# Blow-up for the 3D intercritical inhomogeneous NLS with inverse-square
potential

Luccas Campos, Mykael Cardoso and Luiz Gustavo Farah

###### Abstract.

In this paper we study the focusing inhomogeneous 3D nonlinear Schrödinger
equation with inverse-square potential in the mass-supercritical and energy-
subcritical regime. We first establish local well-posedness in
$\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$, with $s_{c}=3/2-(2-b)/2\sigma$.
Next, we prove the blow-up of the scaling invariant Lebesgue norm for radial
solutions and also, with an additional restriction, in the non-radial case.

##  1\. Introduction

We consider the initial value problem (IVP) for the inhomogeneous nonlinear
Schrödinger equation with inverse-square potential $(\mbox{INLS})_{a}$

$\begin{cases}i\partial_{t}u+\Delta
u-\frac{a}{|x|^{2}}u+|x|^{-b}|u|^{2\sigma}u=0,\,\,\,x\in\mathbb{R}^{3},\,t>0,\\\
u(0)=u_{0}.\end{cases}$ (1.1)

The (INLS)a model can be interpreted as an extension to the nonlinear
Schrödinger equation. For instance, when $a=b=0$ we have the classical
nonlinear Schrödinger equation, extensively studied over the last decades (see
Sulem and Sulem [44], Bourgain [5], Cazenave [17], Linares and Ponce [38],
Fibich [25], Tao [45] and the references therein). The case $b=0$ and $a\neq
0$ is known as the NLS equation with inverse square potential, denoted by
(NLS)a,

$i\partial_{t}u+\Delta u-\frac{a}{|x|^{2}}u+|u|^{2\sigma}u=0.$ (1.2)

This equation has also been attracting attention in recent years (see for
instance, [7], [34], [39], [43], [47]). The equation (1.2) appears in several
physical settings, such as in quantum field equations or black hole solutions
of the Einstein’s equations (see e.g. [29]).

Alternatively, if $a=0$ and $b\neq 0$ we have the inhomogeneous NLS equation,
denoted by INLS, i.e.,

$i\partial_{t}u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$ (1.3)

which also has received substantial attention recently (see e.g. [27], [28],
[24], [18],[9], [20], [41], [42], [1], [2], [36], [37]. It can model, for
example, nonlinear optical systems with spatially dependent interactions (see
e.g. [4] and the references therein). In particular, it can be thought of as
modeling inhomogeneities in the medium in which the wave propagates (see for
instance [30]). The (INLS)a can thus be seen as a general model for various
physical contexts, and it has been already studied in [11] and [3].

Here, we are interested in the following range of the parameters:

$a>0\,,\quad 0<b<\frac{3}{2}\quad\text{and}\quad\dfrac{2-b}{3}<\sigma<2-b.$
(1.4)

The first restriction in (1.4) is related to the dispersive estimate (2.13).
It is necessary to deduce Lemma 2.6, which is used, in turn, to prove our
local well-posedness result (Theorem 1.1). The restriction $a>0$ also ensures,
via the sharp Hardy inequality

$\displaystyle\||x|^{-1}u\|_{L^{2}}\leq 2\|\nabla u\|_{L^{2}},\,\,\,\mbox{for
all}\,\,\,u\in H^{1}(\mathbb{R}^{3}).\,\,\,$ (1.5)

the coercivity and boundedness of the linear operator in (1.1).

Indeed, consider the quadratic form

$Q_{a}(u)=\int|\nabla u(x)|^{2}+\frac{a}{|x|^{2}}|u(x)|^{2}\,dx$

defined in $C^{\infty}_{c}(\mathbb{R}^{3}\backslash\\{0\\})$ and its
Friedrichs extension denoted by

$\mathcal{L}_{a}=-\Delta+\frac{a}{|x|^{2}}$

(see Killip, Miao, Visan, Zhang and Zheng [35] and the references therein for
more details). The restriction $a>0$ (or, more broadly, $a>-{1}/{4}$)
guarantees that $\mathcal{L}_{a}>0$ and, defining
$\|f\|^{2}_{\dot{H}^{1}_{a}}=\|\mathcal{L}_{a}^{\frac{1}{2}}f\|^{2}_{L^{2}}=(f,\mathcal{L}_{a}f)_{L^{2}}=Q_{a}(f)$,
we also have

$\displaystyle\|f\|_{\dot{H}^{1}_{a}}\sim\|\nabla f\|_{L^{2}},\,\,\,\mbox{for
all}\,\,\,f\in\dot{H}^{1}(\mathbb{R}^{3}).$ (1.6)

This is a particular case of the $L^{p}$\- based Sobolev spaces adapted to the
Schrödinger operator with inverse-square potential $\mathcal{L}_{a}$ equipped
with the norms

$\|f\|_{H_{a}^{s,p}}=\|(1+\mathcal{L}_{a})^{\frac{s}{2}}f\|_{L^{p}}\,\,\,\mbox{and}\,\,\,\|f\|_{\dot{H}_{a}^{s,p}}=\|\mathcal{L}_{a}^{\frac{s}{2}}f\|_{L^{p}}.$

We use the usual convention for $p=2$, i.e.
$\|f\|_{H_{a}^{s}}=\|f\|_{H_{a}^{s,2}}$ and
$\|f\|_{\dot{H}_{a}^{s,2}}=\|f\|_{\dot{H}_{a}^{s}}$. In [35] the authors
studied the relations between these spaces and those defined via
$(\Delta)^{s/2}$ and we recall their results in Lemma 2.2 below. For instance,
when $s>0$ and we have

$\displaystyle\|\mathcal{L}_{a}^{\frac{s}{2}}f\|_{L^{r}}\sim\|D^{s}f\|_{L^{r}},$
(1.7)

for $0<s<2$ and $1<r<3/s$. Note that the case $s=1$ is a direct consequence of
the sharp Hardy inequality as explained above. The relation (1.7) will be one
of the key ingredients to deduce the local well-posedness result in Theorem
1.1.

On the other hand, the second and third conditions in (1.4) are related to the
local well-posedness of (1.1). Indeed, the scale-invariant Sobolev space for
the initial data in view of the scale symmetry

$\lambda\mapsto u_{\lambda}(x,t):=\lambda^{\frac{2-b}{2\sigma}}u(\lambda
x,\lambda^{2}t),$

corresponds to $\dot{H}^{s_{c}}(\mathbb{R}^{3})$, where
$s_{c}=\frac{3}{2}-\frac{2-b}{2\sigma}$. The equation is called intercritical
if $0<s_{c}<1$, which is equivalent to the third condition in (1.4). The
condition $0<b<3/2$ stems from the more general restriction
$0<b<\min\\{N/2,2\\}$, for $u_{0}\in\dot{H}^{1}(\mathbb{R}^{N})$, and it is
exploited in detail by Correia and the first and third authors [13].

Next, we recall mass and energy conservation along the flow of the
$(\mbox{INLS})_{a}$ equation. If $u(t)$ is a solution to (1.1) on some time
interval $I\subset\mathbb{R}$, $0\in I$, then for any $t\in I$

$M[u(t)]=\int|u(x,t)|^{2}\,dx=M[u_{0}]$ (1.8)

and

$E_{a}[u(t)]=\frac{1}{2}\int|\nabla
u(x,t)|^{2}\,dx+\frac{a}{2}\int|x|^{-2}|u(x,t)|^{2}\,dx-\frac{1}{2\sigma+2}\int|x|^{-b}|u(x,t)|^{2\sigma+2}\,dx=E_{a}[u_{0}].$
(1.9)

Recently, Guzmán and the first author [12] obtained the local well-posedness
for the $(\mbox{INLS})_{a}$ equation in $H^{1}(\mathbb{R}^{N})$ via Kato’s
method, which is based on the contraction mapping principle and the Strichartz
estimates. More precisely, they considered $N\geq 3$ and showed that for
initial data $u_{0}\in H^{1}_{a}(\mathbb{R}^{N})$ and

$\left\\{\begin{array}[]{ll}a>-\frac{(N-2)^{2}}{4},&\mbox{ if
}\frac{1-b}{N}<\sigma\leq\frac{1-b}{N-2}\mbox{ and }0<b<1\\\
a>-\frac{(N-2)^{2}}{4}+\left(\frac{\sigma(N-2)-(1-b)}{2\sigma+1}\right)^{2},&\mbox{if}\max\\{0,\frac{1-b}{N}\\}<\sigma<\frac{2-b}{N-2}\end{array}\right.$

there exists a $T=T(\|u_{0}\|_{\dot{H}^{1}_{a}},N,\sigma,b)$ and a unique
solution of (1.1) satisfying

$\displaystyle u\in C\left([0,T];H^{1}_{a}(\mathbb{R}^{N})\right)\cap
L^{q}\left([0,T];H^{1,r}_{a}(\mathbb{R}^{N})\right)$ (1.10)

where $(q,r)$ is a $L^{2}$-admissible pair.

They also proved the existence of a ground state and a dichotomy between
global existence and blow up below the ground state threshold. Furthermore,
they proved small data scattering, constructed wave operators in $H^{1}_{a}$
and proved that suitable space-time bounds imply scattering in the
intercritical case. Later on, An, Jan and Kim [3] extended the results of
Duyckaerts and Roudenko [21] for the NLS and of the first and second authors
[10] for the INLS to include a description of the global behavior of solutions
to (INLS)a with mass-energy possibly greater than or equal to the ground state
threshold.

We are interested here in studying blow-up rates for (INLS)a. To that end, we
first obtain the local theory in
$\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$. We
exclude the $L^{2}$ norm from the well-posedness theorem since it is
subcritical with respect to the scaling of the equation. Therefore, the
procedure adopted in the proof of Theorem 1.3 produces a sequence of functions
whose mass is unbounded. In order to be able to obtain uniform bounds, we work
with a Morrey-Campanato type semi-norm $\rho$, defined by (4.27), which
somehow mimics the mass and is defined for functions merely in
$\dot{H}^{s_{c}}$. Moreover, the control of this quantity allows us to show
that global solutions must have positive energy in the intercritical case (see
Theorem 1.2 below). Next, we state our local theory result.

######  Theorem 1.1.

Let $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. If
$u_{0}\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$,
then there exist $T=T(\|u_{0}\|_{\dot{H}^{s_{c}}_{a}\cap\dot{H}^{1}_{a}})>0$
and a unique solution $u$ to the IVP (1.1) satisfying

$u\in C\left([-T,T];\dot{H}^{s_{c}}_{a}\cap\dot{H}^{1}_{a}\right)\bigcap
L^{q}\left([-T,T];\dot{H}_{a}^{s_{c},p}\cap\dot{H}_{a}^{1,p}\right)\bigcap
L^{\overline{q}}\left([-T,T];L^{\overline{p}}\right),$

for any ($q,p$) $L^{2}$-admissible and $(\overline{q},\overline{p})$
$\dot{H}^{s_{c}}$-admissible.

We note that the nonlinear estimates used in the proof of the previous theorem
are more refined then the previous ones that appeared in the literature (see,
for instance, [16, Lemmas 3.2-3.4]). Indeed, in Lemmas 3.1, 3.2, 3.5 we are
able to achieve the power of $T$ predicted by the scaling of the equation.
Such precise estimate implies the desired blow-up rate (3.40) used as an
additional assumption in our previous works [14, Theorem 1.2.], [15, Theorem
1.4]. The strategy employed here is inspired by the new ideas introduced in
[13] and we split the spacial domain in a ball of radius $R$ and its
complement and then optimize the choice of $R$.

