Blowup on an arbitrary compact set for a Sch\"odinger equation with nonlinear source term

TL;DR

This paper constructs solutions to the nonlinear Schrödinger equation that blow up precisely on a given compact set, using an ansatz based on ODE solutions and energy estimates.

Contribution

It introduces a method to produce solutions that blow up on arbitrary compact sets for a class of nonlinear Schrödinger equations, extending previous blow-up constructions.

Findings

Solutions blow up exactly on the prescribed compact set.

The construction uses an ansatz refined through ODE techniques.

Energy estimates and compactness arguments establish the solutions' properties.

Abstract

We consider the nonlinear Schr\"odinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i \Delta u + \lambda | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \end{equation*} with $\lambda \in {\mathbb C}$ and $\Re \lambda >0$, for $H^1$-subcritical nonlinearities, i.e. $\alpha >0$ and $(N-2) \alpha < 4$. Given a compact set $K \subset {\mathbb R}^N $, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K $ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( \Re \lambda )^{- \frac {1} {\alpha }} (-\alpha t + A(x) )^{ -\frac {1} {\alpha } - i \frac {\Im \lambda } {\alpha \Re \lambda } }$, where $A\ge 0$ vanishes exactly on $ K $, which is a solution of the ODE $u'= \lambda | u |^\alpha u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4].

Equations

$$\partial_{t}u=i\Delta u+\lambda|u|^{\alpha}u\quad\mbox{on ${\mathbb{R}}^{N}$, $\alpha>0$,}$$

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

$$\alpha >0,(N-2)\alpha <4.$$

$$\lambda =1+i\mu ,\mu \in \mathbb{R}.$$

$$\underset{t\uparrow 0}{\lim }\Vert u(t){\Vert }_{L^2(|x-x_0|<r)}=\infty .$$

$$\underset{t\uparrow 0}{\lim }\Vert \nabla u(t){\Vert }_{L^2(U)}=\infty .$$

$$\underset{t\in (-T,0]}{\sup }\Vert u(t){\Vert }_{H^1(\Omega )}<\infty .$$

$$f(u)=|u{|}^{\alpha }u.$$

$$df(u)v=\frac{\alpha +2}{2}|u{|}^{\alpha }v+\frac{\alpha }{2}|u{|}^{\alpha -2}{u}^2\overline{v}.$$

$$\nabla f(u)=\frac{\alpha +2}{2}|u{|}^{\alpha }\nabla u+\frac{\alpha }{2}|u{|}^{\alpha -2}{u}^2\nabla \overline{u},$$

$$\Re (\nabla (f(u))\cdot \nabla \overline{u})=\frac{\alpha +2}{2}|u{|}^{\alpha }|\nabla u{|}^2+\frac{\alpha }{2}\Re (|u{|}^{\alpha -2}{u}^2(\nabla \overline{u}{)}^2)\ge |u{|}^{\alpha }|\nabla u{|}^2.$$

$$C_{\alpha }=\{\begin{array}{ll}0& 0<\alpha \le 1\\ 1& \alpha >1.\end{array}$$

$$|df(u)|\le M|u{|}^{\alpha }$$

$$|f(u+v)-f(u)|\le M(|u{|}^{\alpha }+|v{|}^{\alpha })|v|$$

$$|df(u+v)-df(u)|\le M(|v{|}^{\alpha }+C_{\alpha }|u{|}^{\alpha -1}|v|)$$

$$|df(u+v)-df(u)|\le M(|u{|}^{\alpha }+|v{|}^{\alpha })$$

$$|f(u+v)-f(u)-df(u)v|\le M(|v{|}^{\alpha +1}+C_{\alpha }|u{|}^{\alpha -1}|v{|}^2)$$

$$|\nabla [f(u+v)-f(u)-f(v)]|\le M(|u{|}^{\alpha }+|v{|}^{\alpha })(|\nabla u|+|\nabla v|)$$

$$f(u+v)-f(u)={\int }_0^1\frac{d}{dt}f(u+tv)dt={\int }_0^{1}df(u+tv)vdt.$$

$$|v{|}^{\alpha }+C_{\alpha }|u{|}^{\alpha -1}|v|\le 2(|u{|}^{\alpha }+|v{|}^{\alpha }).$$

$$f(u+v)-f(u)-df(u)v={\int }_0^1[df(u+\theta v)v-df(u)v]d\theta$$

$$\begin{array}{rl}{\partial }_{x_j}[f(u+v)-f(u)-f(v)]& \\ =[df(u+v)-& df(u)]{\partial }_{x_j}u+[df(u+v)-df(v)]{\partial }_{x_j}v\end{array}$$

$$|u{|}^{\alpha }|\nabla u{|}^2\ge |u{|}^{\alpha }|\nabla |u|{|}^2=\frac{4}{(\alpha +2{)}^2}|\nabla (|u{|}^{\frac{\alpha +2}{2}}){|}^2.$$

$$\theta :=\frac{N\alpha }{4(\alpha +1)}\in (0,1).$$

$$\Vert v{\Vert }_{L^{\frac{4(\alpha +1)}{\alpha +2}}}\lesssim \Vert v{\Vert }_{L^2}^{1-\theta }\Vert \nabla v{\Vert }_{L^2}^{\theta },$$

$$\int|u|^{2\alpha+2}\lesssim\Bigl{(}\int|u|^{\alpha+2}\Bigr{)}^{\frac{4-(N-4)\alpha}{2(\alpha+2)}}\Bigl{(}\int|u|^{\alpha}|\nabla u|^{2}\Bigr{)}^{\frac{N\alpha}{2(\alpha+2)}}.$$

