Modified scattering for the critical nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the long-time behavior of solutions to the critical nonlinear Schr"odinger equation, constructing global solutions with specific decay rates and asymptotic expansions, using pseudo-conformal transformations and Sobolev norm estimates.

Contribution

It introduces a class of initial data leading to global solutions with precise decay and asymptotic profiles, addressing regularity issues at zero and employing novel analytical techniques.

Findings

Constructed solutions with decay like t^{-N/2} or (t log t)^{-N/2}

Provided asymptotic expansions for these solutions

Developed methods to handle non-vanishing solutions and regularity challenges

Abstract

We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty $, like $t^{- \frac {N} {2}}$ if $\Im \lambda =0$ and like $(t \log t)^{- \frac {N} {2}}$ if $\Im \lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty $. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.

Equations

$$i{u}_t+\Delta u=\lambda |u{|}^{\frac{2}{N}}u$$

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

$$\alpha =\frac{2}{N}$$

$$\Im \lambda \le 0$$

$$u(t)=e^{it\Delta }{u}_0-i\lambda {\int }_0^t{e}^{i(t-s)\Delta }|u{|}^{\alpha }uds$$

$$\Sigma =H^1({\mathbb{R}}^N)\cap L^2({\mathbb{R}}^N,|x{|}^{2}dx).$$

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

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

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

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

$$⟨x⟩=(1+|x{|}^2{)}^{\frac{1}{2}}.$$

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

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

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

$$\Phi(t,x)=\frac{b|x|^{2}}{4(1+bt)}-\frac{\lambda}{b}\Bigl{|}w_{0}\Bigl{(}\frac{x}{1+bt}\Bigr{)}\Bigr{|}^{\frac{2}{N}}\log(1+bt).$$

$$t^{\frac{N}{2}}\Vert u(t){\Vert }_{L^{\infty }}{⟶}_{t\to \infty }{b}^{-\frac{N}{2}}\Vert w_0{\Vert }_{L^{\infty }}.$$

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

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

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

$$\Psi(t,y)=\Bigl{(}\frac{1+f_{0}(y)}{1+f_{0}(y)+\frac{2|\Im\lambda|}{Nb}|v_{0}(y)|^{\frac{2}{N}}\log(1+bt)}\Bigr{)}^{\frac{N}{2}}.$$

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

$$e^{-it\Delta }{M}_{\frac{1+bt}{b}}{D}_{1+bt}=M_{\frac{1}{b}}{e}^{-i\frac{t}{1+bt}\Delta }.$$

$$z(t)=e^{-i\frac{\lambda }{b}|w_0(\frac{x}{1+bt}){|}^{\frac{2}{N}}\log (1+bt)}{e}^{it\Delta }{e}^{-it\Delta }{M}_{\frac{1+bt}{b}}{D}_{1+bt}{w}_0$$

$$e^{-it\Delta }[e^{i\frac{\lambda }{b}|w_0(\frac{x}{1+bt}){|}^{\frac{2}{N}}\log (1+bt)}z(t)]{⟶}_{t\to \infty }{M}_{\frac{1}{b}}{e}^{-i\frac{1}{b}\Delta }{w}_0=:u^{+}$$

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

$$\{\begin{array}{l}i{v}_t+\Delta v=\lambda (1-bt{)}^{-1}|v{|}^{\alpha }v\\ v(0)=v_0\end{array}$$

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

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

$$(e^{it\Delta }{)}_{t\in \mathbb{R}}$$

$$\Vert e^{it\Delta }v{\Vert }_{\mathcal{X}}\le C_1\Vert v{\Vert }_{\mathcal{X}}$$

Full text

[TABLE]

in all dimensions $N\geq 1$, where $\lambda\in{\mathbb{C}}$ and $\Im\lambda\leq 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to\infty$, like $t^{-\frac{N}{2}}$ if $\Im\lambda=0$ and like $(t\log t)^{-\frac{N}{2}}$ if $\Im\lambda<0$. Moreover, we give an asymptotic expansion of those solutions as $t\to\infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.

Ivan Naumkin thanks the project ERC-2014-CdG 646.650 SingWave for its financial support, and the Laboratoire J.A. Dieudonné of the Université de Nice Sophia-Antipolis for its kind hospitality.

1. Introduction

In this article, we consider the nonlinear Schrödinger equation

[TABLE]

on ${\mathbb{R}}^{N}$, where

[TABLE]

and

[TABLE]

and its equivalent integral formulation

[TABLE]

where $(e^{it\Delta})_{t\in{\mathbb{R}}}$ is the Schrödinger group.

It is well known that the Cauchy problem for (1.1)–(1.3) is globally well posed in a variety of spaces, for instance in $H^{1}({\mathbb{R}}^{N})$, in $L^{2}({\mathbb{R}}^{N})$, and in

[TABLE]

See e.g. [14]. Concerning the long time asymptotic behavior of the solutions, $\alpha=\frac{2}{N}$ is a limiting case. Indeed, for $\alpha>\frac{2}{N}$, there is low energy scattering, i.e. a solution of (1.1) with a sufficiently small initial value (in some appropriate sense) is asymptotic as $t\to\infty$ to a solution of the free Schrödinger equation. See [21, 7, 8, 5, 6, 16, 4]. On the other hand, if $\alpha\leq\frac{2}{N}$, then low energy scattering cannot be expected, see [20, Theorem 3.2 and Example 3.3, p. 68] and [1].

In the case $\alpha=\frac{2}{N}$, the relevant notion is modified scattering, i.e. standard scattering modulated by a phase. When $\Im\lambda=0$, the existence of modified wave operators was established in [17] in dimension $N=1$. More precisely, for all sufficiently small asymptotic state $u^{+}$, there exists a solution of (1.1) which behaves as $t\to\infty$ like $e^{i\phi(t,\cdot)}e^{t\Delta}u^{+}$, where the phase $\phi$ is given explicitly in terms of $u^{+}$. (See also [2]. See  [12, 19] for extensions in dimension $N=2$.) Conversely, for small initial values, it was proved in [9] that the asymptotic behavior of the corresponding solution has this form when $\Im\lambda=0$, in dimensions $N=1,2,3$. (See also [15].) If $\Im\lambda<0$, then the nonlinearity has some dissipative effect, and an extra log decay appears in the description of the asymptotic behavior of the solutions. This was established in space dimensions $N=1,2,3$ in [18]. (See also [10, 11] for related results.)

