^p}$, $\|\cdot\|: = \|\cdot\|_2.$ Let us denote the Sobolev space $H^1: = \{f\in L^2$ and $\nabla f\in L^2\}$. Finally, if $A$ and $B$ are positive real numbers, $A\lesssim B$ means that $A\leq CB$ for an absolute positive constant $C > 0$.

1.1.   Preliminary

First, let us give some notations. Let the real numbers

Let us define $p_{k, c}: = 1+\frac{4-2\varrho}{N-k}$, the mass critical exponent $p_c: = p_{0, c}$ and the energy critical one

Note that, since the partial confinement breaks the scaling invariance, the mass-critical and energy-critical exponents are taken as the same for the INLS without any potential. The so-called energy space is

endowed with the norm

Hereafter, we denote for $u\in\Sigma,$ the conserved real quantities under the flow of (1.1), which are respectively referred to as the mass and the energy

The problem (1.1) is locally well-posed in the energy space, as demonstrated by the results in ^[8,18,19]. Specifically, by applying the Strichartz estimate from ^[8], we employ the standard fixed-point method introduced in ^[18], while incorporating techniques from ^[19] to address the inhomogeneous term.

Proposition 1.1. Let $N\geq 2$, $0 < \varrho < \tilde 2$, $1 < p < {p^c}$ and $u_{0}\in \Sigma$. Then, the Cauchy problem (1.1) has a unique maximal solution $u\in C\big([0, T^{\max}), \Sigma\big)$, in the sense that

Moreover, the mass and energy are conserved

In order to investigate the problem (1.1), we take the elliptic problem with no potential

The existence and uniqueness of the ground state hold for $0 < \varrho < \tilde{2}$ and $1 < p < p^c$. Specifically, the existence of the ground state is established in ^[20,21,22], while its uniqueness is derived in ^[23,24]. Below, we outline the main contributions of this paper.

1.2.   Main results

First, we give a global existence result, for small data in the intercritical regime.

Theorem 1.1. Let $N\geq 2$, $0 < \varrho < \tilde2$ and $p_{c} < p < p^c$. Let $\phi$ be the positive radial decreasing ground state of (1.2) and $0\neq u_0\in\Sigma$ satisfying

Then, there exists a unique global solution $u\in C(\mathbb{R}, \Sigma)$ to the Schrödinger problem (1.1), which satisfies

Remarks 1.1. 1. Taking $0 < \lambda\ll1$, one checks that $\lambda u_0$ satisfies the above condition. This gives an infinite family of global solutions.

2. When $B\to2$, equivalently $p\to p_c$, the above conditions read $\|u_0\| < \|\phi\|$.

3. The previous result complements ^[25] to the inhomogeneous regime, namely, $\varrho\neq0$.

In order to prove the finite-time blow-up of solutions, one needs the next variance identities.

Proposition 1.2. Let $N\geq2$, $0 < \varrho < \tilde2$ and $1 < p < p^c$. Let $u\in C\big([0, T];\Sigma\big)$ be a local solution to the problem (1.1). Then, for all $t\in[0, T]$, holds

Moreover, if $xu_0\in L^2$, then

Remarks 1.2. 1. In order to use (1.5), we need that the data belongs to the set

2. The previous result is proved in the appendix.

Using the variance-type identities in Proposition 1.2, we present the following blow-up result.

Proposition 1.3. Let $N\geq2$, $0 < \varrho < \tilde2$ and $1 < p < p^c$. Let $u\in C\big([0, T), \Sigma\big)$ be a local solution to the problem (1.1). Then, $u$ is non-global if

1. $p\geq p_c$ and $u_0\in\Sigma'$ satisfies one of the following:

(a) $E(u_0) < 0$;

(b) $E(u_0) = 0$ and $\Im\int_{ \mathbb{R}^N}\bar u_0\big(x\cdot\nabla u_0\big)\, dx < 0$.