$$\int |u{|}^{2\alpha +2}\le \eta \int |u{|}^{\alpha }|\nabla u{|}^2+C_{\eta }\int |u{|}^{\alpha +2}.$$

$${\phi }^{(\nu )}(a(\cdot ))\prod _{\ell =1}^{|\beta |}({\partial }_x^{{\beta }_{\ell }}a(\cdot ){)}^{{\nu }_{\ell }},$$

$$k>\max \{6,N\alpha \}.$$

$$\{\begin{array}{ll}A\ge 0\; and\; |{\partial }_x^{\beta }A|\lesssim A^{1-\frac{|\beta |}{k}}& \text{on }{\mathbb{R}}^N\text{ for }|\beta |\le k-1,\\ A(x)=|x{|}^k& \text{for }x\in {\mathbb{R}}^N,|x|\ge 2.\end{array}$$

$$U_0(t,x)=(-\alpha t+A(x){)}^{-\frac{1}{\alpha }-i\frac{\mu }{\alpha }}t<0,x\in {\mathbb{R}}^N.$$

$$\text{U0 is C∞ in t<0 and Ck−1 in x∈RN},$$

Full text

[TABLE]

with $\lambda\in{\mathbb{C}}$ and $\Re\lambda>0$, for $H^{1}$-subcritical nonlinearities, i.e. $\alpha>0$ and $(N-2)\alpha<4$. Given a compact set $K\subset{\mathbb{R}}^{N}$, we construct $H^{1}$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K$ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_{0}(t,x)=(\Re\lambda)^{-\frac{1}{\alpha}}(-\alpha t+A(x))^{-\frac{1}{\alpha}-i\frac{\Im\lambda}{\alpha\Re\lambda}}$, where $A\geq 0$ vanishes exactly on $K$, which is a solution of the ODE $u^{\prime}=\lambda|u|^{\alpha}u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of [3, 4].

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

We consider the nonlinear Schrödinger equation

[TABLE]

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

[TABLE]

and $\Re\lambda>0$. Note that by scaling invariance of (1.1), we may assume that $\Re\lambda=1$, so we write

[TABLE]

Under assumption (1.2), equation (1.1) is $H^{1}$-subcritical, so that the corresponding Cauchy problem is locally well posed in $H^{1}({\mathbb{R}}^{N})$.

Concerning blowup, it is proved in [1, Theorem 1.1] that, for $\alpha<2/N$, equation (1.1) has no global in time $H^{1}$ solution that remains bounded in $H^{1}$. In other words, every $H^{1}$ solution blows up, in finite or infinite time. Moreover, it is proved in [3] that under assumption (1.2) with the restriction $\alpha\geq 2$, and for $\lambda=1$, finite-time blowup occurs. This result is extended in [8] to the case $\alpha>1$ and $\lambda\in{\mathbb{C}}$ with $(\alpha+2)\Re\lambda\geq\alpha|\lambda|$.

In this paper, we extend the previous blow-up results to the whole range of $H^{1}$ subcritical powers and arbitrary $\lambda\in{\mathbb{C}}$ with $\Re\lambda>0$. Moreover, we prove blowup on any prescribed compact subset of ${\mathbb{R}}^{N}$. Our result is the following.

Theorem 1.1**.**

Let $N\geq 1$, let $\alpha>0$ satisfy (1.2), let $\mu\in{\mathbb{R}}$ and $\lambda$ given by (1.3), and let $K$ be a nonempty compact subset of ${\mathbb{R}}^{N}$. It follows that there exist $T>0$ and a solution $u\in C((-T,0),H^{1}({\mathbb{R}}^{N}))$ of (1.1) which blows up at time [math] exactly on $K$ in the following sense.

1. (i)

If $x_{0}\in K$ then for any $r>0$,

[TABLE] 2. (ii)

If $U$ is an open subset of ${\mathbb{R}}^{N}$ such that $K\subset U$, then

[TABLE] 3. (iii)

If $\Omega$ is an open subset of ${\mathbb{R}}^{N}$ such that $\overline{\Omega}\cap K=\emptyset$, then

[TABLE]

Remark 1.2**.**

Here are some comments on Theorem 1.1.

1. (i)

Estimate (1.4) can be refined. More precisely, it follows from (5.12) that $(-t)^{-\frac{1}{\alpha}+\frac{N}{2k}}\lesssim\|u(t)\|_{L^{2}(|x-x_{0}|<r)}\lesssim(-t)^{-\frac{1}{\alpha}}$ where $k>\frac{N\alpha}{2}$ is given by (4.1)–(4.4). 2. (ii)

From the proof of Theorem 1.1, the blow-up mechanism for $u$ is described as follows. If $U_{0}(t,x)=(-\alpha t+A(x))^{-\frac{1}{\alpha}-i\frac{\mu}{\alpha}}$ where $A\geq 0$ (which vanishes exactly on $K$) is given by (5.1), then $u(t)=U_{0}(t)+V(t)+\varepsilon(t)$, where $\|V(t)\|_{H^{1}}\lesssim(-t)^{\nu}\|U_{0}(t)\|_{L^{2}}$ and $\|\varepsilon(t)\|_{H^{1}}\lesssim(-t)^{\nu}$ for some $\nu>0$, see (3.28) and (5.6).

