Sharp conditions for a class of nonlinear Schrödinger equations

1.   Introduction

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equation

where $u(t, x) : \mathbb{R}\times\mathbb{R}^2 \rightarrow \mathbb{C}$, $g_1\in C^1(\mathbb{R}, \mathbb{R}_+)$ is a positive real function satisfying $g_1(0) = 0$, and

We assume $g(u)$ satisfies the following conditions:

Recently, the qualitative research on the nonlinear fourth-order Schrödinger equations has been widely performed, and the corresponding results have greatly developed the mathematical theory of Schrödinger equations (see for instance ^[1,2,3] and the references therein). Here we restrict our attention to the nonlinear second-order Schrödinger equations. Monomial semilinear Schrödinger equation

is called defocusing if $\mu = -1$ and focusing if $\mu = 1$. The solutions of (1.2) satisfy conservation of mass

and Hamiltonian

For $d = 2$, when $p > 1$ the Cauchy problem for nonlinear Schrödinger equation (1.2) is energy subcritical ^[4]. As is well known, the problems with exponential nonlinear terms have lots of applications, for instance the self-trapped laser beams in plasma ^[5]. Cazenave ^[6] considered the Cauchy problem for the Schrödinger equation

with the decreasing exponential nonlinear term $F\left(u\right)$, and showed global well-posedness and scattering. Generally, the problems with increasing exponential nonlinear terms are more complicated because there are no a priori $L^\infty$-estimates on the nonlinear terms. Furthermore, in view of its relationship with the critical Moser-Trudinger inequality, the two-dimensional case is interesting (see ^[7,8]). For the higher-dimensional case, we refer the readers to ^{[3,9,10,11,12,13]} and the references therein.

Later on, Colliander et al. ^[14] considered the Cauchy problem for the Schrödinger equation

They obtained global well-posedness under the situation that the initial data $u_0$ satisfies

and an instability when $H(u_0) > 1$. Saanouni ^[4] used the Strichartz estimate and some embedding inequalities to get the global existence result of the Cauchy problem for semilinear Schrödinger equation

in the subcritical case

In critical case

he got the blow-up result for the above equation under some assumptions. However, the sharp conditions for global existence and blow-up of solutions of the problem is still unsolved. In the present paper, we aim to consider this by the concavity arguments and the potential well theory (see for instance ^{[3,9,10,11,12,13,15,16,17,18,19,20,21,22,23]} and the references therein).

The outline of our paper is as follows. In Section 2, we show a few propositions and lemmas. Moreover, we introduce some functionals and invariant manifolds. In Section 3, we provide a sharp condition for global existence and blow-up of solutions of problem (1.1).

In this paper, we use $\|\cdot\|_{H^1}$ to stand for the norm of $H^1(\mathbb{R}^2)$ and $\|\cdot\|$ of $L^2(\mathbb{R}^2)$. For simplicity, hereafter, we will denote $\int_{\mathbb{R}^2}\cdot {\rm d}x$ by $\int\cdot$.

2.   Preliminaries

Regarding problem (1.1), we define the energy space in the course of nature by

with the inner product

where $\bar{v}$ denotes the conjugate function of $v$.

Proposition 2.1 (^[24]). Let $\varphi_0\in H$. Then the Cauchy problem (1.1) has a unique solution $u\in C([0, T); H)$, where $T\leq\infty$ is the maximal existence time of the solution. Moreover, we have alternative: $T = \infty$, or $T < \infty$ and

The solution $u$ satisfies

and

Lemma 2.2 (^[20]). Let $g(u)$ satisfy (${\rm H}$). Then

and the equality holds only for $u = 0$.

From ^[24] we have the following lemma.

Lemma 2.3. Let $u$ be the solution of the problem (1.1) with $u_0\in H$. For $J(t): = \int|x|^2|u|^2$, we have

Next, for $\varphi\in H$, we define

and

When $\varphi$ is the solution of problem (1.1) with $\varphi_0\in H$, there holds

Now we consider a constrained variational problem

where

Lemma 2.4. If $\varphi\in M$, then $d > 0$.

Proof. By (${\rm H}$), (2.2) and (2.3), we have

and

Furthermore, combining with (2.6) and (2.5), we can obtain $d > 0$.

Lemma 2.5. Let $\varphi\in H$. Put $\varphi_{\lambda}(x) = \lambda\varphi(x)$ for $\lambda > 0$, then there exists a unique constant $\mu > 0$ (depending on $\varphi$) such that $I(\varphi_\mu) = 0$, $I(\varphi_\lambda) > 0$ for any $0 < \lambda < \mu$, and $I(\varphi_\lambda) < 0$ for any $\lambda > \mu$. Furthermore, $P(\varphi_\mu)\geq P(\varphi_\lambda)$ for any $\lambda > 0$.