Next, we study the existence of blow-up solutions and the lower bound for the
blow-up rate.

######  Theorem 1.2.

Let $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. Consider
$u_{0}\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$
such that $E(u_{0})\leq 0$.Then the maximal time of existence $T^{\ast}>0$ of
the corresponding solution $u(t)$ to (1.1) is finite in each one of the
following cases

  * $(a)$

$u_{0}$ is radial;

  * $(b)$

$\sigma<\min\left\\{2-b,\frac{2}{3}\right\\}$.

It should be emphasized that in case $(b)$ the radial assumption is not
necessary and instead we need an extra restriction for the parameter $\sigma$.
However, if $b>4/3$, then ${2-b}<\frac{2}{3}$ and the result covers all the
intercritical range for $\sigma$ (see (1.4)).

Let $\sigma_{c}=\frac{6\sigma}{2-b}$. In view of the Sobolev embedding
$\dot{H}^{s_{c}}\subset L^{\sigma_{c}}$ and (1.7), we have

$\|u(t)\|_{L^{\sigma_{c}}}\lesssim\|u(t)\|_{\dot{H}^{s_{c}}}\sim\|u(t)\|_{\dot{H}_{a}^{s_{c}}}$
(1.11)

and then $\dot{H}_{a}^{s_{c}}\subset L^{\sigma_{c}}$. The next result states
that this scaling invariant Lebesgue norm blows up with a certain lower bound.

######  Theorem 1.3.

Assume $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. Let
$u_{0}\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$
be such that the maximal time of existence $T^{\ast}>0$ of the corresponding
solution $u$ to (1.1) is finite. Suppose that either $u_{0}$ is radial or
$\sigma<\min\left\\{2-b,\frac{2}{3}\right\\}$. Then there exists
$\alpha=\alpha(N,\sigma,b)>0$ such that

$\displaystyle\|u(t)\|_{L^{\sigma_{c}}}\geq|\log(T^{\ast}-t)|^{\alpha},\,\,\,as\,\,\,t\to
T^{\ast}.$ (1.12)

Note that this phenomenon does not occur in the mass-critical case
($s_{c}=0$), since the scaling invariant $L^{2}$-norm is preserved along the
flow by the mass conservation (1.8).

We emphasize that our previous studies about the INLS model in [14, 15] prove
the lower bound (1.12) upon imposing an additional assumption and only implies
$\limsup_{t\to T^{*}}\|u(t)\|_{\dot{H}^{s_{c}}}=\infty$. The novelty here is
that we prove that this assumption always holds in the intercritical case, see
Proposition 3.8 below. Using similar ideas, the same statement can also be
shown to hold for the INLS setting, improving our previous results in this
case.

We expect that our blow-up results in the last two theorems extend to higher
dimensions, if one can generalize the local theory in Theorem 1.1 to dimension
$N\geq 3$. The main obstacle is the need of Strichartz estimates for exotic
pairs, especially the non $L^{2}$-admissible ones. This can be achieved, for
example, by proving the dispersive $L^{1}$-$L^{\infty}$ decay for the linear
evolution, which can then be combined with the theory developed by Foschi [26]
to prove the desired estimates.

This paper is organized as follows. In Section 2, we introduce some notation
and preliminary estimates. The well-posedness theory in
$\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$ is
presented in Section 3. The last section is devoted to prove the existence of
blow-up solutions and also a lower bound for the blow-up rate.

##  2\. Preliminaries

We start this section by introducing the notation used throughout the paper.
We use $c$ to denote various constants that may vary line by line. Given any
positive quantities $a$ and $b$, the notation $a\lesssim b$ means that $a\leq
cb$, with $c$ uniform with respect to the set where $a$ and $b$ vary. We
denote by $p^{\prime}$ the Hölder conjugate of $1\leq p\leq\infty$ and we use
$a^{+}$ and $a^{-}$ to denote $a+\varepsilon$ and $a-\varepsilon$,
respectively, for a sufficiently small $\varepsilon>0$.

The following fractional Leibniz and Chain rules obtained will be used in the
sequel (see also the generalizations suitable for singular weights proved in
[13]).

######  Lemma 2.1 (Fractional calculus [19, 33]).

Let $N\geq 1$, $s\in(0,1]$, $p,q_{1},q_{2},r_{1},r_{2}\in(1,\infty)$ with
$\frac{1}{p}=\frac{1}{q_{i}}+\frac{1}{r_{i}}$, $i=1,2$ and $G\in
C^{1}(\mathbb{C})$. Then

  * (i)

$\displaystyle\|D^{s}(fg)\|_{L^{p}(\mathbb{R}^{N})}\lesssim\|D^{s}f\|_{L^{q_{1}}(\mathbb{R}^{N})}\|g\|_{L^{r_{1}}(\mathbb{R}^{N})}+\|f\|_{L^{p_{2}}(\mathbb{R}^{N})}\|D^{s}g\|_{L^{q_{2}}(\mathbb{R}^{N})}.$

  * (ii)

$\displaystyle\|D^{s}G(f)\|_{L^{p}(\mathbb{R}^{N})}\lesssim\|G^{\prime}(f)\|_{L^{q_{1}}(\mathbb{R}^{N})}\|D^{s}f\|_{L^{r_{1}}(\mathbb{R}^{N})}.$

Next, we recall an important tool proved by Killip, Miao, Visan, Zhang and
Zheng. It was proved for a large range of parameters, and we only state the
case that interests us here.

######  Lemma 2.2 (Equivalence of Sobolev norms [35]).

If $a>0$, $0<s<1$, and $1<p<3/s$ then

$\|D^{s}f\|_{L^{p}}\lesssim\|\mathcal{L}_{a}^{\frac{s}{2}}f\|_{L^{p}}\lesssim\|D^{s}f\|_{L^{p}}\;\textnormal{for
all}\;f\in\mathbb{C}^{\infty}_{c}(\mathbb{R}^{3}).$ (2.1)

Now, we define the notion of admissible pairs.

######  Definition 2.3.

If $s\in(-1,1)$, the pair $(q,r)$ is called $\dot{H}^{s}$-admissible if it
satisfies the condition

$\frac{2}{q}=\frac{3}{2}-\frac{3}{r}-s,$ (2.2)

where

$2\leq q,r\leq\infty.$

In particular, if $s=0$, we say that the pair is $L^{2}$-admissible.

######  Definition 2.4.

Consider the set

$\mathcal{A}_{0}=\left\\{(q,r)\text{ is }L^{2}\text{-admissible}\left|\,2\leq
r\leq 6\right.\right\\}.$ (2.3)

For $s\in(-1,1)$, consider also

$\mathcal{A}_{s}=\left\\{(q,r)\text{ is
}\dot{H}^{s}\text{-admissible}\left|\,\left(\frac{6}{3-2|s|}\right)^{+}\leq
r\leq 6^{-}\right.\right\\}.$ (2.4)

If $s\in[0,1)$, we define the following Strichartz norm

$\|u\|_{S(\dot{H}^{s},I)}=\sup_{(q,r)\in\mathcal{A}_{s}}\|u\|_{L_{I}^{q}L_{x}^{r}},$
(2.5)

and the dual Strichartz norm

$\|u\|_{S^{\prime}(\dot{H}^{-s},I)}=\inf_{(q,r)\in\mathcal{A}_{-s}}\|u\|_{L_{I}^{q^{\prime}}L_{x}^{r^{\prime}}}.$
(2.6)

For $s=0$, we shall write $S(\dot{H}^{0},I)=S(L^{2},I)$ and
$S^{\prime}(\dot{H}^{0},I)=S^{\prime}(L^{2},I)$. If $I=\mathbb{R}$, we will
omit $I$.

Here, we are mainly interested in the case $a>0$ and $x\in\mathbb{R}^{3}$. In
particular, the relation (1.7) holds for $a>0$, $0<s<2$ and $r<3/s$, which
implies

$\displaystyle\|f\|_{\dot{H}^{s_{c}}_{a}}\sim\|f\|_{\dot{H}^{s_{c}}}\,\,\,\mbox{and}\,\,\,\|f\|_{\dot{H}^{1}_{a}}\sim\|f\|_{\dot{H}^{1}}.$
(2.7)

In this work, we consider the following Strichartz estimates for $s\in[0,1)$
(see Kato [31], Cazenave [17], Keel and Tao [32], Foschi [26])

$\|e^{it\Delta}f\|_{S(\dot{H}^{s})}\lesssim\|f\|_{\dot{H}^{s}},$ (2.8)

and

$\left\|\int_{\mathbb{R}}e^{i(t-\tau)\Delta}g(\cdot,\tau)\,d\tau\right\|_{S(\dot{H}^{s},I)}+\left\|\int_{0}^{t}e^{i(t-\tau)\Delta}g(\cdot,\tau)\,d\tau\right\|_{S(\dot{H}^{s},I)}\lesssim\|g\|_{S^{\prime}(\dot{H}^{-s},I)}.$
(2.9)

######  Lemma 2.5.

Let $s\in[0,1)$ and $a>-\frac{(N-2)^{2}}{4}$. Then,

$\displaystyle\|e^{-it\mathcal{L}_{a}}f\|_{S(\dot{H}^{s};I)}$
$\displaystyle\lesssim$ $\displaystyle\|f\|_{\dot{H}^{s}_{a}}$ (2.10)
$\displaystyle\left\|\int_{t_{0}}^{t}e^{-i(t-t^{\prime})\mathcal{L}_{a}}g(\cdot,t^{\prime})dt^{\prime}\right\|_{S(L^{2};I)}$
$\displaystyle\lesssim$ $\displaystyle\|g\|_{S^{\prime}(L^{2};I)}$ (2.11)

###### Proof.

See Burq, Planchon, Stalker and Tahvildar-Zadeh [6] and Zhang and Zheng [46].
∎

######  Lemma 2.6.

Let $s\in[0,1)$ and $a>0$. Then,

$\left\|\int_{t_{0}}^{t}e^{-i(t-t^{\prime})\mathcal{L}_{a}}g(\cdot,t^{\prime})dt^{\prime}\right\|_{S(\dot{H}^{s};I)}\lesssim\|g\|_{S^{\prime}(\dot{H}^{-s};I)},$
(2.12)

where $I\subset\mathbb{R}$ be an interval and $t_{0}\in I$.

###### Proof.

The Application 2 in Fanelli, Felli, Fontelos and Primo [22] proves

$\|e^{-it\mathcal{L}_{a}}f\|_{L^{\infty}_{x}}\lesssim\frac{1}{|t|^{\frac{3}{2}}}\|f\|_{L^{1}_{x}},$
(2.13)

which corresponds to the same decay as for the free ($a=0$) Schrödinger
evolution. The estimate (2.12) is then a corollary of the main result in
Foschi [26]. ∎

######  Remark 2.7.

The result in Lemma 2.5 also holds in general space dimension $N>3$. On the
other hand, Lemma (2.6) is still an open problem in this general setting.