To prove Theorem 1.1, we follow the strategy, introduced in [3] (see also the references there), of defining the ansatz $U_{0}$, blowing-up solution of the ODE $\partial_{t}U_{0}=\lambda|U_{0}|^{\alpha}U_{0}$, and then using energy estimates and compactness arguments. In [3], restricted to $\alpha\geq 2$ and $\lambda=1$, $U_{0}$ is a sufficiently good approximation and blowup is proved at any finite number of points. To treat any subcritical $\alpha$ and any $\lambda\in{\mathbb{C}}$ with $\Re\lambda>0$, we need to refine the ansatz following the technique developed in [4] for the semilinear wave equation. We emphasize that this technique only uses ODE arguments. See the beginning of Sections 3 and 4 for more details. See also Remark 5.1 below for comments on the restriction (1.2) to $H^{1}$-subcritical powers.

The rest of this paper is organized as follows. In Section 2 we introduce some notation that we use throughout the paper and we recall some useful estimates. In Section 3 we construct the appropriate blow-up ansatz. Section 4 is devoted to the construction of a sequence of solutions of (1.1) close to the blow-up ansatz and to estimates of this sequence. Finally, we complete the proof of Theorem 1.1 in Section 5.

2. Notation and preliminary estimates

2.1. Some Taylor’s inequalities

Let

[TABLE]

In general, $f$ is not $C^{1}$ as a function ${\mathbb{C}}\to{\mathbb{C}}$ (except for $\alpha\in 2{\mathbb{N}}$, when $f$ is analytic). However, $f$ is $C^{1}$ as a function ${\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$. We denote by $df$ the derivative of $f$ is this sense, and we have

[TABLE]

We also have

[TABLE]

so that

[TABLE]

In addition, we have the following estimates.

Lemma 2.1**.**

Set

[TABLE]

There exists a constant $M\geq 1$ such that

[TABLE]

for all $u,v\in{\mathbb{C}}$. Moreover,

[TABLE]

a.e. for all $u,v\in H^{1}({\mathbb{R}}^{N})$.

Proof.

Estimate (2.5) is an immediate consequence of (2.2), and (2.6) follows, using

[TABLE]

Estimate (2.7) is classical, see e.g. [2, formulas (2.26)-(2.27)]; and (2.8) follows from (2.7) and the elementary estimate

[TABLE]

Writing

[TABLE]

and using (2.7), we obtain (2.9). Finally, given $u,v\in H^{1}({\mathbb{R}}^{N})$,

[TABLE]

so that (2.10) follows from (2.8). ∎

2.2. A Sobolev inequality

We have

[TABLE]

Note that by (1.2)

[TABLE]

Since $\frac{\alpha+2}{4(\alpha+1)}=\frac{1}{2}-\frac{\theta}{N}$ , we deduce from Gagliardo-Nirenberg’s inequality that

[TABLE]

so that, letting $v=|u|^{\frac{\alpha+2}{2}}$ and using (2.11),

[TABLE]

Since $\frac{N\alpha}{2(\alpha+2)}<1$ by (1.2), we see that for every $\eta>0$, there exists $C_{\eta}>0$ such that

[TABLE]

2.3. Faà di Bruno’s formula

We recall that by the Faà di Bruno formula (see Corollary 2.10 in [5]), if $\beta$ is a multi-index, $|\beta|\geq 1$, if $a\in C^{|\beta|}(O,{\mathbb{R}})$ where $O$ is an open subset of ${\mathbb{R}}^{N}$, and if $\varphi\in C^{|\beta|}(U,{\mathbb{R}})$ where $U$ is a neighborhood of $a(O)$, then on $O$, $\partial_{x}^{\beta}[\varphi(a(\cdot))]$ is a sum of terms of the form

[TABLE]

with appropriate coefficients, where $\nu\in\{1,\cdots,|\beta|\}$, $\nu_{\ell}\geq 0$, $\displaystyle\sum_{\ell=1}^{|\beta|}\nu_{\ell}=\nu$, $\displaystyle\sum_{\ell=1}^{|\beta|}\nu_{\ell}\beta_{\ell}=\beta$.

3. The blow-up ansatz

In this section, we construct inductively an appropriate blow-up ansatz. The first ansatz is $U_{0}$ defined by (3.3) below. $U_{0}$ is a natural candidate, since it is an explicit blowing-up solution of the ODE $\partial_{t}U_{0}=\lambda f(U_{0})$. Moreover, the error term $i\Delta U_{0}$ is of lower order than both $\partial_{t}U_{0}$ and $f(U_{0})$. (See Lemma (3.1) below.) However, we need at least the error term to be integrable in time near the singularity. Since $\Delta U_{0}$ is of order $(-t)^{-\frac{2}{k}}|U_{0}|\lesssim t^{-\frac{1}{\alpha}-\frac{2}{k}}$, this is not the case for any choice of $k$ if $\alpha\leq 1$. In Section 3.2, we introduce a procedure to reduce the singularity of the error term at any order of $(-t)$ by refining the approximate solution. This is important, not only to obtain blowup for arbitrarily small powers $\alpha$, but also to avoid any condition between $\Re\lambda$ and $\Im\lambda$. We also point out that in this section, there is no condition on the power $\alpha$ other than $\alpha>0$.

