Asymptotic behavior for a Schr\"odinger equation with nonlinear subcritical dissipation

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear Schrödinger equation with subcritical dissipation, providing detailed decay rates and asymptotic profiles for large initial data when the nonlinearity is near the critical exponent.

Contribution

It offers a precise description of the asymptotic behavior and decay rates of solutions to a nonlinear Schrödinger equation with dissipation, especially for large initial data and near the critical nonlinearity.

Findings

Solutions exhibit specific decay rates in $L^2$ and $L^ Infty$ norms.

Asymptotic profiles of solutions are characterized.

Results hold for large initial data when $ ext{Re} \, ext{lambda} < 0$ and $ ext{alpha}$ is close to the critical exponent.

Abstract

We study the time-asymptotic behavior of solutions of the Schr\"odinger equation with nonlinear dissipation \begin{equation*} \partial _t u = i \Delta u + \lambda |u|^\alpha u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C}$, $\Re \lambda <0$ and $0<\alpha<\frac2N$. We give a precise description of the behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that $\alpha $ is sufficiently close to $\frac2N$.

Equations

$${\partial }_{t}u=i\Delta u+\lambda |u{|}^{\alpha }u$$

$$\{\begin{array}{l}{\partial }_{t}u=i\Delta u+\lambda |u{|}^{\alpha }u,\\ u(0,x)=u_0,\end{array}$$

$$\Re \lambda <0$$

$$(t\log t{)}^{\frac{N}{2}}\Vert u(t){\Vert }_{L^{\infty }}{⟶}_{t\to \infty }(\alpha |\Re \lambda |{)}^{-\frac{N}{2}}.$$

$$\frac{\alpha }{2\sqrt{\alpha +1}}|\Im \lambda |\le |\Re \lambda |,$$

$$\Vert u(t){\Vert }_{L^{\infty }}\lesssim t^{-\frac{1}{\alpha }},$$

$${k}>\frac{N}{2},\quad{n}>\max\Bigl{\{}\frac{N}{2}+1,\frac{N(N+1)}{4}\Bigr{\}},\quad 2{m}\geq{k}+{n}+1$$

$$J=2m+2+k+n.$$

$$\begin{array}{rl}\mathcal{X}=\{u\in H^J({\mathbb{R}}^N);& ⟨x{⟩}^n{D}^{\beta }u\in L^{\infty }({\mathbb{R}}^N)\text{ for }0\le |\beta |\le 2m,\\ ⟨x{⟩}^n{D}^{\beta }u\in L^2& ({\mathbb{R}}^N)\text{ for }2m+1\le |\beta |\le 2m+2+k,\\ ⟨x{⟩}^{J-|\beta |}{D}^{\beta }u& \in L^2({\mathbb{R}}^N)\text{ for }2m+2+k<|\beta |\le J\}\end{array}$$

$$\Vert u{\Vert }_{\mathcal{X}}=\sum _{j=0}^{2m}\underset{|\beta |=j}{\sup }\Vert ⟨x{⟩}^n{D}^{\beta }u{\Vert }_{L^{\infty }}+\sum _{\nu =0}^{k+1}\sum _{\mu =0}^n\sum _{|\beta |=\nu +\mu +2m+1}\Vert ⟨x{⟩}^{n-\mu }{D}^{\beta }u{\Vert }_{L^2}$$

$$⟨x⟩=(1+|x{|}^2{)}^{\frac{1}{2}}.$$

$$\underset{x\in {\mathbb{R}}^N}{\inf }⟨x{⟩}^n|v_0(x)|>0.$$

$$\|v_{0}\|_{{\mathcal{X}}}+\Big{(}\inf\limits_{x\in{\mathbb{R}}^{N}}\langle x\rangle^{n}|v_{0}(x)|\Big{)}^{-1}\leq K.$$

$$t^{\frac{1}{\alpha }-\frac{N}{2}}\Vert u(t)-z(t){\Vert }_{L^2}+t^{\frac{1}{\alpha }}\Vert u(t)-z(t){\Vert }_{L^{\infty }}\le C{t}^{-(\frac{N+1}{2}-\frac{1}{\alpha })},$$

$$z(t,x)=(1+bt)^{-\frac{N}{2}}e^{i\Theta(t,x)}\Psi\Bigl{(}t,\frac{x}{1+bt}\Bigr{)}\omega_{0}\Bigl{(}\frac{x}{1+bt}\Bigr{)}$$

$$\Theta(t,x)=\frac{b|x|^{2}}{4(1+bt)}-\frac{\Im\lambda}{\Re\lambda}\log\Bigl{(}\Psi\Bigl{(}t,\frac{x}{1+bt}\Bigr{)}\Bigr{)}$$

$$\Psi(t,y)=\Big{(}\frac{1+f_{0}(y)}{1+f_{0}(y)+\frac{2\alpha|\Re\lambda|}{b(2-N\alpha)}|v_{0}(y)|^{\alpha}[(1+bt)^{\frac{2-N\alpha}{2}}-1]}\Big{)}^{\frac{1}{\alpha}}.$$

$$|{\omega }_0{|}^{\alpha }=\frac{|v_0{|}^{\alpha }}{1+f_0},$$

$$t\Vert u(t){\Vert }_{L^{\infty }}^{\alpha }{⟶}_{t\uparrow \infty }\frac{2-N\alpha }{2\alpha |\Re \lambda |},$$

$$t^{-(\frac{1}{\alpha }-\frac{N}{2})(1-\frac{N}{2n})}\lesssim \Vert u(t){\Vert }_{L^2}\lesssim t^{-(\frac{1}{\alpha }-\frac{N}{2})(1-\frac{N}{2n})}$$

$$\underset{t\to \infty }{\mathrm{liminf}}{t}^{\frac{1}{\alpha }}\Vert u(t){\Vert }_{L^{\infty }}>0,t^{\frac{1}{\alpha }-\frac{N}{2}}\Vert u(t){\Vert }_{L^2}{⟶}_{t\to \infty }\infty .$$