Our purpose in this article is to complete the previous results for (1.1)-(1.2). In order to state our results, we introduce some notation. We consider three integers ${k},{m},{n}$ such that

[TABLE]

and we let

[TABLE]

We consider the Banach space ${\mathcal{X}}$ introduced in [4, formulas (1.6) and (1.7)], i.e.

[TABLE]

with

[TABLE]

where

[TABLE]

Our main results are the following.

Theorem 1.1**.**

Let $\lambda\in{\mathbb{R}}$. Assume (1.2), (1.6), (1.7), let ${\mathcal{X}}$ be defined by $\eqref{n27}$-$\eqref{n28}$, and ${\Sigma}$ by $\eqref{sigma}$. Suppose that $u_{0}(x)=e^{i\frac{b|x|^{2}}{4}}v_{0}(x)$, where $b\in{\mathbb{R}}$ and $v_{0}\in{\mathcal{X}}$ satisfies

[TABLE]

If $b>0$ is sufficiently large, then there exists a unique, global solution $u$ in the class $C([0,\infty),{\Sigma})\cap L^{\infty}((0,\infty)\times{\mathbb{R}}^{N})\cap L^{\infty}((0,\infty),H^{1}({\mathbb{R}}^{N}))$ of (1.4). Moreover, there exist $\delta>0$ and $w_{0}\in L^{\infty}({\mathbb{R}}^{N})$ with $\langle\cdot\rangle^{n}w_{0}\in L^{\infty}({\mathbb{R}}^{N})$ and $h\not\equiv 0$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

In addition,

[TABLE]

Theorem 1.2**.**

Let $\lambda\in{\mathbb{C}}$ with $\Im\lambda<0$. Assume (1.2), (1.6), (1.7), let ${\mathcal{X}}$ be defined by $\eqref{n27}$-$\eqref{n28}$, and ${\Sigma}$ by $\eqref{sigma}$. Suppose $u_{0}(x)=e^{i\frac{b|x|^{2}}{4}}v_{0}(x)$, where $b\in{\mathbb{R}}$ and $v_{0}\in{\mathcal{X}}$ satisfies (1.10). If $b>0$ is sufficiently large, then there exists a unique, global solution $u\in C([0,\infty),\Sigma)\cap L^{\infty}((0,\infty)\times{\mathbb{R}}^{N})\cap L^{\infty}((0,\infty),H^{1}({\mathbb{R}}^{N}))$ of (1.4). Moreover, there exist $\delta>0$ and $f_{0},w_{0}\in L^{\infty},$ with $f_{0}$ real valued, $\|f_{0}\|_{L^{\infty}}\leq\frac{1}{2}$, $w_{0}\not\equiv 0$ and $\langle\cdot\rangle^{n}w_{0}\in L^{\infty}({\mathbb{R}}^{N})$ such that

[TABLE]

where

[TABLE]

with

[TABLE]

and

[TABLE]

In addition,

[TABLE]

Remark 1.3**.**

Here are some comments on the above Theorems 1.1 and 1.2.

1. (i)

The results are valid in any space dimension $N\geq 1$. 2. (ii)

We do not require the initial value $u_{0}$ to have small amplitude. Instead, we require $u_{0}$ to be sufficiently oscillatory (in the sense that $b$ is requested to be sufficiently large). Note also that Theorems 1.1 and 1.2 do not yield any information on the behavior of the solution for $t<0$. 3. (iii)

It is easy to verify that ${\mathcal{S}}({\mathbb{R}}^{N})\subset{\mathcal{X}}$, and that if $\rho\geq n$, then $\langle x\rangle^{-\rho}\in{\mathcal{X}}$. Therefore, if $v_{0}=c\langle\cdot\rangle^{-n}+\varphi$ with $c\in{\mathbb{C}}$, $c\not=0$, and $\varphi\in{\mathcal{S}}({\mathbb{R}}^{N})$, $|\varphi|\leq(|c|-\varepsilon)\langle\cdot\rangle^{-n}$, $\varepsilon>0$, then $v_{0}\in{\mathcal{X}}$ and $v_{0}$ satisfies (1.10). 4. (iv)

The exponent $\delta>0$ that we obtain in (1.11) and (1.13) is provided by Proposition 5.1 below and equals $1-\sigma_{J}$, where $\sigma_{J}$ is given by (4.14). In particular, it is independent of the solution. Moreover, it can be chosen as close to $1$ as we want. 5. (v)

Note that the limit in (1.14) is independent of the initial value $u_{0}$. This is due to the fact that the limit in (5.6) is independent of the initial value $v_{0}$ in (1.16). This last property can be understood by considering the ODE $iz^{\prime}=\lambda(1-bt)^{-1}|z|^{\alpha}z$. One easily verifies that $|z(t)|^{-\alpha}=|z(0)|^{-\alpha}+\frac{\alpha|\Im\lambda|}{b}|\log(1-bt)|$, so that $\lim_{t\uparrow\frac{1}{b}}|\log(1-bt)|^{\frac{1}{\alpha}}|z(t)|=(\frac{b}{\alpha|\Im\lambda|})^{\frac{1}{\alpha}}$ is independent of $z(0)\not=0$. 6. (vi)

One can express formula (1.11) in the form of the standard modified scattering. To see this, let the dilation operator $D_{a}$ and the multiplier $M_{a}$ be defined by $D_{a}\phi(x)=a^{-\frac{N}{2}}\varphi(a^{-1}x)$ and $M_{a}(x)=e^{i\frac{|x|^{2}}{4a}}$, so that (see [13]) $e^{it\Delta}=i^{-\frac{N}{2}}M_{t}D_{t}{\mathcal{F}}M_{t}$, where ${\mathcal{F}}$ is the Fourier transform. Using the relations $D_{a}{\mathcal{F}}={\mathcal{F}}D_{\frac{1}{a}}$ and $M_{a}D_{b}=D_{b}M_{\frac{a}{b^{2}}}$, one obtains

[TABLE]

Since $z$ in (1.11) can be written in the form

[TABLE]

we deduce that

[TABLE]

in $L^{2}({\mathbb{R}}^{N})$. Therefore, (1.11) takes the form of modified scattering. In other words, $u(t)$ behaves like $e^{i\frac{\lambda}{b}|w_{0}(\frac{x}{1+bt})|^{\frac{2}{N}}\log(1+bt)}e^{t\Delta}u^{+}$, i.e. a free solution modulated by a phase.