Throughout this section, we assume

[TABLE]

Let $A\in C^{k-1}({\mathbb{R}}^{N},{\mathbb{R}})$ be piecewise of class $C^{k}$ and satisfy

[TABLE]

Assuming (1.3), (3.1) and (3.2), and set

[TABLE]

It follows that

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

3.1. Estimates of $U_{0}$

Lemma 3.1**.**

Assume (1.3), (3.1), (3.2) and let $U_{0}$ be given by (3.3). If $p\geq 1$ then

[TABLE]

for $-1\leq t<0$. In addition, for every $\rho\in{\mathbb{R}}$, $\ell\in{\mathbb{N}}$ and $|\beta|\leq k-1$,

[TABLE]

for $x\in{\mathbb{R}}^{N}$, $t<0$, and

[TABLE]

Furthermore, for any $x_{0}\in{\mathbb{R}}^{N}$ such that $A(x_{0})=0$, for any $r>0$, $-1\leq t<0$ and $1\leq p\leq\infty$,

[TABLE]

where the implicit constant in (3.14) depend on $r$ and $p$.

Proof.

Estimate (3.9) is a consequence of (3.6), (3.2) and (3.1). Indeed,

[TABLE]

We now set

[TABLE]

and we prove (3.11)-(3.12). We let $|\beta|\geq 1$, and we write $|U_{0}|^{\rho}=\varphi(W)$ with $\varphi(s)=s^{-\frac{\rho}{\alpha}}$. We see that $|\varphi^{(\nu)}(s)|\lesssim s^{-\frac{\rho}{\alpha}-\nu}$ for $s>0$. Moreover, if $1\leq|\gamma|\leq|\beta|$, then

[TABLE]

where we used (3.2), $A\leq W$, and $1-\frac{|\gamma|}{k}\geq 1-\frac{|\beta|}{k}\geq\frac{1}{k}>0$. By the Faà di Bruno formula (see Section 2.3), we deduce that $|\partial_{x}^{\beta}(|U_{0}|^{\rho})|$ is estimated by a sum of terms of the form

[TABLE]

with appropriate coefficients, where $\nu\in\{1,\cdots,|\beta|\}$, $\nu_{\ell}\geq 0$, $\displaystyle\sum_{\ell=1}^{|\beta|}\nu_{\ell}=\nu$, $\displaystyle\sum_{\ell=1}^{|\beta|}\nu_{\ell}\beta_{\ell}=\beta$. It follows that

[TABLE]

and (3.11) follows.

Now we claim that if $1\leq|\widetilde{\beta}|\leq k-1$, then

[TABLE]

Indeed, we write

[TABLE]

Since $(\log s)^{(\nu)}\lesssim s^{-\nu}$ for $\nu\geq 1$, it follows easily from Faà di Bruno’s formula and (3.15) that

[TABLE]

if $1\leq|\gamma|\leq|\widetilde{\beta}|$. Using again Faà di Bruno’s formula together with (3.18), we obtain

[TABLE]

if $1\leq|\gamma|\leq|\widetilde{\beta}|$. Estimate (3.16) follows from (3.17), (3.11) with $\rho=1$, (3.19), and Leibnitz’s formula.

Estimate (3.12) follows from (3.11), (3.16), and Leibnitz’s formula.

To prove (3.10), we observe that $\partial_{t}^{\ell}U_{0}=cW^{-\frac{1}{\alpha}-\ell-i\frac{\mu}{\alpha}}$ for some constant $c$. Therefore, $\partial_{t}^{\ell}U_{0}=c\widetilde{U}_{0}$, where $\widetilde{U}_{0}=W^{-{1}/{\widetilde{\alpha}}-i{\widetilde{\mu}}/{\widetilde{\alpha}}}$ with $\widetilde{\alpha}=\frac{\alpha}{1+\ell\alpha}$ and $\widetilde{\mu}=\frac{\mu}{1+\ell\alpha}$. In particular, $|\widetilde{U}_{0}|=|U_{0}|^{1+\ell\alpha}$. Applying formula (3.16) with $\alpha$ and $\mu$ replaced by $\widetilde{\alpha}$ and $\widetilde{\mu}$, we obtain

[TABLE]

from which (3.10) follows.

Property (3.13) is an immediate consequence of (3.4), (3.10) and (3.9).

Finally, we prove (3.14). Let $x_{0}\in{\mathbb{R}}^{N}$ be such that $A(x_{0})=0$ and $r>0$. We have $|U_{0}(t,x_{0})|=(-\alpha t)^{-\frac{1}{\alpha}}$, and (3.14) follows in the case $p=\infty$. We now assume $p<\infty$. Since $A$ is piecewise $C^{k}$, it follows easily from (3.2) that for any $x$ such that $|x-x_{0}|<r$, we have $|A(x)|\leq C(r)|x-x_{0}|^{k}$; and so

[TABLE]

Estimate (3.14) then follows from

[TABLE]

This completes the proof. ∎

3.2. The refined blow-up ansatz

We consider the linearization of equation (3.5)

[TABLE]

where $df$ is defined by (2.2). Equation (3.21) has the two solutions $w=iU_{0}$ and $w=\partial_{t}U_{0}$, i.e. $w=\lambda|U_{0}|^{\alpha}U_{0}$. Moreover, $\partial_{t}(|U_{0}|^{\alpha}U_{0})=(\lambda+\alpha)|U_{0}|^{2\alpha}U_{0}$. Elementary calculations show that for suitable $G$, the corresponding nonhomogeneous equation

[TABLE]

has the solution $w={\mathcal{P}}(G)$, where

[TABLE]

We define $U_{j},w_{j},{\mathcal{E}}_{j}$ by

[TABLE]

and then recursively

[TABLE]

for $j\geq 1$, as long as this makes sense. We will see that for $j\leq\frac{k-4}{2}$, ${\mathcal{P}}({\mathcal{E}}_{j-1})$ is well defined at each step, on a sufficiently small time interval. We have the following estimates.