$$v(t,x)=(1-bt)^{-\frac{N}{2}}u\Bigl{(}\frac{t}{1-bt},\frac{x}{1-bt}\Bigr{)}e^{-i\frac{b|x|^{2}}{4(1-bt)}}\quad t\geq 0,\ x\in{\mathbb{R}}^{N},$$

$$\{\begin{array}{l}{\partial }_{t}v=i\Delta v+\lambda (1-bt{)}^{-\frac{4-N\alpha }{2}}|v{|}^{\alpha }v,\\ v(0,x)=v_0.\end{array}$$

$$v(t)=e^{it\Delta }{v}_0+\lambda {\int }_0^t(1-bs{)}^{-\frac{4-N\alpha }{2}}{e}^{i(t-s)\Delta }|v(s){|}^{\alpha }v(s)ds$$

$${\int }_0^t(1-bs{)}^{-1-\nu }ds=\frac{1}{b\nu }[(1-bt{)}^{-\nu }-1]\le \frac{1}{b\nu }(1-bt{)}^{-\nu },$$

$$\Vert u(t){\Vert }_{L^2}^2+(-\Re \lambda ){\int }_0^t\Vert u(s){\Vert }_{L^{\alpha +2}}^{\alpha +2}=\Vert u_0{\Vert }_{L^2}^2.$$

$$\{\begin{array}{l}{\partial }_{t}v=i\Delta v+f,\\ v(0,x)=v_0,\end{array}$$

$$\Vert ⟨x{⟩}^n{D}^{\beta }v(t){\Vert }_{L^{\infty }}\le \Vert v_0{\Vert }_{\mathcal{X}}+C_1{\int }_0^t(\Vert v(s){\Vert }_{\mathcal{X}}+\Vert ⟨x{⟩}^n{D}^{\beta }f(s){\Vert }_{L^{\infty }})ds.$$

$$\Vert ⟨x{⟩}^{n-\mu }{D}^{\beta }v(t){\Vert }_{L^2}\le \Vert v_0{\Vert }_{\mathcal{X}}+C_1{\int }_0^t(\Vert v(s){\Vert }_{\mathcal{X}}+\Vert ⟨x{⟩}^{n-\mu }{D}^{\beta }f(s){\Vert }_{L^2})ds.$$

$$\Vert u{\Vert }_{1,\ell }=\underset{0\le |\beta |\le \ell }{\sup }\Vert ⟨\cdot {⟩}^n{D}^{\beta }u{\Vert }_{L^{\infty }}$$

Full text

[TABLE]

in ${\mathbb{R}}^{N}$, $N\geq 1$, where $\lambda\in{\mathbb{C}}$, $\Re\lambda<0$ and $0<\alpha<\frac{2}{N}$. We give a precise description of the behavior of the solutions (including decay rates in $L^{2}$ and $L^{\infty}$, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that $\alpha$ is sufficiently close to $\frac{2}{N}$.

ZH thanks NSFC 11671353,11401153, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010025, and CSC for their financial support; and the Laboratoire Jacques-Louis Lions for its kind hospitality

1. Introduction

In this paper, we consider the Schrödinger equation with nonlinear dissipation

[TABLE]

where $\lambda\in{\mathbb{C}}$ with

[TABLE]

and $0<\alpha<\frac{2}{N}$.

Equation (1.1) is itself a particular case of the more general complex Ginzburg-Landau equation on ${\mathbb{R}}^{N}$: $u_{t}=e^{i\theta}\Delta u+z|u|^{\alpha}u$, where $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$, $z\in{\mathbb{C}}$ and $\alpha>0$, which is a generic modulation equation describing the nonlinear evolution of patterns at near-critical conditions (see e.g. [20, 7, 16]).

Equation (1.1) is mass-subcritical, and is globally well-posed in $L^{2}({\mathbb{R}}^{N})$ and $H^{1}({\mathbb{R}}^{N})$. See Proposotion 2.1 below.

Concerning the large-time asymptotic behavior of the solutions of (1.1) under assumption (1.2), $\alpha=\frac{2}{N}$ is a limiting case. Indeed, if $\alpha>\frac{2}{N}$, $\lambda\in{\mathbb{C}}$, then a large set of initial values produces solutions that scatter as $t\to\infty$, i.e. that are asymptotic to a solution of the free Schrödinger equation. (See [21, 9, 10, 6, 8, 17, 1, 4].)

If $\alpha\leq\frac{2}{N}$, then in many cases solutions are known to decay faster than the solutions of the free Schrödinger equation. If $\alpha=\frac{2}{N}$, then for a large class of initial values, the solutions of (1.1) can be described by an asymptotic formula, and have the decay rate $(t\log t)^{-\frac{N}{2}}$. See [19, 15, 5]. In addition, for some solutions,

[TABLE]

See [5].

In the one-dimensional case $N=1$, if $\alpha<2$ is sufficiently close to $2$ and

[TABLE]

then the large-time asymptotic behavior of solutions can be described for any initial data in $H^{1}({\mathbb{R}})\cap L^{2}({\mathbb{R}},|x|^{2}dx)$, and the solutions satisfy

[TABLE]

see [15]. In addition, in any space dimension $N\geq 1$, under assumption (1.3) and for $\alpha<\frac{2}{N}$ sufficiently close to $\frac{2}{N}$, all solutions with initial value in $H^{1}({\mathbb{R}}^{N})\cap L^{2}({\mathbb{R}}^{N},|x|^{2}dx)$ satisfy $\|u(t)\|_{L^{2}}\lesssim t^{-(\frac{1}{\alpha}-\frac{N}{2})q}$ for all $q<\frac{2}{N+2}$, $q\leq\frac{1}{2}$. See [11].

In space dimensions $N=1,2,3$ without the condition (1.3), and for $\alpha<\frac{2}{N}$ sufficiently close to $\frac{2}{N}$, the upper estimate (1.4), as well as lower estimates, is established for sufficiently small initial data in a certain space. See [12].