2. $k = 1$, $0 < \varrho < \frac{2}{N-1}$, $p\geq 1+\frac{4}{N-k}$ and one of the following holds:

(a) $E(u_0) < 0$;

(b) $E(u_0) = 0$ and $\sum_{j\notin J}\Im\int_{ \mathbb{R}^N}\bar u_0\big(x_j\cdot\nabla u_0\big)\, dx < 0$.

Remarks 1.3. 1. The identity (1.5) reads by use of the energy

So, by (1.7), since $p\geq p_c$ reads $B\geq2$, the first point of Proposition 1.3 follows by time integration.

2. The identity (1.4) reads by use of the energy

So, by (1.8), since $p\geq 1+\frac{4}{N-k}$, the second point of Proposition 1.3 follows by time integration.

3. The assumption $k = 1$ in the second case is because $1+\frac{4}{N-k}\leq p < p^c$.

The ground state of (1.2) gives a threshold of global existence versus finite-time blow-up of mass-critical solutions to (1.1).

Theorem 1.2. Let $N\geq 2$, $0 < \varrho < \tilde2$ and $p = p_c$. Let $u\in C\big([0, T^{\max}), \Sigma\big)$ be a maximal solution to (1.1) and $\phi$ be the positive radial solution to (1.2). Then,

1. ${T^{\max}} < \infty$ if $\|u_0\| > \|\phi\|$ and $u_0\in \Sigma'$;

2. ${T^{\max}} = \infty$ if $\|u_0\| < \|\phi\|$.

In the mass-critical regime, we give a mass-concentration result of non-global solutions.

Theorem 1.3. Let $N\geq2$, $0 < \varrho < \tilde2$ and $p = p_c$. Let $u\in C\big([0, T^{\max}), \Sigma'\big)$ a non-global solution to (1.1) and a positive real function $\mu: = \mu(t)$ such that

Thus, there exists $x(t)\in \mathbb{R}^N$ satisfying

where $\phi$ is the ground state to (1.2).

Remarks 1.4. 1. Thanks to the identity (1.7), the concentration does not occur in the potential quantity, namely $\limsup_{t\to T^{\max }}\|x_ju(t)\|^2 < \infty,$ for any ${j\notin J}$.

2. The assumption $u_0\in\Sigma'$ is imposed because the use of (1.8) needs $p\geq1+\frac{4}{N-1}$, which fails for $p = p_c$.

3. The mass concentration (1.10) implies in particular that the solution has no $L^2$ limit when $t\to T^{\max}$.

Eventually, we study the lower bound for the mass-critical blow-up rate.

Theorem 1.4. Let $N\geq 3$, $0 < \varrho < \tilde2$, $p = p_c$ and $\phi$ be the radial positive solution to (1.2). Let $u\in C\big([0, T^{\max}), \Sigma\big)$ be a maximal non-global solution to (1.1) satisfying $\|u_0\| = \|\phi\|$. Then,

Remarks 1.5. 1. In the case $p = p_c$, in order to study the non-global solutions to (1.1), we use the associated ground state without any potential, namely the solution to (1.2).

2. The restriction on space dimensions $N > 2$ is needed when using Hardy estimate.

3. In the standard NLS case, namely without partial confinement, a classical scaling argument gives $\|\nabla u(t)\|\gtrsim \frac1{\sqrt{T^{\max}-t}}$, which is better than the lower bound (1.11).

In the next sub-section, we gather some standard estimates.

1.3.   Useful estimates

The next compactness result [26, Theorem 1.3] is adapted to the analysis of the blow-up phenomenon of Schrödinger equations.

Lemma 1.1. Let $N\geq2$, $0 < \varrho < 2$, $m, \, M > 0$ and a sequence of $H^1$ satisfying

Then, there exist $V \in H^1$ and a sequence $(x_n)$ in $\mathbb{R}^N$ such that up to a sub-sequence, one has

where $\phi$ is a ground state to (1.2).

The following Gagliardo-Nirenberg inequality ^[27,28] will be useful.