Lemma 3.2**.**

Assume (1.3), (3.1), (3.2), and let $U_{j},w_{j},{\mathcal{E}}_{j}$ be given by (3.3), (3.24) and (3.25). There exists $-1\leq T<0$ such that the following estimates hold for all $0\leq j\leq\frac{k-4}{2}$.

1. (i)

If $0\leq|\beta|\leq k-1-2j$, then

[TABLE] 2. (ii)

If $0\leq|\beta|\leq k-3-2j$, then

[TABLE] 3. (iii)

If $0\leq|\beta|\leq k-1-2j$, then

[TABLE]

Moreover

[TABLE]

In addition,

[TABLE]

Proof.

For $j=0$, estimates (3.26) and (3.27) are immediate consequences of (3.12) and (3.6) (for $T=-\infty$), and estimates (3.28) and (3.29) are trivial. We now proceed by induction on $j$. We assume that for some $1\leq j\leq\frac{k-4}{2}$, estimates (3.26)–(3.29) hold for $0,\cdots,j-1$, and we prove estimates (3.26)–(3.29) for $j$, by possibly assuming $T$ smaller.

Proof of (3.26). Let $0\leq|\beta|\leq k-1-2j$. Given $\rho\in{\mathbb{R}}$, it follows from Leibnitz’s formula, (3.12), and (3.27) for $j-1$ that

[TABLE]

Integrating on $(t,0)$ for $t\in(T,0)$, and using that $|U_{0}|^{-1}$ is a decreasing function of $t$ by (3.7), we see that if $\rho+2\leq 0$, then

[TABLE]

Note that $j\geq 1$ and $\frac{|\beta|}{k}\leq 1-\frac{2j+1}{k}$, so that $j(1-\frac{2}{k})-\frac{|\beta|}{k}\geq 1-\frac{2}{k}-\frac{|\beta|}{k}>0$; and so,

[TABLE]

It follows from Leibnitz’s formula, (3.12), (3.31), (3.11) and (3.6) that

[TABLE]

This proves (3.26).

Proof of (3.28) and (3.29). Since $U_{j}-U_{0}=w_{j}+U_{j-1}-U_{0}$, estimate (3.28) for $j$ follows from (3.26) for $j$ and (3.28) for $j-1$. Estimate (3.29) follows from (3.28) by possibly choosing $T>0$ smaller.

Proof of (3.27). Since $U_{j}-U_{j-1}=w_{j}$, it follows from (3.25) and the definition of ${\mathcal{P}}$ that

[TABLE]

so that

[TABLE]

It follows from (3.26) (for $j$) that if $|\beta|\leq k-3-2j$ (so that $|\beta|+2\leq k-1-2j$), then

[TABLE]

We now estimate $A_{2}$, and we write

[TABLE]

where

[TABLE]

so that

[TABLE]

We write

[TABLE]

where

[TABLE]

Similarly,

[TABLE]

where

[TABLE]

Thus we may write

[TABLE]

Using (3.26), we obtain by choosing $T$ possibly smaller

[TABLE]

so that

[TABLE]

for all $0\leq\tau,\theta\leq 1$. Applying (3.11), (3.12), (3.26), and Leibnitz’s formula, it is not difficult to show that if $|\beta|\leq k-3-2j$ then

[TABLE]

Using now (3.34), (3.35), and the Faà di Bruno formula, we deduce that

[TABLE]

Similarly (using in addition Leibnitz’s formula), we see that

[TABLE]

Next, we deduce from (3.12), (3.26), (3.36), (3.37), (3.6), and Leibnitz’s formula that if $|\beta|\leq k-3-2j$ then

[TABLE]

Using (3.11) with $\rho=\alpha-2$, we obtain similarly

[TABLE]

Estimate (3.27) follows from (3.32), (3.33), (3.38) and (3.39).

Finally, we prove (3.30). For this, we prove by induction on $j$ that

[TABLE]

For $j=0$, (3.40) holds, by (3.13). We assume that for some $1\leq j\leq\frac{k-4}{2}$, property (3.40) holds for $j-1$, and we prove it for $j$. Let $t_{0}\in(T,0)$. It follows from (3.26) and (3.9) that $w_{j}(t_{0})\in H^{k-1-2j}({\mathbb{R}}^{N})$. Moreover, ${\mathcal{E}}_{j-1}\in C((T,0),H^{k-1-2j}({\mathbb{R}}^{N}))$ by the induction assumption. Since $\partial_{t}w_{j}=\lambda df(U_{0})w_{j}+{\mathcal{E}}_{j-1}$, it is not difficult to prove (using Lemma 3.1 for the relevant estimates of $U_{0}$) that $w_{j}\in C^{1}((T,0),H^{k-1-2j}({\mathbb{R}}^{N}))$. Hence $U_{j}\in C^{1}((T,0),H^{k-1-2j}({\mathbb{R}}^{N}))$, and by definition of ${\mathcal{E}}_{j}$, we deduce that ${\mathcal{E}}_{j}\in C((T,0),H^{k-3-2j}({\mathbb{R}}^{N}))$. This proves (3.40). ∎

4. Construction and estimates of approximate solutions

In this section, we construct a sequence $u_{n}$ of solutions of (1.1), close to the ansatz $U_{J}$ of Section 3, which will eventually converge to the blowing-up solution of Theorem 1.1. We estimate $\varepsilon_{n}=u_{n}-U_{J}$ by an energy method. More precisely, we estimate $(-t)^{-\sigma}\|\varepsilon_{n}\|_{L^{2}}+(-t)^{-(1-\theta)\sigma}\|\nabla\varepsilon_{n}\|_{L^{2}}$ for some appropriate parameters $\sigma\geq 0$ and $0\leq\theta\leq 1$. This parameter $\sigma$ is taken large enough to avoid unnecessary condition on $\lambda$, see (4.18) and (4.33). Moreover, the parameter $J$ of the ansatz $U_{J}$ is chosen sufficiently large to absorb the singularity $(-t)^{-\sigma}$, see (4.17) and (4.25).