Our purpose in this article is to complete the previous results for (1.1), and describe the large-time asymptotic behavior of the solutions for a class of arbitrarily large initial data. In order to state our result, we recall the definition of the space ${\mathcal{X}}$ introduced [4], which we use in a essential way. We consider three integers ${k},{m},{n}$ such that

[TABLE]

and we let

[TABLE]

We define the space ${\mathcal{X}}$ by

[TABLE]

and we equip ${\mathcal{X}}$ with the norm

[TABLE]

where

[TABLE]

In particular, $({\mathcal{X}},\|\cdot\|_{\mathcal{X}})$ is a Banach space and ${\mathcal{X}}\hookrightarrow H^{J}({\mathbb{R}}^{N})$.

Our main result is the following.

Theorem 1.1**.**

Let $\lambda\in{\mathbb{C}}$ satisfy (1.2), assume (1.5)-(1.6) and let ${\mathcal{X}}$ be defined by (1.7)-(1.8). Given any $K>1$, there exist $\frac{2}{N+1}<\alpha_{1}<\frac{2}{N}$ and $b_{1}>0$ with the following property. Let $\alpha_{1}\leq\alpha<\frac{2}{N}$. Suppose $u_{0}(x)=e^{i\frac{b|x|^{2}}{4}}v_{0}(x)$, where $b\geq b_{1}$ and $v_{0}\in{\mathcal{X}}$ satisfies

[TABLE]

and

[TABLE]

It follows that the corresponding solution $u\in C([0,\infty),H^{1}({\mathbb{R}}^{N}))$ of (1.1) belongs to $L^{\infty}((0,\infty)\times{\mathbb{R}}^{N})$. Moreover, there exist $f_{0},\omega_{0}\in L^{\infty}({\mathbb{R}}^{N})$, with $f_{0}$ real-valued and $\|f_{0}\|_{L^{\infty}}\leq\frac{1}{2}$, and $\langle\cdot\rangle^{n}\omega_{0}\in L^{\infty}({\mathbb{R}}^{N})$, such that

[TABLE]

where

[TABLE]

with

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

so that $\frac{2}{3}|v_{0}|^{\alpha}\leq|\omega_{0}|^{\alpha}\leq 2|v_{0}|^{\alpha}$. Furthermore,

[TABLE]

and

[TABLE]

as $t\to\infty$, where $n$ is given by (1.5).

Remark 1.2**.**

Here are some comments on Theorem 1.1.

1. (i)

We have $v_{0}\in{\mathcal{X}}\hookrightarrow L^{2}({\mathbb{R}}^{N})$, ${\mathcal{X}}\hookrightarrow H^{1}({\mathbb{R}}^{N})$, and ${\mathcal{X}}\hookrightarrow L^{2}({\mathbb{R}}^{N},|x|^{2}dx)$ (because $n>\frac{N}{2}+1$), so that $u_{0}\in H^{1}({\mathbb{R}}^{N})$. Therefore, the solution $u\in C([0,\infty),H^{1}({\mathbb{R}}^{N}))$ of (1.1) is well defined, see Proposotion 2.1. Moreover, $u$ is smoother than stated. Indeed, $u$ is given by the pseudo-conformal transformation (5.1) in terms of a solution $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ of equation (1.19). In particular, $u$ is a classical solution of (1.1) ($C^{1}$ in $t$ and $C^{2}$ in $x$). 2. (ii)

Theorem 1.1 is valid in any dimension $N\geq 1$ and for any $\lambda\in{\mathbb{C}}$ with $\Re\lambda<0$. In particular, we do not require assumption (1.3). The main restrictions are that $\alpha$ must be sufficiently close to $\frac{2}{N}$ and that the initial value must be bounded from below in the sense (1.9) and sufficiently oscillatory in the sense that $b$ must be sufficiently large. Moreover, how close $\alpha$ must be to $\frac{2}{N}$ depends on a certain bound on the initial value through (1.10). On the other hand, there is no restriction on the size of $u_{0}$. 3. (iii)

A typical initial value which is admissible in Theorem 1.1 is $v_{0}=c\langle\cdot\rangle^{-n}+\varphi$ with $c\in{\mathbb{C}}$, $c\not=0$, and $\varphi\in{\mathcal{S}}({\mathbb{R}}^{N})$, $|\varphi|\leq(|c|-\varepsilon)\langle\cdot\rangle^{-n}$, $\varepsilon>0$. Indeed, it is easy to check that $v_{0}\in{\mathcal{X}}$ and $v_{0}$ satisfies (1.9). Then $K$ must be chosen sufficiently large so that (1.10) holds and $\alpha$ sufficiently close to $\frac{2}{N}$. Note that any value of $n$ sufficiently large so that the second condition in (1.5) is satisfied, is admissible. 4. (iv)

The limit (1.16) gives the exact decay rate of $\|u(t)\|_{L^{\infty}}$. Note that this limit is independent of the initial value $u_{0}$. The reason for this is that (1.16) is equivalent to (4.5), and that the solutions of the ODE $z^{\prime}=\lambda(1-bt)^{-\frac{4-N\alpha}{2}}|z|^{\alpha}z$ satisfy $(1-bt)^{-\frac{2-N\alpha}{2}}|z(t)|^{\alpha}\to\frac{b(2-N\alpha)}{2\alpha|\Re\lambda|}$ as $t\uparrow\frac{1}{b}$, independently of the initial value $z(0)$. 5. (v)

It follows from (1.17) that $\|u(t)\|_{L^{2}}$ is equivalent to $t^{-(\frac{1}{\alpha}-\frac{N}{2})(1-\frac{N}{2n})}$. With respect to the results in [11], (1.17) gives the exact decay rate of $\|u(t)\|_{L^{2}}$. As opposed to the decay rate of $\|u(t)\|_{L^{\infty}}$, which is $t^{-\frac{1}{\alpha}}$ (hence independent of the solution), the decay rate of $\|u(t)\|_{L^{2}}$ does depend on the solution, through the parameter $n$ which can be chosen (as long as it is sufficiently large). 6. (vi)

It follows from (1.16) and (1.17) that

[TABLE]

Thus we see that the asymptotic behavior of $u(t)$ as $t\to\infty$ is described by the function $z(t)$ via the estimate (1.11). Note that the functions $f_{0}$ and $\Psi$ are both real-valued, and that $\frac{1}{2}\leq 1+f_{0}\leq\frac{3}{2}$ and $0<\Psi\leq 1$. The function $\Theta$ is also real-valued. If $\Im\lambda\leq 0$, then $\Theta>0$. If $\Im\lambda>0$, then $\Theta$ takes both positive and negative values.

