Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D ()
1. Introduction
Consider the Cauchy problem for the nonlinear Schrö- dinger equation (NLS) in dimensions d = 3:
(1.1)
where 
is a complex-valued function in 
. The initial-value problem 
is locally well-posed in
.
In this paper we will study the focusing (NLS) problem, which is the mass-supercritical and energy-subcritical, where 
The Equation (1.1) has mass 
where
Energy 
where
and Momentum 
where
.
If
, then u satisfies
(1.2)
Equation (1.2) is said to be the Virial identity.
The Equation (1.1) has the scaling:
and also this scaling is a solution if 
is a solution.
Moreover, u0 is a solution that is globally defined by u, if it is globally defined
, and it does scatter (See [1,2]). We say the solution “blows-up in finite time”. If the solution is not globally defined, as
, we can provide a depiction of the behavior of the solution, where T is the “blow-up time”. It follows from the H1 local theory optimized by scaling, that if blow-up in finite-time T > 0 happens, (see [3] or [4]), then there is a lower-bound on the “blow-up rate”:
(1.3)
for some constant c. Thus, to prove global presence, it suffices to prove a global axiomatic bound on
.
From the Strichartz estimates, there is a constant 
such that if 
, then the solution 
is globally defined and scattered.
Note that the quantities 
and
are also scale-invariant (See also [5]).
Let 
then u solves (1.1) as long as
solves the nonlinear elliptic equation
(1.4)
Equation (1.4) has an infinite number of solutions in
. The solution of minimal mass is denoted by 
and for the properties of 
see [3,5,6].
Under the condition
, solutions to (1.1) globally exist if u0 satisfies;
(1.5)
and there exist 
such that
.
Theorem 1.1. Let
, and let 
be the corresponding solution to (1.1) in H1. Suppose
(1.6)
If 
then u scatters in H1.
The argument of [6] in the radial case followed a strategy introduced by [7] for proving global well-posedness and scattering for the focusing energy-critical NLS. The beginning used a contradiction to the argument: suppose the sill for scattering is strictly below that claimed. This uniform localization enabled the use of a local Virial identity to be established, with the support of the sharp Gagliardo-Nirenberg inequality, an accurately positive lower bound on the convexity (in time) of the local mass of uc Mass conservation is then violated at enough large time.
We show in this paper, that the above program carries over to the non-radial setting with the extension of two key components.
Theorem 1.2. Suppose the radial H1 solution u to (1.1) blows-up at time 
Then either there is a non-absolute 
constant such that, as 
(1.7)
or there exists a sequence of times 
such that for an absolute constant 
(1.8)
From (1.3), we have that the concentration in (1.7) satisfies
, and the concentration in (1.8) satisfies 
(For more additional information see [8-10]).
Notation
Let 
be the free Schrödinger propagator, and let
, with 
be linear equation, a solution in physical space, is given by:
and in frequency space
In particular, they save the Farewell homogeneous Sobolev norms and obey the dispersive inequality
(1.9)
For all times
.
Let 
be a radial function, so that, 
for 
and 
for
, Define the inner and outer spatial localizations of 
at radius 
as
Let 
be a radial function so that,
for 
and 
for 
then
, and define the inner and outer indecision localizations at radius 
of u1 as
and
(the 
and 
radii are chosen to be consistent with the assumption
, since
. In reality, this is for suitability only; the argument is easily proper to the case where 
is any number
). We note that the indecision localization of 
is inaccurate, though decisively we have;
(1.10)
2. Proof of Theorem 1.2
In this section we discuss a proof of Theorem (1.2).
Proposition 2.1. Let u be an H1 radial solution to (1.1) that blows-up in finite
. Let
and
, (Where c1 and c2 are absolute constants), and 
as characterized in the paragraph above.
1) There exists an absolute constant 
such that
(2.1)
2) Let us assume that there exists a constant 
such that
. Then
(2.2)
for some absolute constant c > 0, where 
is a stance function such that
We recall, an “exterior” estimate, usable to radially symmetric functions only, originally due to [11]:
(2.3)
where c is independent of R > 0. We recall the generally usable symmetric functions and for any function 
(2.4)
(2.3), (2.4) are Gagliardo-Nirenberg estimates for functions on
.
Proof of Prop 2.1: Since by (1.3), 
as 
by energy conservation, we have
Thus, for t to be large enough to close to T
(2.5)
By (2.3), the selection of 
and mass conservation;
(2.6)
where c1 in the definition of 
has been selected to obtain the factor 
here. By Sobolev embedding, (1.10), and the selected 
(2.7)
where c2 in the definition of 
has been selected to obtain the factor 
here. Bring together (2.5), (2.6), and (2.7), to obtain
(2.8)
By (2.8) and (2.4), we obtain (2.1), completing the proof of part (1) of the proposition.
To prove part (2), we assume 
by (2.8)
There exists 
for which at least 
of this supremum is attained. Thus,
where we used Hölder’s inequality in the last step. By the selected
, we obtain (2.2). To complete the proof, it keeps to obtain the remind control on 
which will be a consequence of the radial supposition and the supposed bound 
Assume 
along a sequence of times 
Assume the spherical annulus;
And inside A place 
disjoint balls, at radius 
both the radius
, centered on the sphere. By the radiality supposition, on all ball B, we have
, and hence on the annulus A,
.
which contradicts the assumption
. 
We now point out how to obtain Theorem 1.2 as a consequence.