Remark 1.4**.**

Here are some open questions related to Theorems 1.1 and 1.2.

1. (i)

We do not know what happens if $\Im\lambda>0$. Let us observe that if $\alpha<\frac{2}{N}$ and $\Im\lambda>0$, then it follows from [3, Theorem 1.1] that every nontrivial solution of (1.1) either blows up in finite time or else is global with unbounded $H^{1}$ norm. The proof in [3] apparently does not apply to the case $\alpha=\frac{2}{N}$. See also Remark 4.4 below. 2. (ii)

For equation (1.1) with $\Im\lambda>0$, it seems that no finite time blowup result is available (for any dimension $N$ and any $\alpha>0$). Note that for the same equation set on a bounded domain $\Omega$ with Dirichlet boundary conditions, there is no global solution for any $\alpha>0$. See [3, Section 2]. 3. (iii)

If $\alpha<\frac{2}{N}$ and $\Im\lambda\leq 0$, it seems that no precise description of the asymptotic behavior of the solutions of (1.1) is available. When $\lambda\in{\mathbb{R}}$, $\lambda>0$, it is proved in [22] that all $H^{1}$ solutions converge strongly to [math] in $L^{p}({\mathbb{R}}^{N})$, for $2<p<\frac{2N}{N-2}$, but even the rate of decay of these norms seems to be unknown.

For proving Theorems 1.1 and 1.2, we use the strategy of [4]. One main ingredient is the introduction of the space ${\mathcal{X}}$, which is motivated by the observation that one major difficulty in studying equation (1.1)-(1.2) is the lack of regularity of the nonlinearity $|u|^{\frac{2}{N}}u$ (except in dimension $N=1$). However, this lack of regularity is only at $u=0$, so it is not apparent to solutions that do not vanish. The various conditions in the definition of ${\mathcal{X}}$ are here to ensure a control from below of $|u|$, provided the initial value in ${\mathcal{X}}$ satisfies (1.10). See [4, Section 1]. The other main ingredient is the application of the pseudo-conformal transformation. More precisely, given any $b>0$, $u\in C([0,\infty),\Sigma)\cap L^{\infty}((0,\infty)\times{\mathbb{R}}^{N})$ is a solution of (1.1) (and its equivalent formulation (1.4)) if and only if $v\in C([0,\frac{1}{b}),\Sigma)\cap L^{\infty}((0,\frac{1}{b})\times{\mathbb{R}}^{N})$ defined by

[TABLE]

is a solution of the nonautonomous Schrödinger equation

[TABLE]

and its equivalent formulation

[TABLE]

where $v_{0}(x)=u_{0}(x)e^{-i\frac{b|x|^{2}}{4}}$. In [4, Theorem 1.3], a scattering result is established for solutions of (1.1) with $\alpha>\frac{2}{N}$. In this case, (1.15) transforms solutions of (1.1) to solutions of a nonautonomous equation similar to (1.16), but with $(1-bt)^{-1}$ replaced by $(1-bt)^{-\frac{4-N\alpha}{2}}$. Since $\int_{0}^{\frac{1}{b}}(1-bt)^{-\frac{4-N\alpha}{2}}dt=\frac{2}{b(N\alpha-2)}\to 0$ as $b\to\infty$, a solution $v$ can be constructed on the interval $[0,\frac{1}{b})$ by a fixed point argument, provided $b$ sufficiently large. In the present case (1.2), this argument cannot be applied since $(1-bt)^{-1}$ is not integrable at $\frac{1}{b}$. We therefore have to modify the arguments in [4]. Crucial in our analysis is the elementary estimate

[TABLE]

for every $\mu>0$ and $t<\frac{1}{b}$. It follows that if a certain norm of $e^{i(t-s)\Delta}|v(s)|^{\alpha}v(s)$ is estimated by $(1-bs)^{-\mu}$, then the integral in (1.17) is estimated in that norm by the same power $(1-bt)^{-\mu}$. Concretely, this means that we can control a certain growth of $v(t)$ as $t\to\frac{1}{b}$. Technically, this is achieved by introducing an appropriate cascade of exponents. See Section 4, and in particular Remark 4.2.

The rest of this paper is organized as follows. In Sections 2 and 3, we establish estimates of $e^{t\Delta}$ and $|u|^{\alpha}u$, which are refined versions of estimates in [4]. In Section 4, we study equation (1.16). We first obtain a local existence result with a blowup alternative. Then we show that if $b$ is sufficiently large, the solution of (1.16) exists on $[0,\frac{1}{b})$ and satisfies certain estimates as $t\uparrow\frac{1}{b}$. (Proposition 4.3.) This is the crux of the paper, which requires the estimates of Sections 2 and 3, as well as the introduction of an appropriate cascade of exponents. The asymptotics of the corresponding solutions of (1.16) as $t\uparrow\frac{1}{b}$ is determined in Section 5. Finally, the proof of Theorems 1.1 and 1.2 is completed in Section 6, by translating the results of Section 5 in the original variables via the transformation (1.15).

2. An estimate for the linear Schrödinger equation

In this section, we assume (1.6)-(1.7) (where $\alpha>0$ is arbitrary, not necessarily given by (1.2)), and we let ${\mathcal{X}}$ be defined by (1.8)-(1.9). We establish estimates for the solution of the linear, nonhomogeneous Schrödinger equation. We recall that (see [4, Proposition 1])

[TABLE]

and that there exists a constant $C_{1}$ such that

[TABLE]

and

[TABLE]

for all $0\leq t\leq 1$ and $v\in{\mathcal{X}}$.

Proposition 2.1**.**

There exists $C_{2}\geq 1$ such that if $T>0$, $v_{0}\in{\mathcal{X}}$ and $f\in C([0,T],{\mathcal{X}})$, then the solution $v$ of

[TABLE]

satisfies for all $0\leq t\leq T$ the following estimates.

[TABLE]

if $\left|\beta\right|\leq 2m$,

[TABLE]

if $|\beta|={\nu}+{\mu}+2{m}+1$ with $0\leq{\nu}\leq{k}+1$ and $0\leq{\mu}\leq{n}$.

Proof.