Remark 1.3**.**

If $\Re\lambda>0$, then finite-time blowup occurs for equation (1.1), at least for $H^{1}$-subcritical powers $(N-2)\alpha<4$. See [3, 2]. Moreover, if $\alpha<\frac{2}{N}$, then all nontrivial solutions blow up in finite or infinite time, see [1]. Finite-time blowup also occurs if $\Re\lambda=0$, $\Im\lambda>0$, and $\alpha\geq\frac{4}{N}$, since in this case (1.1) is the focusing NLS. If $\Re\lambda<0$, $\alpha>\frac{4}{N}$ and condition (1.3) is not satisfied, then whether or not some solutions of (1.1) blow up in finite time seems to be an open question.

We apply the strategy of [4, 5] to prove Theorem 1.1. We require the non-vanishing condition (1.9), as well as strong decay and regularity of the the initial data to overcome the difficulty of non-smooth nonlinearity and derivative loss in their estimates. This is why the various conditions in the definition of the space ${\mathcal{X}}$ arise. The other main ingredient is the application of the pseudo-conformal transformation. Given any $b>0$, by the pseudo-conformal transformation

[TABLE]

equation (1.1) is equivalent to the nonautonomous equation

[TABLE]

Note that the assumption $\alpha\leq\frac{2}{N}$ implies that $(1-bt)^{-\frac{4-N\alpha}{2}}$ is not integrable at $1/b$. As in [5], we estimate the solution $v(t,x)$ by allowing a certain growth of the various components of the ${\mathcal{X}}$-norm of the solution, see (3.7)-(3.10). Using Duhamel’s formula for (1.19), i.e.

[TABLE]

and the elementary calculation

[TABLE]

we see that if $e^{i(t-s)\Delta}|v(s)|^{\alpha}v(s)$ is estimated in a certain norm by $(1-bs)^{-\mu}$, then $v(t)$ can be controlled in that norm by $(1-bs)^{-\mu+\frac{2-N\alpha}{2}}$. In the case $\alpha=\frac{2}{N}$, one obtains the same power $(1-bs)^{-\mu}$, and this can be used to close appropriate estimates. This is the strategy employed in [5]. In the present case $\alpha<\frac{2}{N}$, we observe that if $e^{i(t-s)\Delta}|v(s)|^{\alpha}v(s)$ is estimated in a certain norm by $(1-bs)^{-\mu+\frac{2-N\alpha}{2}}$, then $v(t)$ can be controlled in that norm by $(1-bs)^{-\mu}$. We obtain the extra decay by monitoring the decay of $|v(s)|$ (see Lemma 3.1). The price to be paid is that the constants that appear in the calculations not only depend on $1/b$, but also on $\frac{2-N\alpha}{|\Re\lambda|}$. Therefore, in order to close the estimates, we are led to require not only that $b$ is large, but also that $\alpha$ is close to $\frac{2}{N}$.

The rest of the paper is organized as follows. in section 2, we recall some estimates and a local well-posedness result in the space ${\mathcal{X}}$ for equation (1.19), taken from [4, 5]. The crucial estimate of the solutions is carried out in Section 3. Using these estimates, we describe in Section 4 the asymptotic behavior of the corresponding solutions of (1.19). Finally, we complete the proof of Theorem 1.1 in section 5, by applying the pseudo-confirmal transformation.

2. Preliminary

We recall some properties of equation (1.1) which will be useful in the next sections. We begin with a global well-posedness result.

Proposition 2.1**.**

Let $0<\alpha<\frac{4}{N}$ and let $\lambda\in{\mathbb{C}}$ satisfy $\Re\lambda\leq 0$. It follows that the Cauchy problem (1.1) is globally well-posed in $L^{2}({\mathbb{R}}^{N})$ and in $H^{1}({\mathbb{R}}^{N})$. More precisely, given any $u_{0}\in L^{2}({\mathbb{R}}^{N})$ there exists a solution $u\in C([0,\infty),L^{2}({\mathbb{R}}^{N}))\cap L^{\alpha+2}([0,\infty),L^{\alpha+2}({\mathbb{R}}^{N}))$ of (1.1). The solution is unique and depends continuously on $u_{0}$ in $C([0,T),L^{2}({\mathbb{R}}^{N}))\cap L^{\alpha+2}((0,T),L^{\alpha+2}({\mathbb{R}}^{N}))$ for every $T>0$. If, in addition, $u_{0}\in H^{1}({\mathbb{R}}^{N})$, then $u\in C([0,\infty),H^{1}({\mathbb{R}}^{N}))$.

Proof.

For the local theory (local existence, uniqueness, continuous dependence, regularity), see e.g. [13, 14]. For global existence, it is sufficient to estimate the $L^{2}$ norm of $u$. Multiplying (1.1) by $\overline{u}$, taking the real part and integrating by parts, we obtain

[TABLE]

(This argument is formal, but (2.1) can be proved by standard approximation arguments, see for instance [18].) It follows that $u$ is bounded in $L^{2}({\mathbb{R}}^{N})$. ∎

Next, we recall some estimates for the Schrödinger equation in the space ${\mathcal{X}}$.

Proposition 2.2** ([5], Propositon 2.1).**

Assume (1.5)-(1.6) and let ${\mathcal{X}}$ be defined by (1.7)-(1.8). There exists $C_{1}\geq 1$ such that if $T\geq 0,v_{0}\in{\mathcal{X}}$ and $f\in C([0,T],{\mathcal{X}})$, then for all $0\leq t\leq T$, the solution $v$ of

[TABLE]

satisfies the following estimates.

