﻿Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D

Vol.3 No.1A(2013), Article ID:27535,8 pages DOI:10.4236/apm.2013.31A023

Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D

Mujahid Abd Elmjed M-Ali1, Amadu Fullah Kamara2

1Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan

2Department of Mathematics, Faculty of Pure and Applied Sciences, Fourah Bay College, University of Sierra Leone, Freetown, Sierra Leone

Received October 2, 2012; revised November 5, 2012; accepted November 13, 2012

Keywords: NLS; Blows-Up in Finite Time; Supremum; Precompactness

ABSTRACT

This paper studies the global behavior to 3D focusing nonlinear Schrödinger equation (NLS), the scaling index here is, which is the mass-supercritical and energy-subcritical, and we prove under some condition the solution 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 we can provide a depiction of the behavior of the solution, where T is the “blow-up time”.

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

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,

.

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

(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

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