We now go into details. We define $\sigma,\theta>0$ by

[TABLE]

where $M$ is given by Lemma 2.1, and we set

[TABLE]

In particular, $k$ satisfies (3.1). We let $A\in C^{k-1}({\mathbb{R}}^{N},{\mathbb{R}})$ be piecewise of class $C^{k}$ and satisfy (3.2), and we consider the ansatz $U_{J}$ constructed in Section 3, and $T<0$ given by Lemma 3.2. (This is possible since $2J\leq k-4$ by (4.4).) For $n>-\frac{1}{T}$, we set

[TABLE]

Since $U_{J}(T_{n})\in H^{2}({\mathbb{R}}^{N})$ (by (3.30) and (4.4)) it follows that there exist $s_{n}<T_{n}$ and a unique solution $u_{n}\in C((s_{n},T_{n}],H^{2}({\mathbb{R}}^{N}))\cap C^{1}((s_{n},T_{n}],L^{2}({\mathbb{R}}^{N}))$ of

[TABLE]

defined on the maximal interval $(s_{n},T_{n}]$, with the blow-up alternative that if $s_{n}>-\infty$, then

[TABLE]

See [6].

We let $\varepsilon_{n}\in C((\max\{s_{n},T\},T_{n}],H^{2}({\mathbb{R}}^{N}))\cap C^{1}((\max\{s_{n},T\},T_{n}],L^{2}({\mathbb{R}}^{N}))$ be defined by

[TABLE]

and we have the following estimate.

Proposition 4.1**.**

Assume (4.1), (4.2), (4.3) and (4.4). If $\varepsilon_{n}$ is given by (4.7), then there exist $T\leq S<0$ and $n_{0}>-\frac{1}{S}$ such that

[TABLE]

Moreover,

[TABLE]

for all $n\geq n_{0}$ and $S\leq t\leq T_{n}$, and

[TABLE]

Proof.

Throughout the proof, we write $\varepsilon$ instead of $\varepsilon_{n}$. Moreover, $C$ denotes a constant that may change from line to line, but that is independent of $n$ and $t$. Unless otherwise specified, all integrals are over ${\mathbb{R}}^{N}$. Using (4.5) and (3.25), we have

[TABLE]

We control $\varepsilon$ by energy estimates. Let

[TABLE]

Since $\varepsilon(T_{n})=0$, we see that $T\leq\tau_{n}<T_{n}$. Moreover, it follows from the blowup alternative (4.6) that

[TABLE]

In addition, by Gagliardo-Nirenberg’s inequality, (4.13) and (4.2),

[TABLE]

for $\tau_{n}\leq t\leq T_{n}$. Moreover, it follows from (3.29), (3.6), (3.28) and (3.10) that

[TABLE]

for all $T<t<0$.

We first estimate $\|\varepsilon\|_{L^{2}}$. Multiplying (4.12) by $\overline{\varepsilon}$ and taking the real part, we obtain

[TABLE]

Using (2.5) and (2.9), we deduce that

[TABLE]

By (4.16),

[TABLE]

The term $\int|\varepsilon|^{\alpha+2}$ is estimated by (4.15). Note that $C_{\alpha}\not=0$ only if $\alpha>1$. In this case $\alpha-1>0$ and $2<3<\alpha+2$, so we deduce from (4.16), Hölder’s inequality, (4.13), (4.15) and (4.2) that

[TABLE]

Next, by (3.27), (3.9) and (4.13),

[TABLE]

Note that by (4.3) and (4.4),

[TABLE]

so that

[TABLE]

It follows from the above inequalities that

[TABLE]

where $\nu=\min\{1,\frac{1}{\alpha},\sigma\}$; and so

[TABLE]

Using (4.1), we obtain

[TABLE]

Integrating on $(t,T_{n})$ and using $\varepsilon(T_{n})=0$, we deduce that

[TABLE]

hence

[TABLE]

for all $t\in(\tau_{n},T_{n})$.

We now define the energy

[TABLE]

Multiplying equation (4.12) by $-\Delta\overline{\varepsilon}-\mu\overline{f(\varepsilon)}$ and taking the real part, we obtain after integrating by parts

[TABLE]

Using (2.3), we have

[TABLE]

Moreover, it follows from (2.6), (4.16), (4.15), (2.12) that

[TABLE]

We let $\widetilde{s}<0$ be defined by

[TABLE]

and we deduce that

[TABLE]

for all $n\geq-\frac{1}{\widetilde{s}}$ and all $\tau_{n}<t\leq T_{n}$ such that $t\geq\widetilde{s}$.