##  3\. Well-posedness in $\dot{H_{a}^{s_{c}}}\cap\dot{H}_{a}^{1}$

In this section we study the well-posedness to $(\mbox{INLS})_{a}$ in
homogeneous Sobolev spaces
$\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$. The
main ingredients are the equivalence of Sobolev norms (Lemma 2.2) and
Strichartz estimates (Lemmas 2.5-2.6).

###  3.1. Nonlinear estimates

We start by proving some preliminary estimates. The novelty here is that we
can optimize the estimates around and outside the origin in order to establish
the exact power of $T$ in the right hand side of these inequalities. This will
be important to deduce a lower bound for the blow-up rate for finite time
solutions (see Proposition 3.8).

######  Lemma 3.1.

Let $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. Then, for
$I=[-T,T]$ we have

$\left\|\mathcal{L}_{a}^{\frac{1}{2}}\left(|x|^{-b}|u|^{2\sigma}u\right)\right\|_{S^{\prime}(L^{2};I)}\lesssim
T^{\sigma(1-s_{c})}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|^{2\sigma+1}_{S(L^{2};I)}.$
(3.1)

###### Proof.

Let $R>0$ to be chosen later and denote by $B_{R}$ the closed ball of radius
$R$ in $\mathbb{R}^{3}$. We consider the sets $B^{R}_{+}=B_{R}$ and
$B^{R}_{-}=\mathbb{R}^{3}\backslash B_{R}$. Let $\eta\in(0,1)$ be such that
$(1-2b)/3<\eta<(3-2b)/2$ (which is possible since $0<b<3/2$). Let also
$0<\varepsilon\ll\min\\{(3-2b)/2-\eta,\eta-(1-2b)/3\\}$. Define

$p=6/(1+2\eta)\quad\mbox{and}\quad\alpha_{\pm}=\frac{6(2\sigma+1)}{5-2b+4\sigma-2\eta\mp
2\varepsilon}.$ (3.2)

From the choices of $\eta$ and $\varepsilon$, together with
$(2-b)/3<\sigma<2-b$ and $0<b<3/2$, one gets that $2<p<6$ and
$2<\alpha_{\pm}<3$. Moreover, the following relation holds

$\displaystyle\frac{1}{p^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{b+1\pm\varepsilon}{3}+(2\sigma+1)\left(\frac{1}{\alpha_{\pm}}-\frac{1}{3}\right)$
(3.3) $\displaystyle=$
$\displaystyle\frac{b\pm\varepsilon}{3}+2\sigma\left(\frac{1}{\alpha_{\pm}}-\frac{1}{3}\right)+\frac{1}{\alpha_{\pm}}.$
(3.4)

Therefore, from the Hölder inequality and Sobolev embedding we obtain

$\displaystyle\left\|\nabla\left(|x|^{-b}|u|^{2\sigma}u\right)\right\|_{L^{p^{\prime}}_{x}(B^{R}_{\pm})}$
$\displaystyle\lesssim\left(\left\||x|^{-b}\right\|_{L^{\frac{3}{b\pm\varepsilon}}_{x}(B^{R}_{\pm})}+\left\||x|^{-b-1}\right\|_{L^{\frac{3}{b+1\pm\varepsilon}}_{x}(B^{R}_{\pm})}\right)\|\nabla
u\|_{L^{\alpha_{\pm}}_{x}}^{2\sigma+1}$ (3.5) $\displaystyle\lesssim
R^{\pm\varepsilon}\|\nabla u\|_{L^{\alpha_{\pm}}_{x}}^{2\sigma+1}.$ (3.6)

Now, let $q$, $\beta_{\pm}$ be such that $(q,p)$, $(\beta_{+},\alpha_{+})$ and
$(\beta_{-},\alpha_{-})$ are $L^{2}$-admissible pairs. One can easily check
that, for $\theta_{\pm}=\sigma(1-s_{c})\mp\varepsilon/2$, the following
condition is satisfied

$\displaystyle\frac{1}{q^{\prime}}=\theta_{\pm}+\frac{2\sigma+1}{\alpha_{\pm}},$
(3.7)

then, by the Hölder inequality in the time variable and the equivalence of
Sobolev norms, we have

$\displaystyle\left\|\mathcal{L}_{a}^{\frac{1}{2}}\left(|x|^{-b}|u|^{2\sigma}u\right)\right\|_{L^{q^{\prime}}_{T}L^{p^{\prime}}_{x}}$
$\displaystyle\lesssim\left(T^{\theta_{+}}R^{\varepsilon}+T^{\theta_{-}}R^{-\varepsilon}\right)\|\mathcal{L}^{\frac{1}{2}}_{a}u\|_{S(L^{2};I)}^{2\sigma+1}.$
(3.8)

Finally, taking $R=T^{\frac{1}{2}}$, we have the desired estimate. ∎

######  Lemma 3.2.

Let $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. Then, for
$I=[-T,T]$ we have

$\left\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}\left(|x|^{-b}|u|^{2\sigma}u\right)\right\|_{S^{\prime}(L^{2};I)}\lesssim
T^{\sigma(1-s_{c})}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S(L^{2};I)}^{2\sigma}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};I)}.$
(3.9)

######  Remark 3.3.

Here, due to the product rule, we are required to estimate quantities
comparable to $(D^{s_{c}}|x|^{-b})|u|^{2\sigma}u$ and to
$|x|^{-b}|u|^{2\sigma}D^{s_{c}}u$. However, unlike the previous Lemma, since
fractional derivatives are not localized in space, one cannot simply split the
estimate on a ball and its exterior. Indeed, it is not expected that

$\|D^{s_{c}}(|x|^{-b}f)\|_{L_{x}^{p}(B_{\pm}^{R})}\lesssim\|D^{s_{c}}|x|^{-b}\|_{L_{x}^{p_{1}}(B_{\pm}^{R})}\|f\|_{L^{q_{1}}}+\||x|^{-b}\|_{L_{x}^{p_{2}}(B_{\pm}^{R})}\|D^{s}f\|_{L^{q_{2}}}$
(3.10)

for general $f$, even for suitable $p,p_{j},q_{j}$, due to the non-local
character of the fractional derivatives. A more appropriate version of the
Leibniz rule, adapted to deal with this case, can be found in [13], and the
nonlinear estimates there can be readily adapted to the proof of this Lemma.
Nevertheless, we present another approach here, involving a well-applied
localization, and a Gagliardo-Nirenberg inequality. Indeed, we let $\psi$ be a
smooth, nonincreasing, radially symmetric function such that

$\psi(x)=\begin{cases}1,&|x|\leq 1\\\ 0,&|x|\geq 2,\end{cases}$ (3.11)

and let $\psi_{R}(x)=\psi(x/R)$ for all $x$ and $R>0$. We then claim

######  Claim 3.4.

We have, for all $0\leq s\leq 1$,

$\|D^{s}(|x|^{-b}\psi_{R})\|_{L^{\frac{3}{b+s+\varepsilon}}}\lesssim
R^{\varepsilon},$ (3.12)

and

$\|D^{s}(|x|^{-b}(1-\psi_{R}))\|_{L^{\frac{3}{b+s-\varepsilon}}}\lesssim
R^{-\varepsilon}.$ (3.13)

###### Proof of the Claim.

We use Sobolev embedding to deduce

$\|D^{s}(|x|^{-b}\psi_{R})\|_{L^{\frac{3}{b+s+\varepsilon}}}\lesssim\|\nabla(|x|^{-b}\psi_{R})\|_{L^{\frac{3}{b+1+\varepsilon}}}\lesssim
R^{\varepsilon},$ (3.14)

since $|\nabla|x|^{-b}||\psi_{R}|=|x|^{-b-1}|\psi_{R}|$ and since
$|x|^{-b}|\nabla\psi_{R}|\lesssim R^{-b-1}$ and $\nabla\psi_{R}$ is supported
on $R\leq|x|\leq 2R$. Similar calculations give the estimate for the term with
$(1-\psi_{R})$. ∎

###### Proof of Lemma 3.2.

As in the proof of Lemma 3.1, consider the $L^{2}$-admissible pairs $(q,p)$,
$(\beta_{+},\alpha_{+})$ and $(\beta_{-},\alpha_{-})$ given by (3.2). From
relation (3.3), we also have

$\displaystyle\frac{1}{p^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{b+s_{c}\pm\varepsilon}{3}+2\sigma\left(\frac{1}{\alpha_{\pm}}-\frac{1}{3}\right)+\left(\frac{1}{\alpha_{\pm}}-\frac{s_{c}}{3}\right).$
(3.15)

The Lemma 2.1 (Leibniz and Chain rules), Hölder and Sobolev inequalities yield

$\displaystyle\|D^{s_{c}}(|x|^{-b}|u|^{2\sigma}u)\|_{L_{x}^{p^{\prime}}}$
$\displaystyle\lesssim\|D^{s_{c}}(|x|^{-b}\psi_{R})\|_{L_{x}^{\frac{3}{b+s_{c}+\varepsilon}}}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{+}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{+}}}$ (3.16)
$\displaystyle\quad+\||x|^{-b}\psi_{R}\|_{L_{x}^{\frac{3}{b+\varepsilon}}}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{+}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{+}}}$ (3.17)
$\displaystyle\quad+\|D^{s_{c}}(|x|^{-b}(1-\psi_{R}))\|_{L_{x}^{\frac{3}{b+s_{c}-\varepsilon}}}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{-}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{-}}}$ (3.18)
$\displaystyle\quad+\||x|^{-b}(1-\psi_{R})\|_{L_{x}^{\frac{3}{b-\varepsilon}}}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{-}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{-}}}.$ (3.19)

Next, from Claim 3.4, we get

$\|D^{s_{c}}(|x|^{-b}|u|^{2\sigma}u)\|_{L_{x}^{p^{\prime}}}\lesssim
R^{\varepsilon}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{+}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{+}}}+R^{-\varepsilon}\|\nabla
u\|^{2\sigma}_{L_{x}^{\alpha_{-}}}\|D^{s_{c}}u\|_{L_{x}^{\alpha_{-}}}.$

Finally, using the relation (3.7), the Hölder inequality in the time variable
and the equivalence of Sobolev norms imply

$\|\mathcal{L}^{\frac{s_{c}}{2}}_{a}(|x|^{-b}|u|^{2\sigma}u)\|_{L^{q^{\prime}}_{T}L^{p^{\prime}}_{x}}\lesssim\left(T^{\theta_{+}}R^{\varepsilon}+T^{\theta_{-}}R^{-\varepsilon}\right)\|\mathcal{L}^{\frac{1}{2}}_{a}u\|_{S(L^{2};I)}^{2\sigma}\|\mathcal{L}^{\frac{s_{c}}{2}}_{a}u\|_{S(L^{2};I)}$
(3.20)

and the choice $R=T^{\frac{1}{2}}$ finishes the proof. ∎

######  Lemma 3.5.

Let $a>0$, $0<b<2$, $\frac{2-b}{3}<\sigma<2-b$. Then, there exists
$\delta\in(0,1)$ such that, for $I=[-T,T]$ we have

$\left\|\mathcal{|}x|^{-b}|u|^{2\sigma}v\right\|_{S^{\prime}(\dot{H}^{-s_{c}};I)}\lesssim\left[\left(T^{\frac{1-s_{c}}{2}}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S(L^{2};I)}\right)^{1-\delta}\left(\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};I)}\right)^{\delta}\right]^{2\sigma}\|v\|_{S(\dot{H}^{s_{c}};I)}.$
(3.21)