It follows from (2.1) that $v\in C([0,T],{\mathcal{X}})$. We first observe that if $|\beta|\leq 2{m}+2$, then

[TABLE]

for all $u\in{\mathcal{X}}$. Indeed, if $|\beta|\leq 2{m}$, then (2.7) follows immediately from (1.9). Moreover, if $2{m}+1\leq|\beta|\leq 2{m}+2$, then by Sobolev’s inequality $\|\langle\cdot\rangle^{n}D^{\beta}u\|_{L^{\infty}}\leq C\|\langle\cdot\rangle^{n}D^{\beta}u\|_{H^{k}}$ since $k>\frac{N}{2}$ by (1.6). Applying [4, formula (2.13)] (with $s=0$), we deduce that $\|\langle\cdot\rangle^{n}D^{\beta}u\|_{L^{\infty}}\leq C\|u\|_{\mathcal{X}}$, and (2.7) follows.

We now prove (2.5). Let $\left|\beta\right|\leq 2{m}$. Applying $\left\langle\cdot\right\rangle^{n}D^{\beta}$ to equation (2.4) we obtain

[TABLE]

so that

[TABLE]

Integrating this last equation on $(0,t)$ with $0<t\leq T$, we deduce that

[TABLE]

Inequality (2.5) follows, by using (2.7).

Next we prove (2.6). Multiplying (2.8) by $\langle\cdot\rangle^{n-2\mu}D^{\beta}\overline{v}$ we obtain

[TABLE]

after integration by parts. If $\mu=n$, then

[TABLE]

If $\mu<n$, then using the estimate $|\nabla\langle x\rangle^{2n-2\mu}|\leq C\langle x\rangle^{2n-2\mu-1}$ (see [4, formula (A.1)]), we see that

[TABLE]

Since $\|\langle\cdot\rangle^{n-\mu-1}\nabla D^{\beta}v\|_{L^{2}}\leq\|v\|_{\mathcal{X}}$, estimate (2.6) easily follows from (2.9), (2.10) and (2.11). ∎

3. A nonlinear estimate

Throughout this section, we consider $\alpha>0$ (not necessarily given by (1.2)), we assume (1.6)-(1.7), and we let ${\mathcal{X}}$ be defined by (1.8)-(1.9). It is proved in [4, Proposition 2] that there exists a constant $C_{3}$ such that if $u\in{\mathcal{X}}$ and $\eta>0$ satisfy

[TABLE]

then $|u|^{\alpha}u\in{\mathcal{X}}$ and

[TABLE]

Moreover, if both $u_{1},u_{2}\in{\mathcal{X}}$ satisfy (3.1), then

[TABLE]

We now establish a refined version of (3.2). The refinement is based on the fact that expanding $D^{\beta}(|u|^{\alpha}u)$, one obtains on the one hand a term that contains derivatives of $u$ of order $|\beta|$ and can be estimated by $C|u|^{\alpha}|D^{\beta}u|$ (see (3.11)); and on the other hand terms that contain products of derivatives of $u$, all of them being of order at most $|\beta|-1$ (see (3.12)). The refined version of (3.2) is essential in our proof of Proposition 4.3 below. (See Remark 4.2.) Given $\ell\in{\mathbb{N}}$, we set

[TABLE]

[TABLE]

and

[TABLE]

and we have the following estimates.

Proposition 3.1**.**

There exists a constant $C_{4}\geq 1$ such that if $u\in{\mathcal{X}}$ and $\eta>0$ satisfy (3.1), then

[TABLE]

for $0\leq|\beta|\leq 1$,

[TABLE]

for $2\leq|\beta|\leq 2{m}$,

[TABLE]

for $2{m}+1\leq|\beta|\leq 2{m}+2+{k}$, and

[TABLE]

for $2{m}+3+{k}\leq|\beta|\leq{J}$.

Proof.

The case $|\beta|\leq 1$ is immediate, so we suppose $|\beta|\geq 2$. We observe that

[TABLE]

with the coefficients $c_{\gamma,\rho}$ given by Leibniz’s rule. Since $|u|^{\alpha}=(u\overline{u})^{\frac{\alpha}{2}}$ we see that the development of $D^{\beta}(|u|^{\alpha}u)$ contains on the one hand the term

[TABLE]

and on the other hand, terms of the form

[TABLE]

where

[TABLE]

It follows from (3.11) that

[TABLE]

Moreover, it follows from (3.1) that $|u|^{-2p}\leq\eta^{2p}\langle x\rangle^{2p{n}}$, so that (3.12) implies

[TABLE]

We begin by proving (3.8). It follows from (3.13) that

[TABLE]

Moreover, we deduce from (3.14) and (3.4) that

[TABLE]

Estimate (3.8) follows from (3.15) and (3.16).

Next, we prove (3.9). It follows from (3.13) that

[TABLE]

Now, we estimate $\langle x\rangle^{n}B$. Suppose first that all the derivatives in the right-hand side of (3.14) are of order $\leq 2{m}$, then each of them is estimated by $\langle x\rangle^{-{n}}\|u\|_{1,2m}$. Since also $|u|\leq\langle x\rangle^{-n}\|u\|_{1,2m}$, we obtain

[TABLE]

Moreover, $n\alpha=\frac{2n}{N}>\frac{N}{2}$ by (1.6), so we deduce from (3.18) that

[TABLE]

Suppose now that one of the derivatives in the right-hand side of (3.14) is of order greater or equal to $2{m}+1$, for instance $|\gamma_{1,1}|\geq 2{m}+1$. Note that $|\gamma_{i,j}|\leq|\beta|-1$, so this may only occur if $|\beta|\geq 2m+2$. Since the sum of all derivatives has order $|\beta|\leq 2m+2+k$, we have

[TABLE]

by the last inequality in (1.6). It follows that all other derivatives have order $\leq 2{m}$. Thus, (3.14) and (3.4) yield

[TABLE]

Since $\|\langle x\rangle^{{n}}D^{\gamma_{1,1}}u\|_{L^{2}}\leq\|u\|_{2,|\beta|-1}$ by (3.5), we see that

[TABLE]

Estimates (3.17), (3.19) and (3.20) imply (3.9). (Recall that $\|u\|_{L^{\infty}}\leq\|\langle\cdot\rangle^{n}u\|_{L^{\infty}}\leq\|u\|_{1,2m}$.)