Next by (3.27), (3.9) and (4.13),

[TABLE]

Moreover, using (3.27), (3.9) and (4.15),

[TABLE]

Using $\min\{1-\theta,\frac{\alpha+1}{\alpha+2}(\alpha+2-\frac{N\alpha}{2}\theta)\}\geq 1-(N\alpha+1)\theta\geq 0$ (by (4.2)), we conclude that

[TABLE]

Note that by (4.3) and (4.4)

[TABLE]

so that

[TABLE]

We now estimate $A_{2}$. We write

[TABLE]

so that

[TABLE]

with

[TABLE]

It follows from (2.7) and (3.29) that

[TABLE]

If $\alpha>1$, then $|\varepsilon|^{\alpha-1}|U_{0}|\leq(8M|\lambda|)^{-1}|\varepsilon|^{\alpha}+(8M|\lambda|)^{\alpha-1}|U_{0}|^{\alpha}$, so that (recall $|\lambda|\geq 1$)

[TABLE]

Using (3.6), we deduce that

[TABLE]

To estimate $B_{2}$, we consider separately the cases $\alpha\leq 1$, $1<\alpha\leq 3$, and $\alpha>3$.

Suppose first $\alpha\leq 1$. Using (4.16), we see that

[TABLE]

Now if $|v|\leq\frac{1}{2}|u|$, then $|u+sv|\geq\frac{1}{2}|u|$ for all $0\leq s\leq 1$. Writing $df(u+v)-df(u)=\int_{0}^{1}\frac{d}{ds}df(u+sv)$, it follows easily that

[TABLE]

It follows that

[TABLE]

where we used (4.28) and (4.13). Moreover, using (2.8) and (4.28),

[TABLE]

Thus we see that

[TABLE]

When $\alpha>1$, we deduce from (2.7), (4.28) and (4.13) that

[TABLE]

Suppose $1<\alpha\leq 3$. By (1.2) we have $0\leq N-(N-2)\alpha<2\alpha$, and by Gagliardo-Nirenberg’s inequality

[TABLE]

Using (4.16) and (4.13),

[TABLE]

Using (4.1), we conclude that in this case

[TABLE]

If $\alpha>3$, then by (4.16),

[TABLE]

Applying (4.15) and (3.9), we obtain

[TABLE]

Since $\alpha\geq 3$ and $N\leq 3$,

[TABLE]

Using (4.2), we deduce that

[TABLE]

and so,

[TABLE]

so that in this case

[TABLE]

Estimates (4.26), (4.27), (4.29), (4.30) and (4.31) imply

[TABLE]

Using (4.4), we see that $-\frac{2}{k}+(2-\theta)\sigma\geq 2(1-\theta)+\frac{\theta\sigma}{2}$, hence

[TABLE]

Combining (4.21), (4.22), (4.24), (4.25) and (4.32), we obtain

[TABLE]

Using (4.15) and (4.1), we deduce that

[TABLE]

It follows from (4.2) that $(1-2\theta)\sigma\geq 0$, so that the power of $-t$ on the right-hand side of the above inequality are (strictly) larger than $-1$. Integrating on $(t,T_{n})$, using $\varepsilon(T_{n})=0$, and multiplying by $(-t)^{\sigma}$, we obtain

[TABLE]

Using (4.15), we deduce that

[TABLE]

for all $n\geq-\frac{1}{\widetilde{s}}$ and all $\tau_{n}<t\leq T_{n}$ such that $t\geq\widetilde{s}$.

We now conclude as follows. By (4.19) and (4.34) (and since $2(1-\theta)\sigma\geq\sigma$), there exists $S<0$ such that for $n$ sufficiently large (so that $S<T_{n}$),

[TABLE]

for all $\tau_{n}<t<T_{n}$ such that $t\geq S$. By the definition (4.13) of $\tau_{n}$, this implies $\tau_{n}\leq S$. Using property (4.14), we conclude that $s_{n}<S$ and that (4.9), (4.10) and (4.11) hold. ∎

5. Proof of Theorem 1.1

Let $K$ be any compact set of ${\mathbb{R}}^{N}$ included in the ball of center [math] and radius $1$ (by the scaling invariance of equation (1.1), this assumption does not restrict the generality). It is well-known that there exists a smooth function $Z:{\mathbb{R}}^{N}\to[0,\infty)$ which vanishes exactly on $K$ (see e.g. Lemma 1.4, page 20 of [9]). For $\alpha$ satisfying (1.2), let $\sigma,\theta,J,k$ be defined by (4.1), (4.2), (4.3) and (4.4). Define the function $A:{\mathbb{R}}^{N}\to[0,\infty)$ by

[TABLE]

where

[TABLE]

It follows that the function $A$ satisfies (3.2) and vanishes exactly on $K$.

We consider the solution $u_{n}$ of equation (4.5), $\varepsilon_{n}$ defined by (4.7), and $n_{0}\geq 1$ and $S<0$ given by Proposition 4.1. Using the estimate (2.6) and the embeddings $H^{1}({\mathbb{R}}^{N})\hookrightarrow L^{\alpha+2}({\mathbb{R}}^{N})$, $L^{\frac{\alpha+2}{\alpha+1}}({\mathbb{R}}^{N})\hookrightarrow H^{-1}({\mathbb{R}}^{N})$, we deduce from equation (4.12) that

[TABLE]

so that, applying (4.9), (4.10), (3.27), (3.28), (3.10) and (3.9), there exists $\kappa>0$ such that

[TABLE]