###### Proof.

For a fixed $\sigma>0$, take $s\in(s_{c},1)$ such that

$\sigma<\frac{1-s_{c}}{s-s_{c}}.$ (3.22)

So, since $2\sigma(s-s_{c})+1<3-2s_{c}$, we can find $\ell$ satisfying

$\max\left\\{\frac{6}{3-2s_{c}},\frac{6}{2\sigma(s-s_{c})+1+\sigma-\varepsilon}\right\\}<\ell<\frac{6}{2\sigma(s-s_{c})+1+\varepsilon},$

for some $0<\varepsilon\ll 1$, such that $(m,\ell)$ is
$\dot{H}^{-s_{c}}$-admissible and $(\tau,\ell)$ is
$\dot{H}^{s_{c}}$-admissible. Also consider the pairs $(\nu_{\pm},\mu_{\pm})$
such that

$\displaystyle\mu_{\pm}=\frac{6\sigma\ell}{\ell(3+2\sigma
s-b\mp\varepsilon)-6}.$ (3.23)

It is straightforward to check that $2<\mu_{\pm}<3$, so we can define the
$L^{2}$-admissible pairs $(\nu_{-},\mu_{-})$ and $(\nu_{+},\mu_{+})$. Now,
setting

$\zeta_{\pm}=\sigma(s-s_{c})\mp\frac{\varepsilon}{2},$

we get

$\displaystyle\frac{1}{\ell^{\prime}}=\frac{b\pm\varepsilon}{3}+2\sigma\left(\frac{1}{\mu_{\pm}}-\frac{s}{3}\right)+\frac{1}{\ell}\,\,\,\,\mbox{and}\,\,\,\,\frac{1}{m^{\prime}}=\zeta_{\pm}+\frac{2\sigma}{\nu_{\pm}}+\frac{1}{\tau}.$
(3.24)

Then, applying Hölder’s inequality and Sobolev embedding we have, using the
same notation of the previous lemmas, that

$\displaystyle\left\||x|^{-b}|u|^{2\sigma}v\right\|_{L^{m^{\prime}}_{T}L^{\ell^{\prime}}_{x}(B^{R}_{\pm})}$
$\displaystyle\lesssim
T^{\zeta_{\pm}}R^{\pm\varepsilon}\|D^{s}u\|_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}^{2\sigma}\|v\|_{L^{\tau}_{T}L^{\ell}_{x}}.$
(3.25)

Moreover, by interpolation and the equivalence of Sobolev norms (1.7) we
obtain

$\displaystyle\|D^{s}u\|_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}$
$\displaystyle\lesssim\|Du\|^{1-\delta}_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}\|D^{s_{c}}u\|^{\delta}_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}$
(3.26)
$\displaystyle\lesssim\|\mathcal{L}^{\frac{1}{2}}_{a}u\|_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}^{1-\delta}\|\mathcal{L}^{\frac{s_{c}}{2}}_{a}u\|_{L^{\nu_{\pm}}_{T}L^{\mu_{\pm}}_{x}}^{\delta},$
(3.27)

for $\delta=(1-s)/(1-s_{c})\in(0,1)$. Collecting the previous two estimates,
we obtain

$\displaystyle\left\||x|^{-b}|u|^{2\sigma}v\right\|_{L^{m^{\prime}}_{T}L^{\ell^{\prime}}_{x}}$
$\displaystyle\lesssim\left(T^{\zeta_{+}}R^{\varepsilon}+T^{\zeta_{-}}R^{-\varepsilon}\right)\left(\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S(L^{2};I)}^{1-\delta}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};I)}^{\delta}\right)^{2\sigma}\|v\|_{S(\dot{H}^{s_{c}};I)}.$
(3.28)

Again setting $R=T^{\frac{1}{2}}$, we find

$T^{\zeta_{+}}R^{\varepsilon}+T^{\zeta_{-}}R^{-\varepsilon}=T^{\sigma(s-s_{c})}=T^{\sigma(1-\delta)(1-s_{c})},$
(3.29)

completing the proof.

∎

###  3.2. Local theory

Next, we use the above nonlinear estimates to obtain the local well-posedness
in $\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$.

###### Proof of Theorem 1.1.

Let

$X=\left(\bigcap_{(q,p)\in\mathcal{A}_{0}}L^{q}\left([-T,T];\dot{H}^{s_{c},p}_{a}\cap\dot{H}^{1,p}_{a}\right)\right)\bigcap\left(\bigcap_{(\overline{q},\overline{p})\in\mathcal{A}_{s_{c}}}L^{\overline{q}}\left([-T,T];L^{\overline{p}}\right)\right)$

and

$\|u\|_{T}=\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S\left(L^{2};I\right)}+\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S\left(L^{2};I\right)}+\|u\|_{S\left(\dot{H}^{s_{c}};I\right)},$

where $I=[-T,T]$.

For $m>1$ and $T>0$ define the set

$S(m,T)=\\{u\in X:\|u\|_{T}\leq m\\}$

with the metric

$d_{T}(u,v)=\|u-v\|_{S\left(\dot{H}^{s_{c}};I\right)}.$

By standards arguments $(S(m,T),d_{T})$ is a complete metric space (see
Cazenave [17, Theorem 4.4.1] and also [16, Appendix A]). We shall show that
$G=G_{u_{0}}$ defined by

$G(u)(t)=e^{-it\mathcal{L}_{a}}u_{0}+i\lambda\int_{0}^{t}e^{-i(t-s)\mathcal{L}_{a}}\left(|x|^{-b}|u(s)|^{2\sigma}u(s)\right)ds,$
(3.30)

has a fixed point in $S(m,T)$ for a suitable choice of $m$ and $T$. Indeed, it
follows from the Strichartz inequalities in Lemma 2.5, and also, from Lemmas
3.1 and 3.2, that, for some $c>1$,

$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}G(u)\|_{S(L^{2};I)}$
$\displaystyle\leq
c\|\mathcal{L}_{a}^{\frac{1}{2}}u_{0}\|_{L^{2}}+cT^{\sigma(1-s_{c})}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|^{2\sigma+1}_{S(L^{2};I)},$
(3.31)

and

$\displaystyle\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}G(u)\|_{S(L^{2};I)}$
$\displaystyle\leq
c\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u_{0}\|_{L^{2}}+cT^{\sigma(1-s_{c})}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S(L^{2};I)}^{2\sigma}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};I)}.$
(3.32)

Moreover, from Lemma 2.6 and Lemma 3.5, we also have

$\|G(u)\|_{S\left(\dot{H}^{s_{c}};I\right)}\leq
c\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u_{0}\|_{L^{2}}+c\left[\left(T^{\frac{1-s_{c}}{2}}\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{S(L^{2};I)}\right)^{1-\delta}\left(\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};I)}\right)^{\delta}\right]^{2\sigma}\|u\|_{S(\dot{H}^{s_{c}};I)}.$

Now, by fixing
$m\geq\max\\{1,2c\|u_{0}\|_{\dot{H}^{s_{c}}_{a}\cap\dot{H}^{1}_{a}}\\}$ and
choosing $T\in(0,1)$ such that

$T^{\sigma(1-s_{c})}<\left(\frac{1}{4cm^{2\sigma}}\right)^{\frac{1}{1-\delta}},$
(3.33)

we obtain that $G(u)\in S(m,T)$, if $u\in S(m,T)$ and therefore $G$ is well
defined. To prove that $G$ is a contraction we first recall the elementary
inequality

$\left||x|^{-b}|u|^{2\sigma}u-|x|^{-b}|v|^{2\sigma}v\right|\lesssim|x|^{-b}\left(|u|^{2\sigma}+|v|^{2\sigma}\right)|u-v|.$
(3.34)

Then, again from Lemma 2.6 and Lemma 3.5, we get for $u,v\in S(m,T)$ that

$\displaystyle d_{T}(G(u),G(v))\leq
2cT^{\sigma(1-s_{c})(1-\delta)}m^{2\sigma}d_{T}(u,v).$

Therefore, for $T\in(0,1)$ satisfying (3.33), $G$ is a contraction on
$S(m,T)$. Finally, by the contraction mapping principle we have a unique $u\in
S(m,T)$ such that $G(u)=u$ and the proof is completed.

∎

###  3.3. A Gagliardo-Nirenberg type inequality and Blow-up alternative

We start recalling the celebrated Caffareli-Khon-Nirenberg inequality in
$\mathbb{R}^{3}$.

######  Proposition 3.6 (Caffareli-Khon-Nirenberg inequality).

There exists $C>0$ such that the following inequality holds for all
$f\in\dot{H}^{1}(\mathbb{R}^{3})\cap L^{p}(\mathbb{R}^{3})$

$\int|x|^{-b}|f|^{2\sigma+2}\,dx\leq C\|\nabla
f\|^{\theta}_{L^{2}}\|f\|^{2\sigma+2-\theta}_{L^{p}}.$ (3.35)

if and only, if the following relations hold

$\displaystyle\frac{3-b}{3}=\frac{\theta}{6}+\frac{2\sigma+2-\theta}{p}$
(3.36)

where $b<3$, $p\geq 1$ and $0<\theta<2\sigma+2$.

###### Proof.

This is the 3D version of the main result in Caffareli, Khon and Nirenberg
[8]. ∎

######  Remark 3.7.

We note that:

  * 1.

Taking $p=2$ in the inequality (3.35), we have the well-known Gagliardo-
Nirenberg inequality

$\displaystyle\int|x|^{-b}|f|^{2\sigma+2}\,dx\leq C\|\nabla
f\|_{L^{2}}^{2\sigma s_{c}+2}\|f\|_{L^{2}}^{2\sigma(1-s_{c})},$ (3.37)

where the best constant was established by the third author in [23].

  * 2.

For $p=\sigma_{c}=\frac{2N\sigma}{2-b}$, the inequality (3.35) is given by

$\int|x|^{-b}|f|^{2\sigma+2}\,dx\leq C\|\nabla
f\|^{2}_{L^{2}}\|f\|^{2\sigma}_{L^{\sigma_{c}}}.$ (3.38)

In [16], joint with Guzmán, the sharp constant for above inequality was
obtained.

  * 3.

Since $a>0$, we have that (see relation (1.6))

$\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{2}}\sim\|\nabla u\|_{L^{2}}.$

Thus, it follows from inequality (3.35) that there exists an optimal constant
$K_{a}>0$ such that

$\int|x|^{-b}|f|^{2\sigma+2}\,dx\leq\frac{\sigma+1}{K_{a}^{2\sigma}}\|\mathcal{L}_{a}^{\frac{1}{2}}f\|^{2}_{L^{2}}\|f\|^{2\sigma}_{L^{\sigma_{c}}}.$
(3.39)

We will refer to the constant $K_{a}$ later in this work (see Proposition
4.3).

Next, we prove a blow-up alternative which follows as a consequence of the
local theory obtained in Theorem 1.1.