Proposition 1.4. Let $N\geq1$, $0 < \varrho < \min\{2, N\}$ and $1 < p < p^c$. Then, for all $f\in H^1$,

where $\phi$ is a the ground state solution to (1.2). Moreover, we have the Pohozaev type identities

Finally, one gives an elementary useful result ^[29].

Lemma 1.2. Let an open interval $I\subset \mathbb{R}$, $t_0\in I$, $\theta > 1$, $a, z > 0$ and $g\in C(I, \mathbb{R}_+)$. Let the real function defined on $\mathbb{R}_+$ by $f(x): = a-x+zx^\theta$, $x_*: = (z\theta)^{-\frac1{\theta-1}}$ and ${z_*}: = \frac{\theta-1}{\theta}x_*$. Then,

provided that

Now, let us establish the main results.

2.   Global well-posedness

In this section, we prove Theorem 1.1. Let us define the quantities

where $t\in[0, T^{\max})$ and $\mathtt{K}_{opt}$ is given in Proposition 1.4. Let also the real function defined on $(0, \infty)$, by

With the conservation laws, we write

So, (2.5) via Proposition 1.4 and the mass conservation law implies that

By (2.6), the real function $f:x\mapsto a-x+zx^{\frac B2}$ satisfies $f\Big(g(t)\Big) > 0$, for any $t < T^{\max}$. Now, the assumption (1.3) reads $\|u_0\|\leq\Big(\frac{B-2}A\Big)^{\frac{B-2}{2A}}\|\phi\|^{\frac{p-1}A}a^{-\frac{B-2}{2A}},$ rewritten as

Let us keep the notations of Lemma 1.2, namely

Taking into account of Proposition 1.4, yields

So, (2.7) and (2.11), imply that $a\leq {z_*} < x_*$. Applying Lemma 1.2, it follows that $\sup_{t\in[0, {T^{\max}})}g(t) < x_*$. Then, $u$ is global, namely ${T^{\max}} = \infty.$ Moreover, the energy reads via Proposition 1.4,

So, by (2.12), we write

Since $g < x_*$, we get by (2.13),

Finally, (2.14) implies that

The proof of Theorem 1.1 is closed by (2.15).

3.   Global/ non-global mass-critical solutions

This section proves Theorem 1.2. So, we fix $p = p_c$ and taking account of Proposition 1.4, we denote the quantities

Thus, by Proposition 1.4, we write

By (3.2), if $\frac{\|u_0\|}{\|\phi\|} < 1$, it follows that ${T^{\max}} = \infty$.

Now, we take for $\lambda, \mu > 0$ the scaling $u_0: = \lambda\phi(\frac\cdot{\mu})$ and we compute

Let us pick $0 < \varepsilon\ll1$ and

Taking account of the Pohozaev identities, namely (1.14),

we write by (3.3) to (3.6),

Thus, by (3.9) via (3.7) and (3.8), we write

The proof is achieved via Remark s 1.3.

4.   Mass-critical concentration

This section proves Theorem 1.3. Let us pick the sequences

Thus, by (3.3) and (3.6), we write

Moreover, by (4.2) because $p = p_c$, we have

Applying (1.7), it follows that $\partial_t^2\Big(\|xu(t)\|^2\Big)\leq 8E(u_0)$, which implies that

So, (4.5) via (4.6) and the fact that $\lambda_n$ vanishes at infinity, implies that

Applying Lemma 1.1, with $\|\nabla\phi\|^2 = M$ and $\frac{1+p_c}2\|\nabla\phi\|^2 = m^{1+p_c},$ there exist $x_n\in \mathbb{R}^N$ and $V\in H^1$ such that $\|\phi\|\leq \|V\|$ and

Thus, for any real number $R > 0$, yields

Now, since

taking $n\gg1$, by (4.9), we write

Then,

With a continuity argument, there exists $x(t)\in \mathbb{R}^N$ satisfying

This concludes the proof of Theorem 1.3.

5.   Blow-up rate

This section proves Theorem 1.4.

5.1.   Weak convergence

We start with the next auxiliary result.