Finally, we prove (3.10). It follows from (3.13) that

[TABLE]

We now estimate $\langle x\rangle^{{J}-|\beta|}B$. We first assume that all the derivatives in the right-hand side of (3.14) are of order $\leq 2{m}$. It follows that they are estimated by $\langle x\rangle^{-{n}}\|u\|_{1,2m}$, and we obtain

[TABLE]

Since $n\alpha=\frac{2n}{N}>\frac{N}{2}$ by (1.6), we obtain

[TABLE]

Suppose now that one of the derivatives in the right-hand side of (3.14) is of order $\geq 2{m}+1$, for example $\left|\gamma_{1,1}\right|\geq 2{m}+1$. Since the sum of all derivatives has order $|\beta|\leq J=2m+2+k+n$, we have

[TABLE]

by the last inequality in (1.6). It follows that all other derivatives have order $\leq 2{m}$, hence are estimated by $\langle x\rangle^{-{n}}\|u\|_{1,2m}$. Therefore, (3.14) yields

[TABLE]

If $2{m}+1\leq|\gamma_{1,1}|\leq 2{m}+2+{k}$, we have $\|\langle\cdot\rangle^{{J}-|\beta|}D^{\gamma_{1,1}}u\|_{L^{2}}\leq\|\langle\cdot\rangle^{{n}}D^{\gamma_{1,1}}u\|_{L^{2}}\leq\|u\|_{2,{2{m}+2+{k}}}$, so we deduce from (3.23) that

[TABLE]

If $2{m}+3+{k}\leq|\gamma_{1,1}|\leq|\beta|-1$, then $\|\langle x\rangle^{{J}-|\beta|}D^{\gamma_{1,1}}u\|_{L^{2}}\leq\|\langle x\rangle^{{J}-|\gamma_{1,1}|}D^{\gamma_{1,1}}u\|_{L^{2}}\leq\|u\|_{3,|\beta|-1}$, and thus

[TABLE]

Estimate (3.10) follows from (3.21), (3.22), (3.24) and (3.25). ∎

4. Local and global existence for (1.16)

Throughout this section, we assume (1.2), (1.6), (1.7) and we consider ${\mathcal{X}}$ defined by (1.8)-(1.9). By using the pseudo-conformal transformation (1.15), we transform equation (1.1) into the initial-value problem (1.16), or its equivalent form (1.17). We begin with a local existence result for solutions of (1.16), which follows from the results in [4].

Proposition 4.1**.**

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 $\eqref{NLS2}$ 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 (4.2) for all $0<T<T_{\mathrm{max}}$; and if $T_{\mathrm{max}}<\frac{1}{b}$, then

[TABLE]

Proof.

Given $S>0$, $f\in C([0,S],{\mathbb{C}})$ and $v_{0}\in{\mathcal{X}}$ satisfying (4.1), we consider the equation

[TABLE]

We first observe that a local solution of (4.4) can be constructed by applying the method of [4, Proof of Proposition 3]. Indeed, let

[TABLE]

Given $0<T\leq S$, set

[TABLE]

so that ${\mathcal{E}}$ with the distance ${\mathrm{d}}(u,v)=\|u-v\|_{L^{\infty}((0,T),{\mathcal{X}})}$ is a complete metric space. Given $v\in{\mathcal{E}}$, we set

[TABLE]

for $0\leq t\leq T$. It follows easily from (2.2), (2.3), (3.2) and (3.3) that if

[TABLE]

then the map $v\mapsto\Psi_{v_{0},v}$ is a strict contraction ${\mathcal{E}}\to{\mathcal{E}}$; and so $\Psi_{v_{0},v}$ has a fixed point, which is a solution of (4.4) on $[0,T]$. (See [4, Proof of Proposition 3] for details.)

We next observe that if $v_{0}\in{\mathcal{X}}$ satisfies (4.1), if $0<T<\frac{1}{b}$, and if $v,w\in C([0,T],{\mathcal{X}})$ are two solutions of (4.4) that both satisfy (4.2), then $u=v$. This follows easily from estimates (2.2) and (3.3), and Gronwall’s inequality.

We now argue as follows. We consider $v_{0}\in{\mathcal{X}}$ satisfying (4.1), and we first apply the local existence result for (4.4) with

[TABLE]

where $\eta$ and $M$ are chosen sufficiently large as to satisfy (4.5) and (4.6), and then $0<T<\frac{1}{b}$ is chosen sufficiently small so that (4.7) holds. This yields a solution $u\in C([0,T],{\mathcal{X}})$ of (1.17) satisfying (4.2). Next, we set

[TABLE]

It follows that $0<T_{\mathrm{max}}\leq\frac{1}{b}$. Moreover, we deduce from the uniqueness property that there exists a solution $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$ of (1.17) which satisfies (4.2) for all $0<T<T_{\mathrm{max}}$. Finally, we prove the blowup alternative (4.3). Assume by contradiction that $T_{\mathrm{max}}<\frac{1}{b}$, and that there exist $B>0$ and a sequence $(t_{n})_{n\geq 1}$ such that $t_{n}\uparrow T_{\mathrm{max}}$ and

[TABLE]

We now set $\eta=2B$ and $M=1+C_{1}B$, so that (4.5)-(4.6) hold with $v_{0}$ replaced by $v(t_{n})$, for all $n\geq 1$. We fix $T_{\mathrm{max}}<\tau<\frac{1}{b}$, then we fix $0<T<\tau-T_{\mathrm{max}}$ sufficiently small so that

[TABLE]

If $0\leq t\leq T$, then $t_{n}+t\leq T_{\mathrm{max}}+T\leq\tau$, so that $\|(1-b(t_{n}+\cdot))^{-1}\|_{L^{\infty}(0,T)}\leq(1-b\tau)^{-1}$. Thus (4.10) implies that (4.7) is satisfied with $f(t)\equiv(1-b(t_{n}+t))^{-1}$ for all $n\geq 1$. It follows from the local existence result that for all $n\geq 1$ there exists $v_{n}\in C([0,T],{\mathcal{X}})$ satisfying (4.2), which is a solution of the equation

[TABLE]

Setting now

[TABLE]

we see that $w_{n}\in C([0,t_{n}+T],{\mathcal{X}})$, that $w_{n}$ satisfies (4.2) with $T$ replaced by $t_{n}+T$, and that $w_{n}$ is a solution of (1.17) on $[0,t_{n}+T]$. Since $t_{n}+T>T_{\mathrm{max}}$ for $n$ large, we obtain a contradiction with (4.8). This completes the proof. ∎

Our next result shows that if $v_{0}\in{\mathcal{X}}$ satisfies (4.1) and $b$ is sufficiently large, then the corresponding solution of (1.16) is defined on $[0,\frac{1}{b})$ and satisfies certain estimates as $t\uparrow\frac{1}{b}$. We first comment on the strategy of our proof in the following remark, then we introduce the required notation and state our result in Proposition 4.3.