Proof of Theorem 1.2. By part (1) of Prop. 2.1 and the standard convolution inequality:
.
If 
is not bounded, then there exists a sequence of times 
such that 
Since
, we have (1.8) in Theorem 1.2;
on the other hand, if
, for some c*, as t ® Twe have (2.2) of Prop. 2.1. Since
, we have
which gives (1.7) in Theorem 1.2. 
3. Strichartz Estimates
In this section we show local theory and Strichartz estimates.
Strichartz Type Estimates
We say the pair 
is 
Strichartz admissible if
, with
, 
and
. And the pair 
is 
-passable if
, 
, 
or
.
As habitual we denote by 
the Hölder conjugates of q and r consecutive (i.e.
).
Let
We consider dual Strichartz norms. Let
where 
is the Hölder dual to
. Also define
The Strichartz estimates are:
and
.
By bring together Sobolev embedding with the Strichartz estimates, we obtain
and
(3.1)
We must also need the Kato inhomogeneous Strichartz estimate [12].
. (3.2)
To point out a restriction to a time subinterval 
, we will write 
or
.
Proposition 3.1 Assume
. There is 
such that if
, then u solving (1.1) is global (in
) and
,
.
(Observe that, by the Strichartz estimates, the assumptions are satisfied if
).
Proof. Define
.
Applying the Strichartz estimates, we obtained
and
We apply the Hölder inequalities and fractional Leibnitz [13] to get
Let
Then 
where
and 
is a contraction on N. 
Proposition 3.2. If 
is global with globally finite 
Strichartz norm 
and a uniformly bounded H1 norm 
then 
scatters in H1 as
.
Meaning that there exist 
such that
Proof. Since 
resolves the integral equation
we have
(3.3)
where
Apply the Strichartz estimates to (3.3), to get
As 
above inequality get the claim. 
4. Some Lemma
4.1. Here We Discuss the Precompactness of the Flow Implies Regular Localization
Let u be a solution to (1.1) such that
(4.1)
is precompact in H1. Then for each 
there exist R > 0 so that 
for all 
We proof (4.2) by contradiction, there exists 
and a sequence of times 
and by changing the variables,
(4.3)
Since K is precompact, there exists
, such that 
in H1, by (4.3),
Which is a contradiction with the fact that 
The proof is complete.
Lemma 4.1. Let u be a solution of (1.1) defined on
, such that 
and K such as in (4.1) is precompact in H1, for some continuous function 
then;
(4.4)
Proof. Suppose that (4.4) does not hold. Then there exists a sequence
, such that 
for some ε0 > 0. Retaining generality, we assume 
For R > 0, let
i.e. 
is the first time when 
arrives at the boundary of the ball of radius R. By continuity of
, the value 
is well-defined. Furthermore, the following hold:
1) 
2) 
3)
.
Let 
and 
We note that
, which combined with
, gives
. Since 
and
, we have 
Thus 
We can disregard
. We will concentrate our work on the time interval
, and we will use in the proof:
1) 
we have 
2) 
3) 
and 
By the precompactness of K and (4.2) it follows that for any
, there exists
, such that for any 
(4.5)
We will select ε later; for 
let 
be such that 
for
, 
for
, 
, 
and 
for
. Let 
Then 
for 
and 
For R > 0, set 
Let 
be the truncation center of mass given by
Then
, where
Observe that 
for
. By the zero momentum property
.
Thus,
By Cauchy-Schwarz, we obtain;
(4.6)
Set 
Observe that for 
and
, we have
, and thus
(4.6), (4.5) give
(4.7)
We now obtain an upper bound for 
and a lower bound for 
Hence, by (4.5) we have
(4.8)
For
, we divide 
as
To deduce the expression for I, we observed that 
And use (4.5) to obtain 
For II we first observe that,
and thus 
We rewrite II as
Trivially, 
and by (4.5)
.
Thus,
Taking
, we can get
(4.9)
Combining (4.7), (4.8), and (4.9), we have
Suppose 
and use 
to obtain
Since 
we have
(Assume
) take
, as 
since 
we get a contradiction. 
4.2. We Now Prove the Following Rigidity Theorem
Lemma 4.2. If (1.5) and (1.6) hold, then for all t
(4.10)
where
. We have also the bound for all t;
(4.11)
The hypothesis here is 
except if 
In fact, 
Theorem 4.3. Assume 
satisfies
,
(4.12)
and
(4.13)
Let u be the global H1 solution of (1.1) with initial data u0 and assume that 
is precompact in H1. Then 
.
Proof. Let 
be redial with
.
For R > 0, we define
Then
By the Hölder inequality:
(4.14)
By calculation, we have the local Virial identity
Since 
is radial we have
(4.15)
where
Thus, we obtain
(4.16)
Now discuss 
for R chosen appropriate large and selection time interval 
where
. By (4.15) and (4.11) we have
(4.17)
Set 
in (4.2), 
, such that 
(4.18)
Choosing 
Then (4.16), (4.17) and
(4.18) imply that for all
,
(4.19)
By Lemma 4.1, there exists 
such that for all 
we have 
with 
By taking R =
, we obtain that (4.18) holds for all 
. Integrating (4.19) over 
we obtain
(4.20)
On the other hand, for all
, by (4.10) and (4.14), we have
(4.21)
Combining (4.20) and (4. 21), we obtained
It is important to mention that 
and 
are constant depending only on
, and
.
Putting 
and setting
, we obtain a contradiction except if
, which implies 