Proposition 5.1. Let $N\geq2$, $p = p_c$ and $\phi$ a ground state of (1.2). Let $u\in C\big([0, T^{\max}), \Sigma'\big)$ be a blowing-up solution to (1.1) satisfying $\|u_0\| = \|\phi\|$. Then, there exists $x_0\in \mathbb{R}^N$ such that

in the sense of distribution.

Proof. By Theorem 1.3, namely (1.10), we have for any $R > 0$,

So, (5.1) via the identity $\|u_0\| = \|u(t)\| = \|\phi\|$ implies that for any $R > 0$,

Now, we take $\psi\in C_0^\infty(\mathbb{R}^N)$, $\lambda_n\rightarrow0$ and $t_n\to T^{\max}$, when $n\to\infty$. We define $w(t): = u(t, \cdot+x(t))$ and $z_n: = \lambda_n^\frac N2w_n(\lambda_n\cdot)$, so by the dominated convergence theorem, we get

Thus, (5.3) implies that in the sense of distribution, when $t\to T^{\max}$,

Now, for a real-valued function $\theta(x)$, we compute

Hence, by (5.5), we get

Moreover, by Proposition 1.4 for any $\tau\geq0$,

Now, (5.7) and (5.6) give a negative discriminant of the polynomial $\tau\mapsto H(ue^{i\tau{\theta}})$, namely

Moreover, (1.1) gives for any $1\leq j\leq N$,

since $H(u)\leq E(u_0)$, an integration by parts via (5.8), (5.9) and the mass conservation law, implies that

Taking $t_n\to T^{\max}$, with (5.10) via the Cauchy criteria, it follows that

So, (5.11) implies that the next limit exists

Moreover, since

keeping in mind (1.5), it follows that

Furthermore, using (5.4), via the equality

we write for $R\gg1$ and $t\to T^{\max}$,

Additionally, by Hölder estimate via (5.10), we have

Hence, by (5.15) and (5.16), it follows that

Thus, by (5.12) and (5.17), we write when $t\to T^{\max}$,

Finally, with (5.4) via (5.18), yields when $t\to T^{\max}$,

The proof of Proposition 5.1 is achieved by (5.19).

5.2.   Proof of Theorem 1.4

Let us take a nonnegative smooth radial function denoted by $\Theta\in C_0^\infty(\mathbb{R}^N)$ satisfying

Using the above function, we define, for $R > 0$ and $x^*$ from Proposition 5.1,

We compute using (1.1) via (5.20) and (5.8),

We integrate in time the identity (5.23) on $[t, t_n]$, where $t_n\rightarrow T^{\max}$, to get via (5.19),

We rewrite (5.24) as follows

Letting $R\to\infty$ in (5.25), yields

Using (5.26) via Hölder and Hardy estimates, we write

We collect (5.26) and (5.27) to get

Finally, (5.28) via the mass conservation law gives (1.11). The proof of Theorem 1.4 is achieved.

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A.   Appendix: Variance identity

Let us give a proof of the first variance identity in Proposition 1.2. The second identity follows similarly. Let a local solution to (1.1) denoted by ${u}\in C\big([0, T^{\max}), \Sigma\big)$ and the real function

Multiplying the equation (1.1) by $2u$ and examining the imaginary parts, we get

We denote by $a(x): = \sum_{j\notin J}|x_j|^2$, $b(x): = \sum_{j\in J}|x_j|^2$. By (A.1) and (A.2), we compute using the convention of sum to repeated index

Denoting the source term by $N: = |x|^{-\varrho}|u|^{p-1}u$ and using the equation (1.1), we write

Using the identity

it follows that

Moreover,

Furthermore

Using integration by parts, we get

Collecting (A.8) and (A.9), we have

Finally, plugging (A.10), (A.7) and (A.6) in (A.3), we get

This proves (1.4). The proof of (1.5) follows similarly by taking account of changing (A.7).

Author contributions

The first author performed the analysis and collected the data. The second author wrote the paper and supervised the work. Both authors investigated equally the paper.

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025)

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Use of Generative-AI tools declaration

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