Remark 4.2**.**

We estimate derivatives of $v$, for instance $\|\langle\cdot\rangle^{n}D^{\beta}v\|_{L^{\infty}}$, by a contraction argument. For this, we assume that

[TABLE]

and

[TABLE]

and we want to recover (4.11)-(4.12) through equation (1.16). It is not too difficult to estimate $\|v(t)\|_{L^{\infty}}$ by using equation (1.16), estimate (4.12), and the assumption $\Im\lambda\leq 0$, so we concentrate on (4.12). We use Proposition 2.1, and then we apply (1.18). This yields an estimate of the form (4.12) provided $\|\langle\cdot\rangle^{n}D^{\beta}(|v|^{\alpha}v)\|_{L^{\infty}}$ is also estimated by $C(1-bt)^{-\mu}$. We now apply Proposition 3.1 to estimate $\|\langle\cdot\rangle^{n}D^{\beta}(|v|^{\alpha}v)\|_{L^{\infty}}$. The right-hand side of (3.8) contains two terms. It follows from (4.11)-(4.12) that the first term is estimated by $C(1-bt)^{-\mu}$. Neglecting the contribution of $\eta$, the second term in (3.8) is essentially of the form $(\sup_{|\gamma|\leq|\beta|-1}\|\langle\cdot\rangle^{n}D^{\gamma}v(t)\|_{L^{\infty}})^{2|\beta|+1}$. If we assume that $\|\langle\cdot\rangle^{n}D^{\gamma}v(t)\|_{L^{\infty}}$ is estimated by $C(1-bt)^{-\mu}$ for $|\gamma|\leq|\beta|-1$, then the second term in (3.8) gives a contribution of the form $C(1-bt)^{-\mu(2|\beta|+1)}$, which is not sufficient to obtain estimate (4.12). Our solution to this difficulty is to assume that derivatives of different orders are estimated by different powers of $(1-bt)$. In other words, we assume that $\mu$ in (4.12) depends on $|\beta|$. Therefore, we need a cascade of exponents, which we introduce below.

Let

[TABLE]

and set

[TABLE]

so that

[TABLE]

Given $0<T<\frac{1}{b}$ and $v\in C([0,T],{\mathcal{X}})$ satisfying (4.2), we define

[TABLE]

where the norms $\|\cdot\|_{j,\ell}$ are defined by (3.4)–(3.6), and we set

[TABLE]

Note that (3.4)–(3.6) imply

[TABLE]

Moreover, one verifies easily that

[TABLE]

where the constant $C_{5}\geq 1$ is independent of $T$.

Proposition 4.3**.**

Suppose $\Im\lambda\leq 0$. Given any $K>0$, there exists $b_{0}>1$ such that if $v_{0}\in{\mathcal{X}}$ satisfies

[TABLE]

then for every $b\geq b_{0}$ the corresponding solution $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$ of (1.17) given by Proposition 4.1, satisfies $T_{\mathrm{max}}=\frac{1}{b}$ and

[TABLE]

where $\Psi_{T}$ is defined by (4.17).

Proof.

Since $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$, we see that $\|v\|_{L^{\infty}((0,T),{\mathcal{X}})}\to\|v_{0}\|_{\mathcal{X}}$ as $T\downarrow 0$. Therefore, it follows from (4.21) and (4.23) that $\|v\|_{T}\leq 2K$ if $T\in(0,T_{\mathrm{max}})$ is sufficiently small, where $K$ is given by (4.23). Moreover, from (4.23) and the property $v\in C([0,T_{\mathrm{max}}),{\mathcal{X}})$, we deduce that

[TABLE]

if $T\in(0,T_{\mathrm{max}})$ is sufficiently small. Therefore, if we set

[TABLE]

then we see that $0<T^{\star}\leq T_{\mathrm{max}}$. We claim that if $b$ is sufficiently large, then

[TABLE]

Assuming (4.26), the conclusion of the theorem follows. Indeed, (4.17) and (4.22) imply that

[TABLE]

If (4.26) holds and $T_{\mathrm{max}}<\frac{1}{b}$, then it follows from (4.27) that

[TABLE]

which contradicts the blowup alternative (4.3). Therefore, we have $T^{\star}=T_{\mathrm{max}}=\frac{1}{b}$, from which the desired conclusion easily follows.

We now prove the claim (4.26), and we assume by contradiction that

[TABLE]

It easily follows from (4.25) and (4.28) that

[TABLE]

We will use the elementary estimate (1.18), as well as the following consequence of (4.22) and (4.29).

[TABLE]

Next, we set

[TABLE]

so that by (4.29)

[TABLE]

for all $0\leq t\leq T^{\star}$. Moreover, it follows from (4.29) that for all $0\leq t\leq T^{\star}$

[TABLE]

If $0\leq j\leq 2m$, then by (4.31) and (4.33) yield

[TABLE]

since $K\geq 1$. Consider now

[TABLE]

and $\ell,q$ such that

[TABLE]

Using the properties $\sigma_{1}\leq\sigma_{\ell-1}$, $\sigma_{j}\leq\sigma_{\ell-1}$, and $(4J+2\alpha+1)\sigma_{\ell-1}\leq\sigma_{\ell}$ (see (4.14)), we deduce from (4.33) (with $q=\ell-1$) and (4.34) that

[TABLE]

for all $0\leq t\leq T^{\star}$.