######  Proposition 3.8 (Blow-up alternative).

Let $a>0$, $0<b<\frac{3}{2}$ and $\frac{2-b}{3}<\sigma<2-b$. If $u$ is a
solution to the IVP (1.1) in
$\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$ with
finite maximal positive time of existence $0<T^{*}<+\infty$, then there exists
a constant $c>0$ such that

$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|_{L^{2}}\geq\frac{c}{(T^{*}-t)^{\frac{1-s_{c}}{2}}},\,\,\,\textit{for
all}\,\,\,t\in[0,T^{*}).$ (3.40)

###### Proof.

Assume, by contradiction, that there exists a sequence $t_{n}\nearrow
T^{\ast}$ such that, for all $n$,

$(T^{*}-t_{n})^{\frac{1-s_{c}}{2}}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}}<\frac{1}{n}.$
(3.41)

Using the same notation as in the proof of Lemma 3.1, we define

$X_{n}(t)=\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\beta_{+}}_{[t_{n},t]}L^{\alpha_{+}}_{x}}+\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\beta_{-}}_{[t_{n},t]}L^{\alpha_{-}}_{x}}.$
(3.42)

From the Duhamel formula and estimate (3.41), for $t\in[t_{n},T^{*})$, there
exists $C_{0}>0$ such that

$\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\infty}_{[t_{n},t]}L^{2}_{x}}+X_{n}(t)\leq
C_{0}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}}+\frac{C_{0}}{(n\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}})^{2\sigma}}[X_{n}(t)]^{2\sigma+1}.$
(3.43)

Consider the function
$f(x)=x-C_{0}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}}-\frac{C_{0}}{(n\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}})^{2\sigma}}x^{2\sigma+1}$.
A simple computation revels that it has a global maximum at
$x_{n}^{\ast}=\frac{n\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}}}{[C_{0}(2\sigma+1)]^{1/2\sigma}}$,
$f(x_{n}^{\ast})=\left(\frac{2\sigma
n}{(2\sigma+1)[C_{0}(2\sigma+1)]^{1/2\sigma}}-C_{0}\right)\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n})\|_{L^{2}}$
and $f(0)<0$. Thus, for
$n_{0}>\frac{\left[C_{0}(2\sigma+1)\right]^{\frac{2\sigma+1}{2\sigma}}}{2\sigma}$
we have that $f(x_{n_{0}}^{\ast})>0$.

Now, from (3.43) we have that $f(X_{n_{0}}(t))\leq 0$, for all
$t\in[t_{n_{0}},T^{*})$. A continuity argument then implies that
$X_{n_{0}}(t)\leq x_{n_{0}}^{\ast}$ and therefore, from inequality (3.43), we
deduce

$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\infty}_{[t_{n_{0}},T^{*})}L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle
C_{0}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n_{0}})\|_{L^{2}}+\frac{C_{0}}{(n_{0}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n_{0}})\|_{L^{2}})^{2\sigma}}(x_{n_{0}}^{*})^{2\sigma+1}$
(3.44) $\displaystyle=$
$\displaystyle\left(1+\frac{n_{0}}{[C_{0}(2\sigma+1)]^{(2\sigma+1)/2\sigma}}\right)C_{0}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t_{n_{0}})\|_{L^{2}},$
(3.45)

which implies
$\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\infty}_{[0,T^{*})}L^{2}_{x}}<\infty$,
by continuity.

On the other hand, the bound $X_{n_{0}}(t)\leq x_{n_{0}}^{\ast}$ and
assumption (3.41) also implies

$(t-t_{n_{0}})^{\frac{1-s_{c}}{2}}X_{n_{0}}(t)<\frac{1}{[C_{0}(2\sigma+1)]^{1/2\sigma}},\,\,\mbox{for
all}\,\,t\in[0,T^{*}).$ (3.46)

Moreover, again from Duhamel and the proof of estimate (3.32), we can write,
for $t_{n_{0}}\leq t<T^{*}$,

$\displaystyle\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};[t_{n_{0}},t])}$
$\displaystyle\leq
C_{0}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u(t_{n_{0}})\|_{L^{2}}+C_{0}\left[(t-t_{n_{0}})^{\frac{1-s_{c}}{2}}X_{n_{0}}(t)\right]^{2\sigma}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};[t_{n_{0}},t])}$
(3.47) $\displaystyle\leq
C_{0}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u(t_{n_{0}})\|_{L^{2}}+\frac{1}{2\sigma+1}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};[t_{n_{0}},t])}.$
(3.48)

We can then absorb the last term of the right-hand side in the left-hand side
and let $t\nearrow T^{*}$ to obtain

$\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{S(L^{2};[t_{n_{0}},T^{*}))}\leq\frac{(2\sigma+1)C_{0}}{2\sigma}\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u(t_{n_{0}})\|_{L^{2}},$
(3.49)

which is a contradiction, since
$\|\mathcal{L}_{a}^{\frac{1}{2}}u\|_{L^{\infty}_{[0,T^{*})}L^{2}_{x}}+\|\mathcal{L}_{a}^{\frac{s_{c}}{2}}u\|_{L^{\infty}_{[0,T^{*})}L^{2}_{x}}$
cannot be finite, otherwise $T^{*}=+\infty$.

######  Remark 3.9.

The constant $c>0$ in the previous proposition can be chosen independently of
$u$. Indeed, from its proof,one can choose

$c=\frac{2\sigma}{\left[C_{0}(2\sigma+1)\right]^{\frac{2\sigma+1}{2\sigma}}}.$
(3.50)

######  Remark 3.10.

The previous proposition can be adapted to the INLS equation (1.3), showing
that the lower bound

$\|\mathcal{\nabla}u(t)\|_{L^{2}}\geq\frac{c}{(T^{*}-t)^{\frac{1-s_{c}}{2}}},\,\,\,\textit{for
all}\,\,\,t\in[0,T^{*}),$

always hold for $\dot{H}^{s_{c}}\cap\dot{H}^{1}$ blow-up solution in the INLS
setting, improving our previous results [14, Theorem 1.2 and Corollary 1.3]
and [15, Theorem 1.4 and Corollary 1.5].

∎

##  4\. Blow-up solutions in $\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$

###  4.1. A virial-type estimate

Consider $u\in C([0,\tau_{*}]:\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1})$ a
solution to (1.1) and define the function

$\displaystyle z(t)=\displaystyle\int\phi|u(t)|^{2}\,dx,$ (4.1)

where $\phi$ is cut-off function in $\mathbb{R}^{3}$. In [12], the second
author and Guzmán obtained

$z^{\prime}(t)=2\,\mbox{Im}\int\nabla\phi\cdot\nabla u(t)\overline{u}(t)\,dx$
(4.2)

and

$\displaystyle z^{\prime\prime}(t)=$ $\displaystyle
4\,\mbox{Re}\sum_{j,k=1}^{3}\int\partial_{j}u(t)\,\partial_{k}\overline{u}(t)\,\partial^{2}_{jk}\phi\,dx-\int|u(t)|^{2}\Delta^{2}\phi\,dx-2a\int\nabla(|x|^{-2})\cdot\nabla\phi|u(t)|^{2}\,dx$
$\displaystyle-\frac{2\sigma}{\sigma+1}\int|x|^{-b}|u(t)|^{2\sigma+2}\Delta\phi\,dx+\frac{2}{\sigma+1}\int\nabla\left(|x|^{-b}\right)\cdot\nabla\phi|u(t)|^{2\sigma+2}\,dx.$
(4.3)

Set $\phi$ to be a non-negative radial function $\phi\in
C^{\infty}_{0}(\mathbb{R}^{3})$, such that

$\displaystyle\phi(x)=\left\\{\begin{array}[]{ll}|x|^{2},&\mbox{ se }|x|\leq
2\\\ 0,&\mbox{ se }|x|\geq 4\end{array}\right.$ (4.6)

satisfying

$\displaystyle\phi(x)\leq c|x|^{2},\quad|\nabla\phi(x)|^{2}\leq
c\phi(x)\quad\mbox{and}\quad\partial_{r}^{2}\phi(x)\leq 2,\quad\mbox{for all
}|x|>2,$ (4.7)

with $r=|x|$. Then, define $\phi_{R}(x)=R^{2}\phi\left(\frac{x}{R}\right)$.
Since $\phi$ is radial, we have

$\displaystyle
z^{\prime}(t)=2\,\mbox{Im}\int\partial_{r}\phi_{R}\frac{x\cdot\nabla
u(t)}{r}\overline{u}(t)\,dx$ (4.8)

and

$\displaystyle z_{R}^{\prime\prime}(t)=$ $\displaystyle
4\int\frac{\partial_{r}\phi_{R}}{r}|\nabla
u(t)|^{2}\,dx+4\int\left(\frac{\partial_{r}^{2}\phi_{R}}{r^{2}}-\frac{\partial_{r}\phi_{R}}{r^{3}}\right)|x\cdot\nabla
u(t)|^{2}\,dx+4a\int\frac{\partial_{r}\phi_{R}}{r}|x|^{-2}|u(t)|^{2}\,dx$
$\displaystyle-\int|u(t)|^{2}\Delta^{2}\phi_{R}\,dx-\frac{2\sigma}{\sigma+1}\int\left[\partial^{2}_{r}\phi_{R}+\left(N-1+\frac{b}{\sigma}\right)\frac{\partial_{r}\phi_{R}}{r}\right]|x|^{-b}|u(t)|^{2\sigma+2}\,dx,$
(4.9)

where $\partial_{r}$ denotes the derivative with respect to $r=|x|$.

In the following lemma we prove a virial type estimate adapted to the
$(\mbox{INLS})_{a}$ setting.

######  Lemma 4.1.

If $u\in
C([0,\tau_{*}]:\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3}))$
is a solution to (1.1) with initial data $u(0)=u_{0}$, then there exists $c>0$
depending only on $N,\sigma,b$ such that for all $R>0$ and
$t\in[0,\tau_{\ast}]$ we have

$\displaystyle 8\sigma s_{c}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|^{2}_{L^{2}}$
$\displaystyle+z^{\prime\prime}_{R}(t)-16(\sigma s_{c}+1)E[u_{0}]$ (4.10)
$\displaystyle\lesssim$ $\displaystyle\left(\frac{1}{R^{2}}\int_{2R\leq|x|\leq
4R}|u(t)|^{2}\,dx+a\int_{|x|\geq R}|x|^{-2}|u(t)|^{2}\,dx+\int_{|x|\geq
R}|x|^{-b}|u(t)|^{2\sigma+2}\,dx\right).$ (4.11)

###### Proof.

Define

$\displaystyle P[u]$ $\displaystyle=\int|\nabla
u|^{2}\,dx+a\int|x|^{-2}|u|^{2}\,dx-\frac{3\sigma+b}{2\sigma+2}\int|x|^{-b}|u|^{2\sigma+2}\,dx,$
(4.12)

then can rewrite $z_{R}^{\prime\prime}(t)$ in (4.1) as

$\displaystyle z_{R}^{\prime\prime}(t)=8P[u(t)]+K_{1}+K_{2}+K_{3}+K_{4},$
(4.13)

where

$\displaystyle K_{1}=$ $\displaystyle
4\int\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|\nabla
u(t)|^{2}\,dx+4\int\left(\frac{\partial^{2}_{r}\phi_{R}}{r^{2}}-\frac{\partial_{r}\phi_{R}}{r^{3}}\right)|x\cdot\nabla
u(t)|^{2}\,dx,$ (4.14) $\displaystyle K_{2}=$ $\displaystyle
4a\int\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|x|^{-2}|u(t)|^{2}\,dx$
(4.15) $\displaystyle K_{3}=$
$\displaystyle-\frac{2\sigma}{\sigma+1}\int\left[\partial^{2}_{r}\phi_{R}+\left(2+\frac{b}{\sigma}\right)\frac{\partial_{r}\phi_{R}}{r}-6-\frac{2b}{\sigma}\right]|x|^{-b}|u(t)|^{2\sigma+2}\,dx,$
(4.16) $\displaystyle K_{4}=$
$\displaystyle-\int|u(t)|^{2}\Delta^{2}\phi_{R}\,dx.$ (4.17)