1. (i)

If $|\beta|\leq 2m$, then

[TABLE] 2. (ii)

If $|\beta|=\nu+\mu+2m+1\ with\ 0\leq\nu\leq k+1\ and\ 0\leq\mu\leq n$, then

[TABLE]

We now recall several estimates of the nonlinearity $|v|^{\alpha}v$. Given $\ell\in{\mathbb{N}}$, we set

[TABLE]

[TABLE]

and

[TABLE]

We have the following estimates of the nonlinearity.

Proposition 2.3** ([5], Proposition 3.1).**

Assume (1.5)-(1.6) and let ${\mathcal{X}}$ be defined by (1.7)-(1.8). Let $\alpha>0$ and suppose that, in addition to (1.5), $n\geq\frac{N}{2\alpha}$. It follows that there exists a constant $C_{2}\geq 1$ such that if $v\in{\mathcal{X}}$ satisfies

[TABLE]

for some $\eta>0$, then the following estimates hold.

1. (i)

If $0\leq|\beta|\leq 1$, then

[TABLE] 2. (ii)

If $2\leq|\beta|\leq 2m$, then

[TABLE] 3. (iii)

If $2m+1\leq|\beta|\leq 2m+2+k$, then

[TABLE] 4. (iv)

If $2m+3+k\leq|\beta|\leq J$, then

[TABLE]

Remark 2.4**.**

Estimates (2.9)–(2.11) are not exactly the estimates of [5, Proposition 3.1]. First, $1+\eta\|v\|_{1,\ell}$ is replaced by $\eta\|v\|_{1,\ell}$ (with $\ell=|\beta|-1$ in (2.9) and $\ell=2m$ in (2.10) and (2.11)). The two quantities are indeed equivalent, since by (2.7), $\eta\|v\|_{1,\ell}\geq 1$. Next, the term $(\eta\|v\|_{1,2m})^{2J+\alpha}\|v\|_{2,|\beta|-1}$ in [5, formula (3.9)] is replaced in formula (2.10) here by $\|v\|_{L^{\infty}}^{\alpha}(\eta\|v\|_{1,2m})^{2J}\|v\|_{2,|\beta|-1}$. This is in fact what the proof in [5] shows, see in particular [5, formula (3.20)]. Finally, the term $(\eta\|v\|_{1,2m})^{2J+\alpha}(\|v\|_{2,2m+2+k}+\|v\|_{3,|\beta|-1})$ in [5, formula (3.10)] is replaced in formula (2.11) here by $\|v\|_{L^{\infty}}^{\alpha}(\eta\|v\|_{1,2m})^{2J}(\|v\|_{2,2m+2+k}+\|v\|_{3,|\beta|-1})$. Again, this is what the proof in [5] shows, see in particular [5, formulas (3.24) and (3.25)]. The term $\|v\|_{L^{\infty}}^{\alpha}$ in these estimates is important in our proof of Proposition 3.2 below.

Finally, we recall the local well-posedness of  (1.19) in the space ${\mathcal{X}}$, see [4, Theorem 1] and [5, Proposition 4.1].

Proposition 2.5**.**

Assume (1.5)-(1.6) and let ${\mathcal{X}}$ be defined by (1.7)-(1.8). Let $\alpha>0$ and suppose that, in addition to (1.5), $n\geq\frac{N}{2\alpha}$. Let $\lambda\in{\mathbb{C}}$ and $b\geq 0$. If $v_{0}\in{\mathcal{X}}$ satisfies

[TABLE]

then there exist $0<T<\frac{1}{b}$ and a unique solution $v\in C([0,T],{\mathcal{X}})$ of  (1.19) satisfying

[TABLE]

Moreover, $v$ can be extended on a maximal existence interval $[0,T_{\mathrm{max}})$ with $0<T_{\mathrm{max}}\leq\frac{1}{b}$ to a solution $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$ satisfying (2.13) for all $0<T<T_{\mathrm{max}}$; and if $T_{\mathrm{max}}<\frac{1}{b}$, then

[TABLE]

3. Estimates for (1.19)

Throughout this section, we assume (1.5)-(1.6) and we let ${\mathcal{X}}$ be defined by (1.7)-(1.8). We derive estimates for certain solutions of (1.19). We first introduce several indices and seminorms. Let

[TABLE]

and set

[TABLE]

It follows that

[TABLE]

Moreover, it follows from (3.3) that

[TABLE]

We deduce from (3.5) and (3.2) that

[TABLE]

Given $0<T<\frac{1}{b}$ and $v\in C([0,T],{\mathcal{X}})$ satisfying (2.13), we define

[TABLE]

where the norms $\|\cdot\|_{l,j}$ are given by  (2.4)–(2.6), and we denote

[TABLE]

From these definitions, it is easy to verify that

[TABLE]

where the constant $C_{3}\geq 1$ is independent of $t\in[0,T]$.

Lemma 3.1**.**

Suppose $\Re\lambda<0$ and $\frac{3}{2N}\leq\alpha\leq\frac{2}{N}$. Let $K\geq 1$ and set

[TABLE]

Let $b>0$, let $v_{0}\in{\mathcal{X}}$ satisfy (2.12), and let $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$ be the solution of  (1.19) given by Proposition 2.5. If $v$ satisfies

[TABLE]

for some $0<T<T_{\mathrm{max}}$ and if $b\geq b_{0}$, then

[TABLE]

for all $0<t\leq T$.

Proof.