Proof. From (2.2) and (2.3), we have

and

It is easy to see that there exists a unique constant $\mu > 0$ (depending on $\varphi$) such that $I(\varphi_\mu) = 0$,

and

Combining

and Lemma 2.2, we get

Integrating (2.7) with respect to $x$ in $\mathbb{R}^2$ and dividing its both sides by $\lambda^2$, we derive

which, together with

yields

Hence

Theorem 2.6. Define

then $V$ is an invariant manifold of (1.1), that is, if $u_0\in V$, then the solution $u$ of problem (1.1) also satisfies $u \in V$ for all $t\in [0, T)$.

Proof. By Proposition 2.1, problem (1.1) admits a unique solution $u\in C([0, T); H)$ with $T\leq\infty$. As (2.4) shows

we conclude that $P(u_0) < d$ implies $P(u) < d$ for all $t\in[0, T)$.

Next, we demonstrate $I(u) < 0$ for all $t\in [0, T)$. If it is not true, then from the continuity of $I\left(u(t)\right)$ in $t$, there exists a $t_1\in[0, T)$ such that $I(u(t_1)) = 0$. By (2.2), (2.3) and

we have $u(t_1) \neq0$. If it is not true, then $P(u(t_1)) = 0$, which contradicts $P(u(t_1)) > 0$. From (2.5) we get $P(u(t_1))\geq d$. This contradicts $P(u) < d$ for all $t\in[0, T)$. Therefore, $I(u) < 0$ for all $t \in [0, T)$, i.e., $u \in V$ for all $t\in [0, T)$. So $V$ is an invariant manifold of problem (1.1).

By the same arguments as Theorem 2.6, we have the following theorem.

Theorem 2.7. Define

Then $W$ is an invariant manifold of problem (1.1).

3.   Sharp conditions

Theorem 3.1. If $u_0\in W$, then the solution $u$ of problem (1.1) globally exists on $t\in[0, \infty)$.

Proof. Theorem 2.7 shows that the solution $u$ of problem (1.1) satisfies $u\in W$ for all $t\in [0, T)$. Hence $P(u) < d$ and $I(u) > 0$. By (${\rm H}$), (2.2) and (2.3), we get

which gives

Therefore, by Proposition 2.1, (3.1) shows that $u$ globally exists.

Let $u_0 = 0$. Thanks to (2.1), we get $u = 0$, which shows that $u$ is the trivial solution of problem (1.1). The proof of Theorem 3.1 is completed.

By the similar arguments in ^[10], we have the following lemma.

Lemma 3.2. Let $\varphi\in H$ and $\mu > 0$ satisfy $I(\varphi_\mu) = 0$. Suppose that $\mu < 1$, then

Theorem 3.3. If $u_0\in V$, then the solution $u$ of problem (1.1) blows up in finite time.

Proof. Suppose that $T = \infty$. Since $u_0\in V$, we conclude from Theorem 2.6 that $u\in V$, i.e., $I(u) < 0$ for all $t\in[0, \infty)$. Thus

From Lemma 2.3 we get

Let $\mu > 0$ satisfy

From $I(u) < 0$ and Lemma 2.5 we obtain $\mu < 1$. Note that

From Lemma 3.2 we have

Let

and $\delta > 0$ be a constant independent of $t$. From (3.2) and (3.3) we obtain

where $c_0 > 0$ is a constant. Furthermore, we get

Hence there exists a $t_0\geq0$ such that $J'(t) < J'(0) < 0$ for all $t\in(t_0, \infty)$, and so

Since $I(u_0) < 0$ implies $J(0) > 0$, we conclude from (3.4) that there exists a $T_1 > 0$ such that $J(t) > 0$ for all $t\in[0, T_1)$ and

From (3.5) and

it follows that

This contradicts $T = \infty$. Thus

4.   Conclusions

It is clear that

Thus, by means of the location of the initial data, Theorems 3.1 and 3.3 provide a sharp condition for global existence and blow-up of solutions of problem (1.1), i.e., $u_0\in W$ vs $u_0\in V$.

The fractional Schrödinger equations may have a lot of interesting phenomena like the fractional version of other partial differential equations explored in ^[25,26], hence we shall focus on these models to investigate the corresponding sharp conditions.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 31920220062), the Science and Technology Plan Project of Gansu Province in China (Grant No. 21JR1RA200), the Talent Introduction Research Project of Northwest Minzu University (Grant No. xbmuyjrc2021008), the Innovation Team Project of Northwest Minzu University (Grant No. 1110130131), the First-Rate Discipline of Northwest Minzu University (Grant No. 2019XJYLZY-02), and the Key Laboratory of China's Ethnic Languages and Information Technology of Ministry of Education at Northwest Minzu University.

Conflict of interest

The authors declare there is no conflict of interest.