First note, by definition of $\phi_{R}$, that

$\displaystyle K_{4}\lesssim\frac{1}{R^{2}}\int_{2R\leq|x|\leq
4R}|u(t)|^{2}\,dx.$ (4.18)

Moreover, if $0\leq|x|\leq 2R$, then

$\displaystyle\partial_{r}\phi_{R}(x)=2|x|=2r,\,\,\,\,\,\,\partial^{2}_{r}\phi_{R}(x)=2,$
(4.19)

which implies

$\displaystyle\mbox{supp}\left[\partial^{2}_{r}\phi_{R}+\left(2+\frac{b}{\sigma}\right)\frac{\partial_{r}\phi_{R}}{r}-6-\frac{2b}{\sigma}\right]\,\cup\,\,\mbox{supp}\left[\frac{\partial_{r}\phi_{R}}{r}-2\right]\subset(2R,\infty).$
(4.20)

Therefore, there exists $c>0$ such that

$\displaystyle K_{3}\lesssim$ $\displaystyle\int_{|x|\geq
R}|x|^{-b}|u(t)|^{2\sigma+2}\,dx$ (4.21)

and

$\displaystyle K_{2}\lesssim a\int_{|x|\geq R}|x|^{-2}|u(t)|^{2}\,dx.$ (4.22)

Now, we set

$\displaystyle\Omega=\left\\{x\in\mathbb{R}^{3};\,\,\frac{\partial^{2}_{r}\phi_{R}(x)}{r^{2}}-\frac{\partial_{r}\phi_{R}(x)}{r^{3}}\leq
0\right\\}.$ (4.23)

Since $\partial^{2}_{r}\phi_{R}\leq 2$ it follows that
$\partial_{r}\phi_{R}(|x|)\leq 2|x|$, for all $x\in\mathbb{R}^{3}$. Then,
using the Cauchy-Schwartz inequality, we get

$\displaystyle K_{1}=$
$\displaystyle\,\,4\int\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|\nabla
u(t)|^{2}\,dx+4\int\left(\frac{\partial_{r}^{2}\phi_{R}}{r^{2}}-\frac{\partial_{r}\phi_{R}}{r^{3}}\right)|x\cdot\nabla
u(t)|^{2}\,dx$ $\displaystyle=$
$\displaystyle\,\,4\int_{\Omega}\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|\nabla
u(t)|^{2}\,dx+4\int_{\Omega}\left(\frac{\partial_{r}^{2}\phi_{R}}{r^{2}}-\frac{\partial_{r}\phi_{R}}{r^{3}}\right)|x\cdot\nabla
u(t)|^{2}\,dx$
$\displaystyle+4\int_{\mathbb{R}^{3}\backslash\Omega}\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|\nabla
u(t)|^{2}\,dx+4\int_{\mathbb{R}^{3}\backslash\Omega}\left(\frac{\partial_{r}^{2}\phi_{R}}{r^{2}}-\frac{\partial_{r}\phi_{R}}{r^{3}}\right)|x\cdot\nabla
u(t)|^{2}\,dx$ $\displaystyle\leq$
$\displaystyle\,\,4\int_{\mathbb{R}^{3}\backslash\Omega}\left(\frac{\partial_{r}\phi_{R}}{r}-2\right)|\nabla
u(t)|^{2}\,dx+4\int_{\mathbb{R}^{3}\backslash\Omega}\left(\partial_{r}^{2}\phi_{R}-\frac{\partial_{r}\phi_{R}}{r}\right)\frac{|x|^{2}}{r^{2}}|\nabla
u(t)|^{2}\,dx$ $\displaystyle=$
$\displaystyle\,\,4\int_{\mathbb{R}^{3}\backslash\Omega}\left(\partial_{r}^{2}\phi_{R}-2\right)|\nabla
u(t)|^{2}\,dx\leq 0,$ (4.24)

where in the last inequality we have used the assumption (4.7).

Collecting (4.18), (4.21), (4.22) and (4.18), there exists $c>0$ such that

$\displaystyle
z_{R}^{\prime\prime}(t)-8P[u(t)]\lesssim\frac{1}{R^{2}}\int_{2R\leq|x|\leq
4R}|u(t)|^{2}\,dx+a\int_{|x|\geq R}|x|^{-2}|u(t)|^{2}\,dx+\int_{|x|\geq
R}|x|^{-b}|u(t)|^{2\sigma+2}\,dx.$ (4.25)

Finally, from the energy conservation (1.9), we can write

$\displaystyle P[u(t)]=-\sigma
s_{c}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|_{L^{2}}^{2}+2(\sigma
s_{c}+1)E[u_{0}]$ (4.26)

and combining the last two relations, we deduce the estimate (4.11). ∎

###  4.2. Gagliardo-Nirenberg inequality

We first recall the following scaling invariant Morrey-Campanato type semi-
norm, used in Merle and Raphaël [40]

$\rho(u,R)=\sup_{R^{\prime}\geq
R}\frac{1}{(R^{\prime})^{2s_{c}}}\int_{R^{\prime}\leq|x|\leq
2R^{\prime}}|u|^{2}\,dx.$ (4.27)

It is easy to see that $\rho(u,R)$ is non-increasing in $R>0$. Moreover, by
Hölder’s inequality, there exists a universal constant $c>0$ such that for all
$u\in L^{\sigma_{c}}$ and $R>0$

$\frac{1}{R^{2s_{c}}}\int_{|x|\leq R}|u|^{2}\,dx\leq
c\|u\|_{L^{\sigma_{c}}}^{2}$ (4.28)

and

$\lim_{R\to+\infty}\frac{1}{R^{2s_{c}}}\int_{|x|\leq R}|u|^{2}\,dx=0$ (4.29)

(c.f. Merle and Raphäel in [40, Lemma 1] and also [14, Lemma 2.1]). In [14]
and [15], the last two authors proved some interpolation inequalities adapted
to the inhomogeneous case in $\dot{H}^{s_{c}}\cap\dot{H}^{1}$. The relations
(2.7) imply that these results also hold in the space
$\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$, for $a>0$ and $x\in\mathbb{R}^{3}$.
Below we recall such estimates.

######  Lemma 4.2 (Gagliardo-Nirenberg type inequality [14], [15]).

Assume $0<b<3/2$. Then, for all $\eta>0$, there exists a constant $C_{\eta}>0$
such that

  * (i)

[Radial] for all
$u\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$
with radial symmetry the following inequality holds

$\int_{|x|\geq R}|x|^{-b}|u|^{2\sigma+2}\,dx\leq\eta\|\nabla
u\|_{L^{2}}^{2}+\frac{C_{\eta}}{R^{2(1-s_{c})}}\left\\{[\rho(u,R)]^{\frac{2+\sigma}{2-\sigma}}+[\rho(u,R)]^{\sigma+1}\right\\},$
(4.30)

if $\frac{2-b}{3}<\sigma<2-b$.

  * (ii)

[Non-radial] for all
$u\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$
the following inequality holds

$\int_{|x|\geq R}|x|^{-b}|u|^{2\sigma+2}\,dx\leq\eta\|\nabla
u\|_{L^{2}}^{2}+\frac{C_{\eta}}{R^{2(1-s_{c})}}[\rho(u,R)]^{\frac{2-\sigma}{2-3\sigma}},$
(4.31)

if $\frac{2-b}{3}<\sigma<\min\\{2-b,\frac{2}{3}\\}$.

Next, we use the previous lemma to refine the estimate (4.11). Indeed, since
$\rho$ non-increasing, for all $u\in\dot{H}_{a}^{s_{c}}\subset L^{\sigma_{c}}$
we have

$\displaystyle\int_{|x|\geq R}|x|^{-2}|u|^{2}\,dx$
$\displaystyle=\sum_{j=0}^{\infty}\int_{2^{j}R\leq|x|\leq
2^{j+1}R}|x|^{-2}|u|^{2}\,dx\leq\sum_{j=0}^{\infty}\dfrac{1}{(2^{j}R)^{2}}\int_{2^{j}R\leq|x|\leq
2^{j+1}R}|u|^{2}\,dx$ (4.32)
$\displaystyle\leq\frac{1}{R^{2(1-s_{c})}}\rho(u,R)\sum_{j=0}^{\infty}\frac{1}{(2^{2(1-s_{c})})^{j}}\leq\frac{C}{R^{2(1-s_{c})}}\rho(u,R).$
(4.33)

Define the number $\gamma=\gamma(\sigma,N)$ by

$\displaystyle\displaystyle\gamma=\left\\{\begin{array}[]{cc}\dfrac{2+\sigma}{2-\sigma},&\mbox{
if $\frac{2-b}{3}<\sigma<2-b$ and $u\in\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$
is radial},\\\ &\\\ \dfrac{2-\sigma}{2-3\sigma},&\mbox{ if
$\frac{2-b}{3}<\sigma<\min\\{2-b,\frac{2}{3}\\}$ and
$u\in\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$}.\end{array}\right.$ (4.37)

Note that in both cases, we have

$\displaystyle\gamma>\sigma+1>1.$ (4.38)

Thus, inserting (4.30), (4.31), (4.33) in (4.11) we deduce

$\displaystyle 8\sigma
s_{c}\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|^{2}_{L^{2}}+z^{\prime\prime}_{R}(t)$
$\displaystyle-16(\sigma s_{c}+1)E[u_{0}]$ (4.39) $\displaystyle\leq$
$\displaystyle\eta\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|_{L^{2}}^{2}+\frac{C_{\eta}}{R^{2(1-s_{c})}}\left\\{[\rho(u(t),R)]^{\gamma}+\rho(u(t),R)\right\\}.$
(4.40)

Now, assume that $v\in C([0,\tau_{*}]:\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1})$
and fix $\tau_{0}\in[0,\tau_{*}]$. Choosing $\eta>0$ we can absorb the term
$\eta\|\mathcal{L}_{a}^{\frac{1}{2}}v(t)\|_{L^{2}}^{2}$ in the left hand side
of the above inequality, moreover integrating in $t$ from $0$ to $\tau$ and
then in $\tau$ from $0$ to $\tau_{0}$, we obtain111See the proof of estimate
(4.15) in [14] for more details.