Multiplying (1.19) by $\overline{v}$, taking the real part and using that $|v|>0$ on $[0,T_{\mathrm{max}})\times{\mathbb{R}}^{N}$ by Proposition 2.5, we obtain

[TABLE]

where

[TABLE]

and so

[TABLE]

Integrating (3.21) in $t$, we obtain

[TABLE]

so that

[TABLE]

where

[TABLE]

It follows from the definitions of $\Psi_{T}$ and $L(t,x)$ that, for any $0\leq s\leq t\leq T$

[TABLE]

where in the last inequality we used

[TABLE]

by (3.3). Using $\sigma_{3}\leq\sigma_{J}\leq\frac{1}{2}$ (see (3.6)), we obtain

[TABLE]

We deduce from (3.25) and (3.16) that

[TABLE]

for $b\geq b_{0}$, $0\leq t\leq T$ and $x\in{\mathbb{R}}^{N}$. In particular, $1+f\geq 0$ and estimate (3.18) follows. ∎

Proposition 3.2**.**

Suppose $\Re\lambda<0$. Given $K\geq 1$, let $\alpha_{1}\in(\max\{\frac{3}{2N},\frac{2}{N+1}\},\frac{2}{N})$ be given by

[TABLE]

and let

[TABLE]

where $\sigma_{1}$ is given by (3.2), $C_{1}$ by Proposition 2.2, $C_{2}$ by Proposition 2.3, and $C_{3}$ by (3.13). If $v_{0}\in{\mathcal{X}}$ satisfies (1.10), then for every $\alpha\in[\alpha_{1},\frac{2}{N})$ and $b\geq b_{1}$, the corresponding solution $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$ of (1.19) given by Proposition 2.5 satisfies $T_{\mathrm{max}}=\frac{1}{b}$ and

[TABLE]

Proof.

We set

[TABLE]

Since $\Psi_{0}\leq K$ and $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$, we see that $0<T^{\star}\leq T_{\mathrm{max}}$. We claim that if $\alpha\in[\alpha_{1},\frac{2}{N})$ and $b\geq b_{1}$, then

[TABLE]

We note that, since $\alpha\geq\alpha_{1}\geq\frac{2}{N+1}$, the second condition in (1.5) implies that ${n}>\max\{\frac{N}{2}+1,\frac{N}{2\alpha}\}$, so that we may apply Propositions 2.3 and 2.5. Assuming (3.31), it follows from (3.13), (3.15) and (3.30) that for any $t\in[0,T_{\mathrm{max}})$

[TABLE]

If $T_{\mathrm{max}}<\frac{1}{b}$, then we deduce from (3.32) that

[TABLE]

which contradicts the blowup alternative (2.14). Therefore, we have $T^{\star}=T_{\mathrm{max}}=\frac{1}{b}$ and (3.29) follows.

Now we prove the claim (3.31). We assume by contradiction that

[TABLE]

then by the definition of $T^{\star}$, we have

[TABLE]

It follows from (3.13), (3.34) and (3.6) that

[TABLE]

Using also (1.10) and (3.28), we see that

[TABLE]

Next, we set

[TABLE]

so that (by definition of $\Phi_{4,T^{\star}}$)

[TABLE]

If $2\leq|\beta|\leq 2m$, we deduce from (3.37), (3.14) and (3.34) that

[TABLE]

since $(\sigma_{1}+\sigma_{|\beta|-1})2|\beta|+\sigma_{|\beta|-1}\leq(8m+1)\sigma_{|\beta|-1}=\sigma_{|\beta|}$ by (3.3). Similarly,

[TABLE]

where the last equality follows from the definition of $\sigma_{2m+1}$ by (3.3). As well, if $2m+2\leq|\beta|\leq J$ and $\ell\in\{2,3\}$, then

[TABLE]

where we used $2J(\sigma_{1}+\sigma_{2m})+\sigma_{|\beta|-1}\leq 4J\sigma_{2m}+\sigma_{|\beta|-1}=\sigma_{|\beta|}$ by (3.3).

Since $\|v\|_{L^{\infty}}\leq 4K$ by (3.14), it follows from (1.21) that, given any $\sigma>0$ and $0\leq t<\frac{1}{b}$,

[TABLE]

Let $0<t^{\prime}<\frac{1}{b}$ be defined by $(1-bt^{\prime})^{\frac{2-N\alpha}{2}}=\frac{1}{2}$, i.e. $(1-bt^{\prime})=2^{-\frac{2}{2-N\alpha}}$. It follows from the above inequality that if $0\leq t\leq t^{\prime}$, then

[TABLE]

Moreover, if $b\geq b_{0}$ and $t^{\prime}\leq t<T^{\star}$, then it follows from (3.18) and (1.21) that

[TABLE]

Using $\int_{0}^{t}=\int_{0}^{t^{\prime}}+\int_{t^{\prime}}^{t}$ if $t^{\prime}<t<T^{\star}$, we deduce from (3.42) and (3.43) that for all $b\geq b_{0}$ and all $0\leq t<T^{\star}$,

[TABLE]

Now, we are ready to estimate $\Psi_{T^{\star}}$ and the process is divided into four steps. We first estimate $\|\langle x\rangle^{n}v\|_{L^{\infty}}$. Since $\Re\lambda<0$, it follows from (3.19) and (3.20) that

[TABLE]

so that

[TABLE]

where we used (3.35) in the last inequality. Since $\langle x\rangle^{n}|v_{0}|\leq K$, we deduce that if $b\geq b_{1}$ with $b_{1}$ given by (3.28), then

[TABLE]

We next estimate $\|\langle x\rangle^{n}D^{\beta}v\|_{L^{\infty}}$ for $1\leq|\beta|\leq 2m$. Applying (2.2) and (3.36), we obtain

[TABLE]

Using (3.38), (2.8)-(2.9), (3.14) and (3.39) and setting $\kappa=0$ if $|\beta|=1$ and $\kappa(\beta)=1$ if $|\beta|\geq 2$, we see that

[TABLE]

Applying now (3.44) and using $\sigma_{|\beta|}\geq\sigma_{1}$, we deduce from (3.46)-(3.47) that

[TABLE]

It follows, using also (3.27), (3.28) and (3.45), that

[TABLE]

We next estimate similarly $\|\langle x\rangle^{n}D^{\beta}v\|_{L^{2}}$ for $2m+1\leq|\beta|\leq 2m+2+k$. It follows from (2.3) (with $\mu=0$) and (3.36) that

[TABLE]

Using (2.10), (3.34), (3.40) and (3.41), we see that

[TABLE]

so that

[TABLE]

