This paper studies the global behavior to 3D focusing nonlinear Schrodinger equation (NLS), the scaling index here is (0＜sc＜1), which is the mass-supercritical and energy-subcritical, and we prove under some condition the solution u(t) is globally well-posed and scattered. We also show that the solution “blows-up in finite time” if the solution is not globally defined, as t→T we can provide a depiction of the behavior of the solution, where T is the “blow-up time”.
NLS; Blows-Up in Finite Time; Supremum; Precompactness1. Introduction
Consider the Cauchy problem for the nonlinear Schrö- dinger equation (NLS) in dimensions d = 3:
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
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”:
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
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;
and there exist such that
.
Theorem 1.1. Let, and let be the corresponding solution to (1.1) in H1. Suppose
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
or there exists a sequence of times such that for an absolute constant
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
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;
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) Let us assume that there exists a constant such that. Then
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]:
where c is independent of R > 0. We recall the generally usable symmetric functions and for any function
(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
By (2.3), the selection of and mass conservation;
where c1 in the definition of has been selected to obtain the factor here. By Sobolev embedding, (1.10), and the selected
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
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
We must also need the Kato inhomogeneous Strichartz estimate [12].
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
where
Apply the Strichartz estimates to (3.3), to get
As above inequality get the claim.
4. Some Lemma4.1. Here We Discuss the Precompactness of the Flow Implies Regular Localization
Let u be a solution to (1.1) such that
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,
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;
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
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;
Set Observe that for and
, we have, and thus
(4.6), (4.5) give
We now obtain an upper bound for and a lower bound for
Hence, by (4.5) we have
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
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
where. We have also the bound for all t;
The hypothesis here is except if In fact,
Theorem 4.3. Assume satisfies,
and
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:
By calculation, we have the local Virial identity
Since is radial we have
where
Thus, we obtain
Now discuss for R chosen appropriate large and selection time interval where. By (4.15) and (4.11) we have
Set in (4.2), , such that
Choosing Then (4.16), (4.17) and
(4.18) imply that for all,
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
On the other hand, for all, by (4.10) and (4.14), we have
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
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is a complex-valued function in 
. The initial-value problem 
is locally well-posed in
.
where
where
where
.
, then u satisfies
is a solution.
, 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 [
.
such that if 
, then the solution 
is globally defined and scattered.
and
are also scale-invariant (See also [
then u solves (1.1) as long as
solves the nonlinear elliptic equation
. The solution of minimal mass is denoted by 
and for the properties of 
see [3,5,6].
, solutions to (1.1) globally exist if u0 satisfies;
such that
.
, and let 
be the corresponding solution to (1.1) in H1. Suppose
then u scatters in H1.
Then either there is a non-absolute 
constant such that, as 
such that for an absolute constant 
, and the concentration in (1.8) satisfies 
(For more additional information see [8-10]).
be the free Schrödinger propagator, and let
, with 
be linear equation, a solution in physical space, is given by:
and in frequency space
.
be a radial function, so that, 
for 
and 
for
, Define the inner and outer spatial localizations of 
at radius 
as
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;
. Let
, (Where c1 and c2 are absolute constants), and 
as characterized in the paragraph above.
such that
such that
. Then
is a stance function such that
.
as 
by energy conservation, we have
Thus, for t to be large enough to close to T
and mass conservation;
has been selected to obtain the factor 
here. By Sobolev embedding, (1.10), and the selected 
has been selected to obtain the factor 
here. Bring together (2.5), (2.6), and (2.7), to obtain
by (2.8)
for which at least 
of this supremum is attained. Thus,
, 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 
along a sequence of times 
Assume the spherical annulus;
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,
.
. 
.
is not bounded, then there exists a sequence of times 
such that 
Since
, we have (1.8) in Theorem 1.2;
, for some c*, as t ® Twe have (2.2) of Prop. 2.1. Since
, we have
is 
Strichartz admissible if
, with
, 
and
. And the pair 
is 
-passable if
, 
, 
or
.
the Hölder conjugates of q and r consecutive (i.e.
).
is the Hölder dual to
. Also define
.
, we will write 
or
.
. There is 
such that if
, then u solving (1.1) is global (in
) and
,
.
).
.
where
is a contraction on N. 
is global with globally finite 
Strichartz norm 
and a uniformly bounded H1 norm 
then 
scatters in H1 as
.
such that
resolves the integral equation
above inequality get the claim. 
there exist R > 0 so that 
for all 
and a sequence of times 
and by changing the variables,
, such that 
in H1, by (4.3),
The proof is complete.
, such that 
and K such as in (4.1) is precompact in H1, for some continuous function 
then;
, such that 
for some ε0 > 0. Retaining generality, we assume 
For R > 0, let
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:
.
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:
we have 
and 
, there exists
, such that for any 
let 
be such that 
for
, 
for
, 
, 
and 
for
. Let 
for 
and 
For R > 0, set 
Let 
be the truncation center of mass given by
, where
for
. By the zero momentum property
.
Observe that for 
and
, we have
, and thus
and a lower bound for 
, we divide 
as
and by (4.5)
.
, we can get
and use 
to obtain
we have
) take
, as 
since 
we get a contradiction. 
. We have also the bound for all t;
except if 
In fact, 
satisfies
,
is precompact in H1. Then 
.
be redial with
.
is radial we have
for R chosen appropriate large and selection time interval 
where
. By (4.15) and (4.11) we have
in (4.2), 
, such that 
Then (4.16), (4.17) and
,
such that for all 
we have 
with 
By taking R =
, we obtain that (4.18) holds for all 
. Integrating (4.19) over 
we obtain
, by (4.10) and (4.14), we have
and 
are constant depending only on
, and
.
and setting
, we obtain a contradiction except if
, which implies 