We now estimate $\Phi_{4,T^{\star}}$. It follows from (1.16) that (recall that $|v|>0$ on $[0,T^{\star}]\times{\mathbb{R}}^{N}$)

[TABLE]

where

[TABLE]

It follows from (4.36) that

[TABLE]

Setting

[TABLE]

we deduce from (4.38) that

[TABLE]

We note that for $0\leq t<T^{\star}$

[TABLE]

by (4.29). Integrating (4.39) in $t$, and applying (4.40), we obtain

[TABLE]

Since $\frac{1}{w(0,x)}\leq K$ by (4.23) and $\frac{1}{w(t,x)}\leq 4K(1-bt)^{-\sigma_{1}}$ by (4.31)-(4.32), the above estimate implies

[TABLE]

Note that $\sigma_{2}+(\alpha+1)\sigma_{1}\leq 2\sigma_{2}\leq\sigma_{J}$ by (4.14), so that (4.41) yields

[TABLE]

from which it follows that

[TABLE]

We next estimate $\|\langle\cdot\rangle^{n}v\|_{L^{\infty}}$. It follows from (4.36) that

[TABLE]

Note that by (4.30)

[TABLE]

Applying (4.43), (4.44) and (4.23), we obtain

[TABLE]

We now estimate $\|\langle\cdot\rangle^{n}D^{\beta}v\|_{L^{\infty}}$ for $1\leq|\beta|\leq 2m$, and we use the estimates of Propositions 2.1 and 3.1. Applying (2.5), (4.32), (3.7) and (3.8), we deduce that

[TABLE]

with $\kappa=0$ if $|\beta|=1$ and $\kappa=1$ if $|\beta|\geq 2$. Moreover, $\|\langle\cdot\rangle^{n}D^{\beta}v\|_{L^{\infty}}\leq\|v\|_{1,|\beta|}$, so that by (4.29)

[TABLE]

We deduce from (4.47) and (1.18) that

[TABLE]

Next, assuming $|\beta|\geq 2$, we apply (4.35) with $j=|\beta|-1$, $\rho=2|\beta|$, $p=1$ and $\ell=|\beta|$, to obtain

[TABLE]

so that

[TABLE]

Applying (1.18), we deduce that

[TABLE]

It follows from (4.46), (4.23), (4.30), (4.48) and (4.49) that

[TABLE]

We next estimate $\|\langle\cdot\rangle^{n}D^{\beta}v\|_{L^{2}}$ for $2{m}+1\leq|\beta|\leq 2m+2+k$. Estimates (2.6) (with $\mu=0$ and $\nu=|\beta|-2m-1$), (4.32) and (3.9) imply

[TABLE]

We have $\|\langle\cdot\rangle^{n}D^{\beta}v\|_{L^{2}}\leq\|v\|_{2,|\beta|}$, so that by (4.29)

[TABLE]

Applying (1.18), we deduce that

[TABLE]

Next, we have by applying (4.35) with $j=2m$, $\rho=2J+\alpha$, and successively $p=1$ and $\ell=2m+1$, then $p=2$ and $\ell=|\beta|$

[TABLE]

It then follows from (1.18) that

[TABLE]

Applying (4.23), (4.30), (4.52) and (4.53), we deduce from (4.51) that

[TABLE]

Now, we estimate $\|\langle\cdot\rangle^{J-|\beta|}D^{\beta}v\|_{L^{2}}$ for ${m}+3+{k}\leq|\beta|\leq J$. It follows from (2.6) (with $\mu=-|\beta|+n-J$ and $\nu=k+1$), (4.32), and (3.10) that

[TABLE]

We have $\|\langle\cdot\rangle^{J-|\beta|}D^{\beta}v\|_{L^{2}}\leq\|v\|_{3,|\beta|}$, hence

[TABLE]

by (4.29). Applying (1.18), we obtain

[TABLE]

Next, we apply (4.35) with $j=2m$, $\rho=2J+\alpha$, and successively $p=1$ and $\ell=2m+1$, then $p=2$ and $\ell=2m+3+k$, then $p=3$ and $\ell=|\beta|$

[TABLE]

Therefore, we deduce from (1.18) that

[TABLE]

Applying (4.23), (4.30), (4.57) and (4.58), we deduce from (4.55) that

[TABLE]

It follows from (4.18)–(4.20), (4.45), (4.50), (4.54), and (4.59) that

[TABLE]

Finally, we assume that $b_{0}$ is sufficiently large so that

[TABLE]

and

[TABLE]

We deduce from (4.42) and (4.62), that if $b\geq b_{0}$, then

[TABLE]

Moreover, we deduce from (4.60) and (4.61), that if $b\geq b_{0}$, then

[TABLE]

Inequalities (4.63) and (4.64) yield $\Psi_{T^{\star}}\leq 2K$, which contradicts (4.29), thus completing the proof. ∎

Remark 4.4**.**

Note that the only place in the proof of Proposition 4.3 where we use the assumption $\Im\lambda\leq 0$ is estimate (4.43). Yet, the conclusion of Proposition 4.3 fails if $\Im\lambda>0$. More precisely, if $v_{0}\in{\mathcal{X}}$ satisfies (4.23) and $b>0$, then there is no solution $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ of (1.17) satisfying (4.24). Indeed, suppose that $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ satisfies (1.17) and (4.24). Applying identity (4.39) with $x=0$ and integrating in $t$ yields

[TABLE]

Since the integral on the right-hand side of the above inequality is bounded as $t\uparrow\frac{1}{b}$ by (4.24), we obtain a contradiction by letting $t\uparrow\frac{1}{b}$.

5. Asymptotics for (1.16)

We now turn to the study of the asymptotic of the solution $v$ as $t\rightarrow\frac{1}{b}$. We prove the following:

Proposition 5.1**.**

Suppose $\Im\lambda\leq 0$. Assume (1.2), (1.6), (1.7) and let ${\mathcal{X}}$ be defined by (1.8)-(1.9). Let $K\geq 1$, and let $b_{0}$ be given by Proposition 4.3. Suppose $b\geq b_{0}$, let $v_{0}\in{\mathcal{X}}$ satisfy (4.23), and let $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ be the solution of (1.16) given by Proposition 4.3. There exists $b_{1}\geq b_{0}$ such that if $b\geq b_{1}$, then there exist $f_{0},w_{0}\in L^{\infty},$ with $f_{0}$ real valued, $\|f_{0}\|_{L^{\infty}}\leq\frac{1}{2}$, $w_{0}\not\equiv 0$ and $\langle\cdot\rangle^{n}w_{0}\in L^{\infty}({\mathbb{R}}^{N})$ such that

[TABLE]

for all $0\leq t<\frac{1}{b}$, where

[TABLE]

and

[TABLE]

In addition, if $\Im\lambda=0$, then

[TABLE]

Furthermore,

[TABLE]

if $\Im\lambda=0$ and

[TABLE]

if $\Im\lambda<0$.

Proof.