Applying (1.21) and (3.44) to estimate the integrals, we obtain

[TABLE]

Using $\sigma_{1}\leq\sigma_{2m+1}\leq\sigma_{|\beta|}$, it follows that

[TABLE]

Using also (3.27) and (3.28), we conclude that

[TABLE]

Now we estimate $\|\langle x\rangle^{J-|\beta|}D^{\beta}v\|_{L^{2}}$ for $2m+k+3\leq|\beta|\leq J$. It follows from (2.3) (with $\mu=n+|\beta|-J$, $\nu=k+1$) and (3.36) that

[TABLE]

Using (2.11), (3.34), (3.40) and (3.41), we see that

[TABLE]

so that

[TABLE]

The right-hand side of (3.51) is similar to the right-hand side of (3.49), and we conclude as above that

[TABLE]

Finally, we estimate $\Phi_{4,T^{\star}}$, we we set

[TABLE]

Multiplying (3.21) by $\langle x\rangle^{-n\alpha}$ and integrating in $t$, we obtain

[TABLE]

Applying (3.34), we see that $\langle x\rangle^{n}|L|\leq\langle x\rangle^{n}|\Delta v|\leq 4K(1-bs)^{-\sigma_{2}}$, and $|w|^{-\alpha-1}\leq(4K)^{\alpha+1}(1-bs)^{-(\alpha+1)\sigma_{1}}$. Since $(\alpha+1)\sigma_{1}+\sigma_{2}\leq\sigma_{3}$ by (3.3), we deduce that

[TABLE]

Since $-\frac{4-N\alpha}{2}=-1-\alpha\sigma_{1}+(\alpha\sigma_{1}-\frac{2-N\alpha}{2})$, and since by (3.27) $\alpha\sigma_{1}\geq\frac{2-N\alpha}{2}$, we see that $-\frac{4-N\alpha}{2}\geq-1-\alpha\sigma_{1}$; and so, using (1.21) and $\sigma_{3}\leq\sigma_{J}\leq\frac{1}{2}$,

[TABLE]

It follows that

[TABLE]

Using (3.28), we deduce that for $b\geq b_{1}$,

[TABLE]

since $\alpha\geq\frac{1}{N}$. Estimates (3.48), (3.50), (3.52) and (3.53) yield $\Psi_{T^{\star}}\leq 3K$, which leads to a contradiction with (3.34). This completes the proof. ∎

4. Asymptotics for (1.19)

Throughout this section, we assume (1.5)-(1.6) and we let ${\mathcal{X}}$ be defined by (1.7)-(1.8). We describe the asymptotic behavior as $t\to\frac{1}{b}$ of the solutions of (1.19) given by Proposition 3.2. More precisely, we have the following result.

Proposition 4.1**.**

Suppose $\Re\lambda<0$. Let $K\geq 1$, let $\alpha_{1}\in(0,\frac{2}{N})$ be given by (3.27) and let $b_{1}>0$ be given by (3.28). Let $v_{0}\in{\mathcal{X}}$ satisfy (1.10), and let $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ be the solution of (1.19) given by Proposition 3.2. There exist $f_{0},\omega_{0}\in L^{\infty}({\mathbb{R}}^{N})$, with $f_{0}$ real-valued, $\|f_{0}\|_{L^{\infty}}\leq\frac{1}{2}$, and $\langle\cdot\rangle^{n}\omega_{0}\in L^{\infty}({\mathbb{R}}^{N})$, such that

[TABLE]

for all $t\in[0,\frac{1}{b})$, where

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

so that $\frac{2}{3}|v_{0}|^{\alpha}\leq|\omega_{0}|^{\alpha}\leq 2|v_{0}|^{\alpha}$. In addition,

[TABLE]

and

[TABLE]

as $t\to\frac{1}{b}$, where $n$ is given by (1.5).

Proof.

We let $f$ be defined by (3.23). It follows from (3.24) that $f(t,\cdot)$ is convergent in $L^{\infty}({\mathbb{R}}^{N})$ as $t\uparrow\frac{1}{b}$. Then $f$ can be extended to a continuous function $[0,\frac{1}{b}]\rightarrow L^{\infty}({\mathbb{R}}^{N})$ and we set

[TABLE]

By using (3.24), (3.25), (3.16) and $\sigma_{3}\leq\sigma_{J}\leq\frac{1}{2}$ (see (3.6)), we have for all $0\leq t\leq\frac{1}{b}$

[TABLE]

In particular, $1+f_{0}>0$, so that by (4.2),

[TABLE]

for all $0\leq t<\frac{1}{b}$ and $x\in{\mathbb{R}}^{N}$. Moreover, it follows from (4.8) that

[TABLE]

for all $0\leq t<\frac{1}{b}.$ We set

[TABLE]

It follows from (1.10) and (4.10) that

[TABLE]

and we deduce from (3.22), (4.7) and (4.10) that

[TABLE]

for all $0\leq t<\frac{1}{b}$. Next, we introduce the decomposition

[TABLE]

where $\psi(t,x)$ and $\theta(t,x)$ are defined by (4.2) and (4.3) respectively. Differentiating (4.13) with respect to $t$, we obtain

[TABLE]

On the other hand, it follows easily from (4.2), (4.3) and (4.11) that

[TABLE]

Therefore, we deduce from (4.14), (4.13) and (1.19) that

[TABLE]

Next, it follows from (4.2) and the property $1+f_{0}\geq\frac{1}{2}$ that

[TABLE]

Moreover, we deduce from (3.18) that if $t^{\prime}\leq t<\frac{1}{b}$ where $t^{\prime}\in(0,\frac{1}{b})$ is defined by $(1-bt^{\prime})^{\frac{2-N\alpha}{2}}=\frac{1}{2}$, then

[TABLE]

hence

[TABLE]

Therefore, it follows from (4.16) that $\psi(t,x)^{-\alpha}\leq 1+2|v_{0}(x)|^{\alpha}|v(t,x)|^{-\alpha}$. Applying (3.15), (1.10) and (3.29), we conclude that

