Global Self-similar Solutions of a Class of Nonlinear Schrödinger Equations

For a certain range of the value in the nonlinear term , in this paper we mainly study the global existence and uniqueness of global self-similar solutions to the Cauchy problem for some nonlinear Schrodinger equations using the method of harmonic analysis.


Introduction
This paper is devoted to study the initial value problem for the nonlinear Schr ödinger equation where λ ∈ R, p > 0 are constants, m ≥ 1 is a positive integer, u t, x is a complex-valued function defined in R × R n , the initial value f x is a complex-valued function defined in R n .When m 1, 1.1 is a classical nonlinear Schr ödinger equation of the second order: For the Cauchy problem of 1.2 , the existence and the scattering theorem of solutions have been studied extensively by many authors with various methods and techniques 1-5 , Cazenave and Weissler 6 also Ribaud and Youssfi 7 established existence of global selfsimilar solutions by introducing new function space.When m ≥ 1, Pecher and von Wahl 8 established the existence of classical solution of the Cauchy problem 1.1 employing the related L p estimate of the elliptic equation and the compact method.Sj ölin and Sj ögren in 9, 10 recently discussed the local smooth effect of solutions of the Cauchy problem 1.1 applying the Strichartz estimate in the nonhomogeneous Sobolev space.In 11 , by constructing a timeweighted space and using the contractive mapping method, the author established global solutions of the problem 1.1 in the possible range of p, and further got the continuous dependence of the solution on the initial value together with its strong decay estimate.In addition, there are also much more efforts working for studying the scattering theorem and the existence of global strong solutions of the problem 1. 1 12, 13 .In this paper, we mainly investigate the existence of global self-similar solutions basing on the existence and uniqueness of global solution for the Cauchy problem 1.1 .
In the following discussion, we suppose that p satisfies where p 0 is a positive solution of the equation nx 2 n − 2m x − 4m 0, which also can be interpreted as a positive integer satisfying p 2 / p 1 np/2m.In fact, condition 1.3 is equivalent to For p which satisfies 1.3 or 1.4 , let then we may introduce our work space X as follows.Let X be a space consisting of all Bochner measurable functions: In order to prove our main result, we should transform the Cauchy problem 1.1 into the following equivalent integral equation: where S t e i −Δ m t F −1 e i|ξ| 2m t F• is a free group produced by the free Schr ödinger equation iv t −Δ m v 0. Besides, we denote, respectively, by F and F −1 the Fourier transformation and the inverse Fourier transformation with respect to the space variables.
For convenience, we provide some useful symbols.L r R n denotes the usual Lebesgue space on R n with the norm • r , 1 ≤ r ≤ ∞.For any q > 0, q stands for the dual to q, that is, Yaojun YE 3 1/q 1/q 1. C which may be different when appeared every time is a constant depending on the dimension or any other constant.
In the end, we will review the definition of the homogeneous Besov space, the details on the properties, and the embedding theorems reference 1, 14 .
Let ϕ ξ ∈ S be a symmetric Bump function with real values satisfying the conditions are also symmetric Bump functions.Denote by Δ j and S j the convolution operator of ψ j ξ and ϕ j ξ , respectively, that is, is called a homogeneous Besov space and Ḃs,∞

Lemmas and main results
The linear Schr ödinger group S t e i −Δ m t satisfies the following L q − L q estimate 14, 15 : We first provide two lemmas that may be useful in in the following.
Proof.According to the property of the Fourier transformation and f x λ 2m/p f λx , we get Thus It is easy to see that from 2.4 and 2.5 , The detailed proof can be referred to [16].
In order to prove the main results, we need the following known theorems 11 .
Theorem 2.3 existence of global solutions .Suppose that p satisfies 1.
then the Cauchy problem 1.1 has a unique solution u x, t ∈ X which satisfies u X ≤ 2ε.
Theorem 2.4 the continuous dependence of the solution on the initial value .Suppose that f x and g x both satisfy the condition 2.9 , u, v are two solutions of the Cauchy problem 1.1 corresponding to the initial value f x and g x , then
In this paper, our object is to study the global self-similar solutions of the Cauchy problem 1.1 .At first, we introduce the definition of the self-similar solution.

2.15
In particular, if existing ε ε/C > 0 such that Ω C n ≤ ε , then there exists a unique self-similar solution of 1.1 with the initial value 2.14 .

The proof of main result
To prove Theorem 2.6, we should provide the following two propositions.
Proof.By Lemma 2.1, we only illustrate that the following inequality is valid:

3.3
It follows that from the embedding Ḃ0,1

3.4
Denote F S 1 f, and then F can be decomposed as follows: where ϕ is referred in the introduction.
Making use of the estimate 2.1 and noting that Δ j F −1 1 − ϕ 0 for all j ≤ −1, then we have where Δ j l 1 l −1 Δ j l .For l ±1, 0, we have Since ψ j l x 2 j l n ψ 0 2 j l x , then ψ j l 1 ψ 0 1 .Thus, it follows that from the Young inequality Besides, as f λx λ −2m/p f x , so that

3.10
By 3.6 together with the Young inequality, we obtain

3.11
We know that from the left side of the inequality 1. ≤ C Δ 0 f p 2 .

3.13
On the other hand, Δ j ϕ 0 for j ≥ 2, thus

3.14
It follows that by the Young inequality,

3.15
We get that from 3.15 and Correspondingly, 3.17 The right side of 1.4 shows that 2m/p − n/ p 2 > 0, consequently

3.18
From p 2 ≤ p 2 and the Bernstein inequality, we get Combining 3.13 with 3.19 , we have

3.20
The proof of Proposition 3.1 is finished.

3.21
Proof.Since p 2 ≥ 1, then n 2m/p p 2 > n.Accordingly, we obtain by Lemma 2.2 that Definition 2.5.Suppose that u t, x is a solution of the Cauchy problem 1.1 , if S t f X ≤ C Δ 0 f p 2 .3.25However, noting that Ω ∈ C n S n−1 as well as Proposition 3.2, we getΔ 0 f p 2 ≤ C Ω C n ., then we have u 0 X ≤ ε for any Ω C n ≤ ε .From Theorem 2.3, we conclude that there is a unique global solution u x, t of the equation in 1.1 with the initial value 2.14 .Besides, Thus, u x, t is just a self-similar solution of the problem 1.1 .This completes the proof of Theorem 2.6.