$\displaystyle 4\sigma s_{c}\int_{0}^{\tau_{0}}(\tau_{0}-\tau)$
$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}^{2}\,d\tau+z_{R}(\tau_{0})$
(4.41) $\displaystyle\leq$ $\displaystyle
z_{R}(0)+\tau_{0}\left(z^{\prime}_{R}(0)+8\tau_{0}(\sigma
s_{c}+1)E[v_{0}]\right)$ (4.42)
$\displaystyle+\frac{1}{R^{2(1-s_{c})}}\int_{0}^{\tau_{0}}\int_{0}^{\tau}\left\\{[\rho(v(t),R)]^{\gamma}+\rho(v(t),R)\right\\}\,dtd\tau$
(4.43) $\displaystyle\leq$ $\displaystyle
cR^{2(1+s_{c})}\|v_{0}\|_{L^{\sigma_{c}}}^{2}+\tau_{0}\left(z^{\prime}_{R}(0)+8\tau_{0}(\sigma
s_{c}+1)E[v_{0}]\right)$ (4.44)
$\displaystyle+\frac{1}{R^{2(1-s_{c})}}\int_{0}^{\tau_{0}}\int_{0}^{\tau}\left\\{[\rho(v(t),R)]^{\gamma}+\rho(v(t),R)\right\\}\,dtd\tau.$
(4.45)

where we have used, from inequality (4.28), that

$\displaystyle z_{R}(0)\leq cR^{2}\int_{|x|\leq 4R}|v_{0}|^{2}\,dx\leq
cR^{2(1+s_{c})}\|v_{0}\|_{L^{\sigma_{c}}}^{2}.$ (4.46)

Next, observe that for all $R>0$ the definition of $\phi$ (see (4.6)) yields

$\displaystyle z_{R}(\tau_{0})$ $\displaystyle\geq\int_{R\leq|x|\leq
2R}|x|^{2}|v(\tau_{0})|^{2}\,dx\geq{R^{2}}\int_{R\leq|x|\leq
2R}|v(\tau_{0})|^{2}\,dx.$ (4.47)

Thereby, injecting (4.47) into (4.45) we have

$\displaystyle 4\sigma s_{c}\int_{0}^{\tau_{0}}(\tau_{0}-\tau)$
$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}^{2}\,d\tau+{R^{2}}\int_{R\leq|x|\leq
2R}|v(\tau_{0})|^{2}\,dx$ (4.48) $\displaystyle\leq$ $\displaystyle
cR^{2(1+s_{c})}\|v_{0}\|_{L^{\sigma_{c}}}^{2}+\tau_{0}\left(z^{\prime}_{R}(0)+8\tau_{0}(\sigma
s_{c}+1)E[v_{0}]\right)$ (4.49)
$\displaystyle+\frac{1}{R^{2(1-s_{c})}}\int_{0}^{\tau_{0}}\int_{0}^{\tau}\left\\{[\rho(v(t),R)]^{\gamma}+\rho(v(t),R)\right\\}\,dtd\tau.$
(4.50)

###  4.3. Preliminary lemmas

We now recall two results that we will be used in the proof of our main
theorems. The first one provides a uniform control of the $\rho$ norm and a
global dispersive estimate.

######  Proposition 4.3.

Let $0<b<\frac{3}{2}$. Assume either $\frac{2-b}{3}<\sigma<2-b$ and
$v_{0}\in\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})$
is radial or $\frac{2-b}{3}<\sigma<\min\\{2-b,\frac{2}{3}\\}$ and
$v_{0}\in\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1}$. Let $v\in
C\left([0,\tau_{\ast}]:\dot{H_{a}^{s_{c}}}(\mathbb{R}^{3})\cap\dot{H}_{a}^{1}(\mathbb{R}^{3})\right)$
be a solution to (1.1) such that the initial data $v(0)=v_{0}$ satisfies

$\tau_{\ast}^{1-s_{c}}\max\\{E[v_{0}],0\\}<1$ (4.51)

and

$\displaystyle M_{0}:=\frac{4\|v_{0}\|_{L^{\sigma_{c}}}}{K_{a}}\geq 2,$ (4.52)

where $K_{a}$ is a optimal constant in (3.39). Then, there exist universal
constants $C_{1},\alpha_{1},\alpha_{2}>0$ depending only on $N,\sigma$ and $b$
such that, for all $\tau_{0}\in[0,\tau_{\ast}]$, the following uniform
estimates hold

$\displaystyle\rho(v(\tau_{0}),M_{0}^{\alpha_{1}}\sqrt{\tau_{0}})\leq
C_{1}M_{0}^{2}$ (4.53)

and

$\displaystyle\int_{0}^{\tau_{0}}(\tau_{0}-\tau)\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}^{2}\,d\tau\leq
M_{0}^{\alpha_{2}}\tau_{0}^{1+s_{c}}.$ (4.54)

The proof of Proposition 4.3 follows exactly the same outline as the proof of
Proposition 3.2 in [15], just using the new estimate (4.50) and the
equivalence (1.6). The next result shows that restricting the initial data in
a neighborhood of the origin we can deduce a lower bound for its $L^{2}$ norm.

######  Proposition 4.4.

Under the same assumptions of Proposition 4.3, let

$\displaystyle\tau_{0}\in\left[0,\frac{\tau_{*}}{2}\right].$ (4.55)

Define
$\lambda_{v}(\tau)=\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|^{-\frac{1}{1-s_{c}}}_{L^{2}}$
and assume that

$\displaystyle
E[v_{0}]\leq\frac{\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau_{0})\|^{2}_{L^{2}}}{4}=\frac{1}{4\lambda_{v}^{2(1-s_{c})}(\tau_{0})}.$
(4.56)

Then, there exist universal constants $C_{2},\alpha_{3}>0$ depending only on
$N,\sigma$ and $b$ such that if

$\displaystyle F_{*}=\frac{\sqrt{\tau_{0}}}{\lambda_{v}(\tau_{0})}\,\,\,\mbox{
and }\,\,\,D_{*}=M_{0}^{\alpha_{3}}\max[1,F_{*}^{\frac{1+s_{c}}{1-s_{c}}}],$
(4.57)

then

$\displaystyle\frac{1}{\lambda_{v}^{2s_{c}}(\tau_{0})}\int_{|x|\leq
D_{*}\lambda_{v}(\tau_{0})}|v_{0}|^{2}\,dx\geq C_{2}.$ (4.58)

###### Proof.

Given $v(\tau_{0})$ we define

$\displaystyle
w(x)=\lambda_{v}^{\frac{2-b}{2\sigma}}(\tau_{0})v(\lambda_{v}(\tau_{0})x,\tau_{0}),$
(4.59)

where
$\lambda_{v}(\tau_{0})=\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau_{0})\|^{-\frac{1}{1-s_{c}}}_{L^{2}}$,
and

$\displaystyle
J_{*}=C_{*}\max[M_{0}^{\alpha_{1}}F_{*},M_{0}^{\frac{\gamma}{1-s_{c}}}],$
(4.60)

for $C_{*}>1$ large enough. Following the same arguments as in the proof of
Proposition 3.3 in [15], there exists a constant $c>0$ such that

$\displaystyle c\leq\int_{|y|\leq
2J_{*}}|w|^{2}\,dx=\frac{1}{\lambda_{v}^{2s_{c}}(\tau_{0})}\int_{|x|\leq
2J_{*}\lambda_{v}(\tau_{0})}|v(\tau_{0})|^{2}\,dx.$ (4.61)

Now, we want to verify that the above estimate is also verified for the
initial data $v_{0}$. Indeed, for $R>0$, applying Hölder’s inequality and
equivalence (1.6) we have

$\displaystyle\left|\frac{d}{d\tau}\int\varphi_{R}|v|^{2}\,dx\right|=2\left|\textit{Im}\left(\int\nabla\varphi_{{R}}\cdot\nabla
v\overline{v}\,dx\right)\right|$
$\displaystyle\leq\frac{c}{{R}}\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}\left(\int_{{R}\leq|x|\leq
2{R}}|v(\tau)|^{2}\,dx\right)^{\frac{1}{2}}$ (4.62) $\displaystyle\leq
c\frac{\rho(v(\tau),{R})^{\frac{1}{2}}}{{{R}}^{1-s_{c}}}\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}.$
(4.63)

Next, integrating the above inequality from $0$ to
$\tau_{0}\in[0,\frac{\tau_{*}}{2}]$ and using Cauchy-Schwarz inequality and
(4.54) we get

$\displaystyle\left|\int\varphi_{{R}}|v(\tau_{0})|^{2}\,dx\right.$
$\displaystyle\left.-\int\varphi_{{R}}|v_{0}|^{2}\,dx\right|$ (4.64)
$\displaystyle\leq
c\frac{\rho(v(\tau),{R})^{\frac{1}{2}}}{{R}^{1-s_{c}}}\int_{0}^{\tau_{0}}\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}\,d\tau\leq
c\frac{\rho(v(\tau),{R})^{\frac{1}{2}}}{{R}^{1-s_{c}}}\left(\tau_{0}\int_{0}^{\tau_{0}}\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}^{2}\,d\tau\right)^{\frac{1}{2}}$
(4.65) $\displaystyle\leq
c\frac{\rho(v(\tau),{R})^{\frac{1}{2}}}{{R}^{1-s_{c}}}\left(\int_{0}^{2\tau_{0}}(2\tau_{0}-\tau)\|\mathcal{L}_{a}^{\frac{1}{2}}v(\tau)\|_{L^{2}}^{2}\,d\tau\right)^{\frac{1}{2}}\leq
c\rho(v(\tau),{R})^{\frac{1}{2}}M_{0}^{\frac{\alpha_{2}}{2}}{R}^{2s_{c}}\left(\frac{\sqrt{\tau_{0}}}{R}\right)^{1+s_{c}}.$
(4.66)

Given $0<\varepsilon\ll 1$, define

$\displaystyle
D_{\varepsilon}=\frac{1}{\varepsilon}\max\left\\{F_{*},F_{*}^{\frac{1+s_{c}}{1-s_{c}}}\right\\}\max\left\\{M_{0}^{\alpha_{1}},M_{0}^{\frac{2+\alpha_{2}}{2(1-s_{c})}}\right\\},$
(4.67)

and for $D\geq D_{\varepsilon}$ consider

$\displaystyle\widetilde{R}=\widetilde{R}(D,\tau_{0})=D\lambda_{v}(\tau_{0}).$
(4.68)

From (4.57) and (4.68), we get

$\displaystyle\tilde{R}=D\lambda_{v}(\tau_{0})\geq
D\frac{\lambda_{v}(\tau_{0})}{\sqrt{\tau_{0}}}\sqrt{\tau_{0}}\geq\frac{D}{F_{*}}\sqrt{\tau},\,\,\,\mbox{for
all}\,\,\,\tau\in[0,\tau_{0}],$ (4.69)

and thus, using (4.67), the monotonicity of $\rho$ and (4.53), we have

$\displaystyle\rho(v(\tau),\tilde{R})\leq\rho(v(\tau),M_{0}^{\alpha_{1}}\sqrt{\tau})\leq
cM_{0}^{2},\,\,\,\mbox{for all}\,\,\,\tau\in[0,\tau_{0}].$ (4.70)

Thereby, from (4.66)

$\displaystyle\frac{1}{\lambda^{2s_{c}}_{v}(\tau_{0})}\left|\int\varphi_{\tilde{R}}|v(\tau_{0})|^{2}\,dx-\int\varphi_{\tilde{R}}|v_{0}|^{2}\,dx\right|\leq
cM_{0}^{1+\frac{\alpha_{2}}{2}}D^{2s_{c}}\left(\frac{F_{*}}{D}\right)^{1+s_{c}}=cM_{0}^{\frac{2+\alpha}{2}}\frac{F_{*}^{1+s_{c}}}{D^{1-s_{c}}}.$
(4.71)

Finally, since $D\geq
D_{\varepsilon}\geq\frac{1}{\varepsilon}F_{*}^{\frac{1+s_{c}}{1-s_{c}}}M_{0}^{\frac{2+\alpha_{2}}{2(1-s_{c})}}$
by (4.67), we obtain

$\displaystyle\frac{1}{\lambda^{2s_{c}}_{v}(\tau_{0})}\left|\int\varphi_{\tilde{R}}|v(\tau_{0})|^{2}\,dx-\int\varphi_{\tilde{R}}|v_{0}|^{2}\,dx\right|<\varepsilon$
(4.72)

which concludes the proof of Proposition 4.4.