Given $\tau\in(S,0)$, it follows from (4.9), (4.10) and (5.2) that $\varepsilon_{n}$ is bounded in $L^{\infty}((S,\tau),H^{1}({\mathbb{R}}^{N}))\cap W^{1,\infty}((S,\tau),H^{-1}({\mathbb{R}}^{N}))$. Therefore, after possibly extracting a subsequence, there exists $\varepsilon\in L^{\infty}((S,\tau),H^{1}({\mathbb{R}}^{N}))\cap W^{1,\infty}((S,\tau),H^{-1}({\mathbb{R}}^{N}))$ such that

[TABLE]

Since $\tau\in(S,0)$ is arbitrary, a standard argument of diagonal extraction shows that there exists $\varepsilon\in L^{\infty}_{\mathrm{loc}}((s,0),H^{1}({\mathbb{R}}^{N}))\cap W^{1,\infty}_{\mathrm{loc}}((S,0),H^{-1}({\mathbb{R}}^{N}))$ such that (after extraction of a subsequence) (5.3), (5.4) and (5.5) hold for all $S<\tau<0$. Moreover, (4.9), (4.10) and (5.5) imply that

[TABLE]

for $S\leq t<0$, and (5.2) and (5.4) imply that

[TABLE]

for all $S<\tau<0$. In addition, it follows easily from (4.12) and the convergence properties (5.3)–(5.5) that

[TABLE]

in $L^{\infty}_{\mathrm{loc}}((S,0),H^{-1}({\mathbb{R}}^{N}))$. Therefore, setting

[TABLE]

we see that $u\in L^{\infty}_{\mathrm{loc}}((S,0),H^{1}({\mathbb{R}}^{N}))\cap W^{1,\infty}_{\mathrm{loc}}((S,0),H^{-1}({\mathbb{R}}^{N}))$ and, using (3.25), that

[TABLE]

in $L^{\infty}_{\mathrm{loc}}((S,0),H^{-1}({\mathbb{R}}^{N}))$. By local existence in $H^{1}({\mathbb{R}}^{N})$ and uniqueness in $L^{\infty}_{t}H^{1}_{x}$, we conclude that $u\in C((S,0),H^{1}({\mathbb{R}}^{N}))\cap C^{1}((S,0),H^{-1}({\mathbb{R}}^{N}))$.

We now prove properties (i), (ii) and (iii). Let $\Omega$ be an open subset of ${\mathbb{R}}^{N}$ such that $\overline{\Omega}\cap K=\emptyset$. It follows from (5.1) that $A>0$ on $\Omega$; and so there exists a constant $c>0$ such that $A(x)\geq c(1+|x|)^{k}$ on $\Omega$. Using (3.6), we deduce that $|U_{0}|\leq C(1+|x|)^{-\frac{k}{\alpha}}$ on $\Omega$. Since $(1+|x|)^{-\frac{k}{\alpha}}\in L^{2}({\mathbb{R}}^{N})$ by (4.4), we conclude, applying (3.28) and (3.10), that

[TABLE]

Property (iii) follows, using (5.6). Let now $x_{0}\in K$ and $r>0$, and set $\omega=\{|x-x_{0}|<r\}$. Let $p\geq 2$ satisfy $(N-2)p\leq 2N$, so that $H^{1}(\omega)\hookrightarrow L^{p}(\omega)$. It follows from (3.29), (3.9) and (3.14) that

[TABLE]

Using (5.6) and the embedding $H^{1}(\omega)\hookrightarrow L^{p}(\omega)$ we deduce that

[TABLE]

Property (i) follows by letting $p=2$. Next, we prove that

[TABLE]

If $N\geq 3$, this follows from (5.12) with $p=\frac{2N}{N-2}$ and Sobolev’s inequality

[TABLE]

If $N=1$, we apply (5.12) with $p=\infty$ and obtain using Gagliardo-Nirenberg’s inequality

[TABLE]

so that $\|\nabla u(t)\|_{L^{2}({\mathbb{R}})}\gtrsim(-t)^{-\frac{1}{\alpha}}$. If $N=2$, we apply (5.12) and Gagliardo-Nirenberg to obtain

[TABLE]

For $p>2+\frac{N\alpha}{2}$, we deduce that $\|\nabla u(t)\|_{L^{2}({\mathbb{R}}^{2})}\gtrsim(-t)^{-\nu}$ with $\nu>0$. This completes the proof of (5.13). Property (ii) is an immediate consequence of (5.13) and (1.6).

The proof of Theorem 1.1 is now complete.

Remark 5.1**.**

As observed at the beginning of Section 3, the construction of the blow-up ansatz does not require any upper bound on the power $\alpha$. Theorem 1.1 is restricted to $H^{1}$-subcritical powers because the energy estimates of Section 4 only provide $H^{1}$ bounds. It is not too difficult to see that a similar result holds in the $H^{1}$-critical case $N\geq 3$ and $\alpha=\frac{4}{N-2}$. Indeed, in this case, the blow-up alternative is not that $\|u(t)\|_{H^{1}}$ blows up, but that certain Strichartz norms blow up, for instance $\|u\|_{L^{\frac{2N}{N-2}}_{t}L^{\frac{2N^{2}}{(N-2)^{2}}}_{x}}$. Control of this norm is given by estimate (4.35) and the inequality

[TABLE]

For $H^{1}$ supercritical powers, higher order estimates would be required. It is not unlikely that a result similar to Theorem 1.1 can be proved in the $H^{2}$-subcritical case $(N-4)\alpha<4$, by establishing $H^{2}$ estimates through $L^{2}$ estimates of $\partial_{t}u$, in the spirit of [7].