We first determine the asymptotic behavior of $|v|$. Integrating equation (4.38) on $(0,t)$ with $0\leq t<\frac{1}{b}$, we obtain

[TABLE]

where $L$ is defined by (4.37), so that

[TABLE]

with

[TABLE]

Since $\|v_{0}\|_{\mathcal{X}}\leq K$, we have $\langle x\rangle^{n}|v_{0}(x)|\leq K$. Moreover,

[TABLE]

by (4.24). Since $(\alpha+1)\sigma_{1}+\sigma_{2}\leq(\alpha+2)\sigma_{2}\leq\sigma_{3}<1$ by (4.14), we deduce that

[TABLE]

Thus we see that the integral in (5.9) is convergent in $L^{\infty}({\mathbb{R}}^{N})$ as $t\uparrow\frac{1}{b}$. It follows that $f$ can be extended to a continuous function $[0,\frac{1}{b}]\to L^{\infty}({\mathbb{R}}^{N})$ and we set

[TABLE]

We note that by (5.9), (5.10) and (5.11),

[TABLE]

and

[TABLE]

for $0\leq t\leq\frac{1}{b}$. In particular, if $b_{1}\geq b_{0}$ is sufficiently large and $b\geq b_{1}$, then

[TABLE]

for all $0\leq t\leq\frac{1}{b}$. Therefore, $1+f_{0}>0$ by (5.12), and it follows from formula (5.2) that

[TABLE]

Moreover, $1-f(t)\geq\frac{1}{2}$ so that

[TABLE]

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

[TABLE]

It follows from (4.23) and (5.15) that

[TABLE]

In addition, we deduce from (5.8), (5.16), (5.13) and (5.15) (with $t$ and with $t=\frac{1}{b}$) that

[TABLE]

for $0\leq t<1/b$. Next, we introduce the decomposition

[TABLE]

where $\psi$ and $\theta$ are defined by (5.2) and (5.3). Differentiating (5.19) with respect to $t$, we obtain

[TABLE]

Moreover, it follows from (5.2) and (5.16) that

[TABLE]

and from (5.3) and (5.16) that

[TABLE]

Thus we see that

[TABLE]

Formulas (5.20), (5.21) and (1.16) yield

[TABLE]

It follows that

[TABLE]

Note that by (5.2)

[TABLE]

Since $\|f_{0}\|_{L^{\infty}}\leq\frac{1}{2}$ by (5.12), we deduce that

[TABLE]

Moreover, $\|\langle\cdot\rangle^{n}\Delta v\|_{L^{\infty}}\leq 4K(1-bt)^{-\sigma_{2}}$ and $\|\langle\cdot\rangle^{n}v\|_{L^{\infty}}\leq 4K$ by (4.24). Therefore, it follows from (5.23), (5.24) and (5.18) that

[TABLE]

since $\sigma_{2}<\sigma_{3}<\sigma_{J}$. We deduce that

[TABLE]

for all $0\leq s<t<\frac{1}{b}$, so that there exists $w_{0}$ such that $\langle\cdot\rangle^{n}w_{0}\in L^{\infty}({\mathbb{R}}^{N})$ and

[TABLE]

for all $0\leq t<\frac{1}{b}$. It follows from (5.19), (5.14), and (5.25) that

[TABLE]

which yields (5.1). We next prove that $w_{0}\not=0$. (Note that if $\Im\lambda=0$, this is obvious by conservation of the $L^{2}$ norm.) Assuming by contradiction that $w_{0}=0$, we deduce from (5.26) and the property $n>\frac{N}{2}$ that

[TABLE]

On the other hand, it follows from equation (1.16) that

[TABLE]

for some $c>0$, by using the $L^{\infty}$ estimate of (5.27). Therefore,

[TABLE]

This is absurd, since $\|v(t)\|_{L^{2}}\to 0$ as $t\uparrow\frac{1}{b}$ by the $L^{2}$ estimate of (5.27).

We now prove (5.4), so we assume $\Im\lambda=0$. The first identity is an immediate consequence of (5.2). Moreover, it follows from (5.3) that

[TABLE]

On the other hand, we deduce from (5.16) that $\widetilde{v}(t,x)=(1+f_{0}(x))^{-\frac{1}{\alpha}}|v_{0}(x)|$, so that (5.18) yields

[TABLE]

in $L^{\infty}({\mathbb{R}}^{N})$. Since $|v(t,x)|=|w(t,x)|$ by (5.19) and the first identity in (5.4), and $|w(t,x)|\to|w_{0}(x)|$, we conclude that

[TABLE]

The second identity in (5.4) now follows from (5.28).

If $\Im\lambda=0$, then (5.5) is an immediate consequence of (5.1) and (5.4). Assuming now $\Im\lambda<0$, we deduce from (5.16) that

[TABLE]

Since $1+f_{0}\geq 0$ by (5.12), it follows in particular that

[TABLE]

Moreover, since $1+f_{0}\leq 2$, we deduce from (5.29) that

[TABLE]

Since $|v_{0}(0)|>0$ by (4.23), it follows that

[TABLE]

Inequalities (5.30) and (5.31) yield

[TABLE]

and (5.6) follows by applying (5.18). This completes the proof. ∎

6. Proof of Theorems 1.1 and 1.2

Let $v_{0}\in{\mathcal{X}}$ satisfy (1.10), let $K>0$ be sufficiently large so that (4.23) holds, and let $b_{1}$ be given by Proposition 5.1. Given $b\geq b_{1}$, let $v\in C([0,\frac{1}{b}),{\mathcal{X}})$ be the corresponding solution of (1.17) given by Proposition 4.3. It is easy to verify that $u$ given by the pseudo-conformal transformation (1.15) satisfies $u\in C([0,\infty),{\Sigma})\cap L^{\infty}((0,\infty)\times{\mathbb{R}}^{N})$, and is a solution of (1.4) with $u_{0}(x)=e^{i\frac{b|x|^{2}}{4}}v_{0}(x)$. Moreover, it follows easily from (4.24) and formula (1.15) that $u\in L^{\infty}((0,\infty),H^{1}({\mathbb{R}}^{N}))$. (Here we use the property $n-1>\frac{N}{2}$.) We now apply Proposition 5.1 and, since $n>\frac{N}{2}$, we deduce from (5.1) that

[TABLE]

If $\Im\lambda=0$, then (1.12) follows from (5.5) and (1.15); and (1.11) follows from (6.1), (5.4), and formula (1.15). This proves Theorem 1.1.

It $\Im\lambda<0$, then (1.14) follows from (5.6) and (1.15). Moreover, it follows from (5.3) and (5.2) that

[TABLE]

Estimate (1.13) follows from (6.1), (6.2), and formula (1.15). This proves Theorem 1.2.