[TABLE]

for $t^{\prime}\leq t<\frac{1}{b}$. It follows from (4.15) and (4.17) that

[TABLE]

Since $\|\langle\cdot\rangle^{n}\Delta v\|_{L^{\infty}}\leq 4K(1-bt)^{-\sigma_{2}}$ and $\|\langle\cdot\rangle^{n}v\|_{L^{\infty}}\leq 4K$ by (3.29), we deduce using (4.12) that

[TABLE]

where we used $\sigma_{1}+\sigma_{2}\leq\sigma_{1}+\frac{2-N\alpha}{2}+\sigma_{3}\leq\sigma_{J}\leq\frac{1}{2}$ by (3.3) and (3.6). It follows from (4.18) that if $t^{\prime}\leq t<s<\frac{1}{b}$, then

[TABLE]

so that there exists $\omega_{0}$ such that $\langle x\rangle^{n}\omega_{0}\in L^{\infty}({\mathbb{R}}^{N})$ and

[TABLE]

for all $t^{\prime}\leq t<\frac{1}{b}$. Using (4.13), (4.9) and (4.19), we obtain

[TABLE]

which proves (4.1).

Next, we prove (4.4). It follows from (4.1) (recall that $0\leq\psi\leq 1$) that

[TABLE]

Using the elementary inequalities $|x^{\alpha}-y^{\alpha}|\leq|x-y|^{\alpha}$ if $\alpha\leq 1$ and $|x^{\alpha}-y^{\alpha}|\leq\alpha(x^{\alpha-1}+y^{\alpha-1)}|x-y|$ if $\alpha\geq 1$, and the boundedness of $\|\langle\cdot\rangle^{n}v\|_{L^{\infty}}$, we deduce that

[TABLE]

Moreover, it follows from (4.12) and $\sigma_{3}\leq\frac{1}{2}$ that

[TABLE]

Thus we see that

[TABLE]

where $\rho=\min\{\alpha,1\}$. Using the explicit expressions (4.2) and (4.11), we obtain

[TABLE]

For $\frac{1}{2b}\leq t<\frac{1}{b}$, we have

[TABLE]

so that

[TABLE]

since $\alpha>\min\{\frac{1}{N},\frac{2}{N+1}\}$ by (3.27), we see that $\frac{\rho}{2}-\frac{2-N\alpha}{2}>0$. Letting $t\uparrow\frac{1}{b}$ in the above inequality, we obtain (4.4).

Now, we prove (4.5). Set

[TABLE]

It follows from (4.2) and (4.4) that

[TABLE]

Since $1+f_{0}\geq 0$ by (4.8), we obtain

[TABLE]

so that

[TABLE]

Moreover, $1+f_{0}\leq 2$, so that

[TABLE]

Since $|v_{0}(0)|>0$ by (1.10), we deduce that

[TABLE]

Thus we see that $\|Z(t)\|_{L^{\infty}}\to\frac{b(2-N\alpha)}{2\alpha|\Re\lambda|}$ as $t\uparrow\frac{1}{b}$. On the other hand, it follows from (4.12) and (4.4) that

[TABLE]

Since $\frac{N\alpha}{2}\geq\frac{1}{2}>\sigma_{3}$, (4.5) follows.

Finally, we prove (4.6). It follows from (4.2) and (4.4) that

[TABLE]

Recall that $\frac{1}{2}\leq 1+f_{0}\leq\frac{3}{2}$ and $\frac{1}{K}\langle x\rangle^{-n}\leq|v_{0}(x)|\leq K\langle x\rangle^{-n}$. Therefore, for $\frac{1}{2b}\leq t<\frac{1}{b}$, we have

[TABLE]

for some constants $0<a\leq A<\infty$. If $|x|\geq(1-bt)^{-\frac{2-N\alpha}{2n\alpha}}$, then $|\omega_{0}|^{2}\psi^{2}\gtrsim|x|^{-2n}$ by the first inequality in (4.20). Since also $|\omega_{0}|^{2}\psi^{2}\lesssim|x|^{-2n}$ by the second inequality in (4.20), we deduce that

[TABLE]

for some constants $0<a_{1}\leq A_{1}<\infty$. If $|x|\leq(1-bt)^{-\frac{2-N\alpha}{2n\alpha}}$, then $|\omega_{0}|^{2}\psi^{2}\gtrsim(1-bt)^{\frac{2-N\alpha}{\alpha}}$ by the first inequality in (4.20). Since also $|\omega_{0}|^{2}\psi^{2}\lesssim(1-bt)^{\frac{2-N\alpha}{\alpha}}$ by the second inequality in (4.20), we deduce that

[TABLE]

for some constants $0<a_{2}\leq A_{2}<\infty$. It follows that

[TABLE]

for some constants $0<a_{3}\leq A_{3}<\infty$. On the other hand, estimate (4.1) implies (since $n>\frac{N}{2}$)

[TABLE]

Since $\alpha>\frac{2}{N+1}$, we have

[TABLE]

and (4.6) follows from (4.21)-(4.22). ∎

5. Proof of Theorem 1.1

Let $\Re\lambda<0$ and $K\geq 1$, and let $v_{0}\in{\mathcal{X}}$ satisfy (1.10). Let $\alpha_{1}$ and $b_{1}$ be given by Proposition 3.2. Given $\alpha_{1}\leq\alpha<\frac{2}{N}$ and $b\geq b_{1}$, let $v\in C([0,1/b),{\mathcal{X}})$ be the corresponding solution of (1.19) given by Proposition 3.2. Let

[TABLE]

It follows that $u\in C([0,\infty),H^{1}({\mathbb{R}}^{N}))$ is the solution of (1.1) with the initial condition $u_{0}(x)=e^{i\frac{b|x|^{2}}{4}}v_{0}(x)$. Since $n>\frac{N}{2}$, we deduce from (4.1) in Proposition 4.1 that

[TABLE]

This proves (1.11), while (1.16) and (1.17) follow from (4.5) and (4.6), respectively. This completes the proof of Theorem 1.1.