∎

###  4.4. Existence of blow-up solutions and lower bound for the blow-up rate

We are now in position to prove our main results. We start with the existence
of blow-up solutions.

###### Proof of Theorem 1.2.

Let $u\in C([0,+\infty):\dot{H}_{a}^{s_{c}}\cap\dot{H}_{a}^{1})$ be a global
solution with $E[u_{0}]\leq 0$. In view of the Gagliardo-Nirenberg type
inequality (3.39) and $a>0$, we deduce that

$\displaystyle 0\geq
E[u_{0}]\geq\frac{1}{2}\|\mathcal{L}_{a}^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2}\left[1-\left(\frac{\|u_{0}\|_{L^{\sigma_{c}}}}{K_{a}}\right)^{2\sigma}\right].$
(4.73)

and therefore the assumptions of Proposition 4.3 are satisfied which implies

$\displaystyle\int_{0}^{\tau_{*}}(\tau_{*}-\tau)\|\mathcal{L}_{a}^{\frac{1}{2}}u(\tau)\|_{L^{2}}^{2}\,d\tau\leq
C_{u_{0}}\tau_{*}^{1+s_{c}},\,\,\mbox{for all}\,\,\tau_{*}>0.$ (4.74)

Define
$\lambda_{u}(\tau)=\|\mathcal{L}_{a}^{\frac{1}{2}}u(\tau)\|^{-\frac{1}{1-s_{c}}}_{L^{2}}$.
By previous inequality there exists a sequence $\tau_{n}\nearrow+\infty$ such
that

$\displaystyle\lambda_{u}(\tau_{n})\geq
C_{u_{0}}\sqrt{\tau_{n}},\,\,\,\mbox{for all}\,\,\,n\in\mathbb{N},$ (4.75)

and in particular $\lambda_{u}(\tau_{n})\nearrow+\infty$. Therefore, since
$u_{0}\in\dot{H}_{a}^{s_{c}}\subset L^{\sigma_{c}}$, the relation (4.29)
implies

$\displaystyle\lim_{n\to+\infty}\frac{1}{(D\lambda_{u}(\tau_{n}))^{2s_{c}}}\int_{|x|\leq
D\lambda_{u}(\tau_{n})}|u_{0}|^{2}\,dx=0,\,\,\,\mbox{for all}\,\,\,D\geq 0.$
(4.76)

On the other hand, we can apply Proposition 4.4 at time $\tau_{n}$, for all
$n\in\mathbb{N}$. Indeed, since
$F_{n}=\frac{\sqrt{\tau_{n}}}{\lambda_{u}(\tau_{n})}\leq C_{u_{0}}$, there
exists $D_{*}$, independent of $n$, such that

$\displaystyle\frac{1}{\lambda_{u}^{2s_{c}}(\tau_{n})}\int_{|x|\leq
D_{*}\lambda_{u}(\tau_{n})}|u_{0}|^{2}\,dx\geq C_{2}>0,$ (4.77)

reaching a contradiction with (4.76). Therefore the solution cannot exist for
all positive time completing the proof. ∎

Next, we refine the analysis to show a lower bound for the blow-up rate of the
critical Sobolev norm.

###### Proof of Theorem 1.3.

Let $u_{0}\in\dot{H_{a}^{s_{c}}}\cap\dot{H}_{a}^{1}$ so that the maximal time
of existence $T^{\ast}>0$ of the corresponding solution $u$ to (1.1) is
finite. The inequality (3.40) yields

$\displaystyle\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|_{L^{2}}\geq\frac{c}{(T^{*}-t)^{\frac{1-s_{c}}{2}}},\,\,\,\textit{for
all}\,\,\,t\in[0,T^{*}).$ (4.78)

For $\sigma_{c}=\frac{6\sigma}{2-b}$, we shall prove that

$\displaystyle\|u(t)\|_{L^{\sigma_{c}}}\geq|\log(T^{\ast}-t)|^{\alpha},\,\,\,as\,\,\,t\to
T^{\ast},$ (4.79)

for a universal constant $\alpha=\alpha(N,\sigma,b)>0$, which implies the same
lower bound for the critical Sobolev norm $\dot{H}_{a}^{s_{c}}$. The proof
follows the ideas introduced by Merle and Raphäel [40] and is very similar to
the proof of Theorem 1.2 in [14]. Below we sketch the main steps. First, for
$t$ close enough to $T^{*}$, define the renormalization of $u$ given by

$\displaystyle
v^{(t)}(x,\tau)=\lambda_{u}^{\frac{2-b}{2\sigma}}(t)\bar{u}(\lambda_{u}(t)x,t-\lambda^{2}_{u}(t)\tau),$
(4.80)

where
$\lambda_{u}(t)=\|\mathcal{L}_{a}^{\frac{1}{2}}u(t)\|^{-\frac{1}{1-s_{c}}}_{L^{2}}$.
One can check that $v^{(t)}(\tau)\in
C([0,\tau^{(t)}]:\dot{H}^{s_{c}}\cap\dot{H}^{1})$ is a solution to (1.1) with
$\tau^{(t)}={t}/{\lambda_{u}^{2}(t)}$ and also Propositions 4.3 and 4.4 can be
applied to $v^{(t)}(\tau_{0})$ for any
$\tau_{0}\in\left[0,{1}/{\lambda_{u}(t)}\right]$ (see [14, Lemma 6.1]). From
now on, to simplify the notation, we write $v=v^{(t)}$.

Define $N(t)=-\log\lambda_{u}(t)$. Since
$\|u(t)\|_{L^{\sigma_{c}}}=\|v(0)\|_{L^{\sigma_{c}}}$, in view of inequality
(4.78), the problem is reduced to find $\alpha=\alpha(N,\sigma,b)>0$ such
that, for $t$ close enough to $T^{\ast}$, we have

$\displaystyle\|v(0)\|_{L^{\sigma_{c}}}\geq[N(t)]^{\alpha}.$ (4.81)

Recalling the constants $K_{a}$ and $\alpha_{2}$ given in (3.39) and (4.54),
respectively, we define

$\displaystyle
L(t)=[100[M(t)]^{\alpha_{2}}]^{\frac{1}{2(1-s_{c})}},\,\,\,\mbox{where}\,\,\,M(t)=\frac{4\|v(0)\|_{L^{\sigma_{c}}}}{K_{a}}\geq
2.$ (4.82)

If $L(t)\geq e^{\frac{\sqrt{N(t)}}{2}}$, then it is easy to see that (4.81)
holds with $\alpha=1$. On the other hand, if $L(t)<e^{\frac{\sqrt{N(t)}}{2}}$,
we pick times $\tau_{i}\in[0,e^{i}]$, for all $i\in[\sqrt{N(t)},N(t)]$, such
that

$\displaystyle F(\tau_{i})\leq L(t)\,\,\,\,\mbox{ and
}\,\,\,\,\,\frac{1}{10L(t)}e^{\frac{i-1}{2}}\leq\lambda_{v}(\tau_{i})\leq\frac{10}{L(t)}e^{\frac{i}{2}},$
(4.83)

where $F(\tau)=\sqrt{\tau}\left/\right.\lambda_{v}(\tau)$ (see the proof of
inequality (6.10) in [14]). Moreover, for this choice of $\tau_{i}$, there
exist universal positive constants $\alpha_{4}(N,\sigma,b)$ and
$c_{4}(N,\sigma,b)$ such that

$\displaystyle\int_{\mathcal{C}_{i}}|v(0)|^{\sigma_{c}}\,dx\geq\frac{c_{4}}{[M(t)]^{\alpha_{4}s_{c}\sigma_{c}}}$
(4.84)

where

$\displaystyle\mathcal{C}_{i}=\left\\{x\in\mathbb{R}^{N};\,\frac{\lambda_{v}(\tau_{i})}{[M(t)]^{\alpha_{4}}}\leq|x|\leq[M(t)]^{\alpha_{4}}\lambda_{v}(\tau_{i})\right\\},$
(4.85)

(see the proof of inequality (6.12) in [14]).

To finish the proof, let $p(t)>1$ an integer such that

$\displaystyle 10^{3}[M(t)]^{2\alpha_{4}}\leq e^{\frac{p(t)}{2}}\leq
10^{7}[M(t)]^{2\alpha_{4}}.$ (4.86)

If $p(t)\geq\sqrt{N(t)}$, then simple computations imply (4.81) with
$\alpha=1/4\alpha_{4}$. In the remaining case $p(t)<\sqrt{N(t)}$ consider
$(i,i+p(t))\in[\sqrt{N(t)},N(t)]\times[\sqrt{N(t)},N(t)]$. From(4.83) and
(4.86) we get

$\displaystyle\lambda_{v}(\tau_{i+p(t)})$
$\displaystyle\geq\frac{e^{\frac{i+p(t)-1}{2}}}{10L(t)}\geq\frac{10^{3}[M(t)]^{2\alpha_{4}}e^{\frac{i-1}{2}}}{10L(t)}\geq[M(t)]^{2\alpha_{4}}\frac{10e^{\frac{i}{2}}}{L(t)}\geq[M(t)]^{2\alpha_{4}}\lambda_{v}(\tau_{i}),$
(4.87)

which implies that the annuli $\mathcal{C}_{i}$ and $\mathcal{C}_{i+p(t)}$
given by (4.85) are disjoints. In this case, for $t$ close enough to
$T^{\ast}$, there are at least $\frac{N(t)}{10p(t)}$ disjoint annuli, thus
(4.84) yields

$\displaystyle\|v(0)\|_{L^{\sigma_{c}}}^{\sigma_{c}}\geq\sum_{k=0}^{\frac{N(t)}{10p(t)}}\int_{\mathcal{C}_{kp(t)+\sqrt{N(t)}}}|v(0)|^{\sigma_{c}}\,dx\geq\frac{c_{4}}{[M(t)]^{\alpha_{4}\sigma_{c}s_{c}}}\frac{\sqrt{N(t)}}{10}=C\frac{\sqrt{N(t)}}{\|v(0)\|_{L^{\sigma_{c}}}^{\alpha_{4}\sigma_{c}s_{c}}},$
(4.88)

which implies (4.81), finishing the proof of Theorem 1.3. ∎

Acknowledgments. L.C. was partially supported by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - CAPES. M.C. was partially
supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
CAPES and Fundação de Amparo a Pesquisa do Estado do Piauí - FAPEPI. L.G.F.
was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - CAPES, Conselho Nacional de Desenvolvimento Científico e
Tecnológico - CNPq and Fundação de Amparo a Pesquisa do Estado de Minas Gerais
- FAPEMIG.

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LUCCAS CAMPOS, Department of Mathematics, UFMG, Brazil

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MYKAEL A. CARDOSO, Department of Mathematics, UFPI, Brazil

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LUIZ G. FARAH, Department of Mathematics, UFMG, Brazil

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