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

Authors: Yaojun YE
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

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


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## Body

## 1. Introduction

This paper is devoted to study the initial value problem for the nonlinear Schrödinger equation(1.1)iut+(−Δ)mu=λ|u|pu,x∈Rn,t∈R+,u(x,0)=f(x),x∈Rn,where λ∈R,p>0 are constants, m≥1 is a positive
integer, u(t,x) is a
complex-valued function defined in R+×Rn,
the initial value f(x) is a
complex-valued function defined in Rn.Whenm=1, (1.1) is a classical nonlinear Schrödinger equation of
the second order:(1.2)iut−Δu=λ|u|pu.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
self-similar 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 Lp estimate of the
elliptic equation and the compact method. Sjölin and Sjögren 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 time-weighted 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 thatp satisfies(1.3)p0<p<4mn−2m,n>2m,p0<p<+∞,n≤2m,where p0 is a positive
solution of the equation nx2+(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(1.4)p+2p+1<np2m<p+2.For p which satisfies
(1.3) or (1.4), let(1.5)θ=4m−(n−2m)p2mp(p+2),then we may introduce our work
space X as follows. Let X be a space
consisting of all Bochner measurable functions:(1.6)u(t):(0,+∞)⟶Lp+2(Rn),such that(1.7)∥u∥X=sup⁡t>0tθ∥u(t)∥p+2<+∞.In order to prove our main result, we should transform
the Cauchy problem (1.1) into the following equivalent integral equation:(1.8)u(t)=S(t)f(x)−iλ∫0tS(t−s)(|u(s)|pu(s))ds,where S(t)=ei(−Δ)mt=ℱ−1(ei|ξ|2mtℱ⋅) is a free group
produced by the free Schrödinger equation ivt+(−Δ)mv=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.Lr(Rn) denotes the
usual Lebesgue space on Rn with the norm ∥⋅∥r, 1≤r≤+∞. For any q>0, q′ stands for the
dual to q, that is, (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].Letφ^(ξ)∈𝒮 be a symmetric
Bump function with real values satisfying the conditions φ^(ξ)=1, |ξ|≤1, φ^(ξ)=0, |ξ|>2, then(1.9)φ^j(ξ)=φ^(2−jξ),ψ^j(ξ)=ψ^(2−jξ)=φ^(2−jξ)−φ^(2−j+1ξ),j∈Zare also symmetric Bump
functions. Denote by Δj and Sj the convolution
operator of ψ^j(ξ) and φ^j(ξ), respectively, that is,
(1.10)Δjf=ℱ−1ψ^jℱf=ψj∗f,Sjf=ℱ−1φ^jℱf=φj∗f∀j∈Z.If s∈R, 1≤p≤+∞, 1≤q<+∞, then(1.11)B˙ps,q={f∈𝒮′:∥f∥B˙ps,q=[∑j∈Z2sjq∥Δjf∥pq]1/q<+∞}is called a homogeneous Besov
space and(1.12)B˙ps,∞={f∈𝒮′:∥f∥B˙ps,∞=sup⁡j∈Z2js∥Δjf∥p<+∞}.


## 2. Lemmas and Main Results

The linear Schrödinger groupS(t)=ei(−Δ)mt satisfies the
following Lq′−Lq estimate [14, 15]:(2.1)∥S(t)f(x)∥q=∥ℱ−1(ei|ξ|2mtℱf)∥q≤C|t|−(n/m)(1/2−1/q)∥φ∥q′,2≤q≤+∞∀t>0.We first provide two lemmas that
may be useful in in the
following.Lemma 2.1.
Letf(x)=Ω(x/|x|)|x|−2m/p, θ=(4m−(n−2m)p)/2mp(p+2), then ∥u0∥X=∥S(t)f∥X=∥S(1)f∥p+2.Proof.
According to the property of the Fourier transformation andf(x)=λ2m/pf(λx), we get(2.2)S(t)f=λ2m/p[S(λ2mt)]f(λx)∀λ>0.Let λ=1/t2m, then(2.3)S(t)f=t−1/p[S(1)]f(xt2m).Thus(2.4)tθ∥S(t)f∥p+2=tθ−1/p∥[S(1)f](xt2m)∥p+2=tθ−1/p+n/2m(p+2)∥S(1)f∥p+2.Since(2.5)θ−1p+n2m(p+2)=0,It is easy to see that from (2.4)
and (2.5),(2.6)sup⁡t>0tθ∥S(t)f∥p+2=∥S(1)f∥p+2,namely,(2.7)∥u0∥X=∥S(t)f∥X=∥S(1)f∥p+2.Lemma 2.2.
LetΩ∈Ck(Sn−1), k≥0, f(x)=Ω(x/|x|)|x|−d, 0<d<n, then(2.8)|Δ0(f)(x)|≤C∥Ω∥Ck(1+|x|)−k−d.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 thatp satisfies (1.3)
or (1.4), θ=(4m−(n−2m)p)/(2mp(p+2)), u0(t,x)=[S(t)f](x) if there is ε>0, such that(2.9)∥u0∥X=∥S(t)f∥X≤ε,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 thatf(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
(2.10)∥u−v∥X≤C∥S(t)(f−g)∥X.
In addition, if
(2.11)sup⁡t>0tθ(1+t)δ∥S(t)(f−g)∥p+2<+∞,
then
(2.12)∥u−v∥p+2≤Ct−θ(1+t)−δ,
where (p+1)θ+δ<1, δ>0.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.Definition 2.5.
Suppose thatu(t,x) is a solution
of the Cauchy problem (1.1), if(2.13)u(t,x)=uλ(t,x)=λ2m/pu(λ2mt,λx)∀λ>0,
then u(t,x) is called the
self-similar solution of the problem (1.1).
One easily knows from the above definition thatuλ(t,x)=λ2m/pu(λ2mt,λx)∀λ>0 is a solution
of the problem (1.1) which satisfies the initial value λ2m/pf(λx), provide that u(t,x) is just a
solution of the Cauchy problem (1.1).
Now, we give our main result.Theorem 2.6.
Letp
satisfy (1.3) or
(1.4), θ=(4m−(n−2m)p)/2mp(p+2), Ω∈Cn(Sn−1), and
(2.14)f(x)=Ω(x/|x|)|x|2m/p,u0(t,x)=S(t)f(x), then(2.15)∥u0∥X≤C∥Ω∥Cn.In particular, if existing ε′=ε/C>0 such that ∥Ω∥Cn≤ε′, then there exists a unique self-similar solution of
(1.1) with the initial value (2.14).


## 3. The Proof of Main Result

To prove Theorem2.6, we should provide the following
two propositions.Proposition 3.1.
Let(3.1)f(x)=Ω(x|x|)|x|−2m/p,θ=4m−(n−2m)p2mp(p+2),
then
(3.2)∥u0∥X=∥S(t)f∥X≤C∥Δ0(f)∥(p+2)′.Proof.
By Lemma2.1, we
only illustrate that the following inequality is valid:(3.3)∥S(1)f∥p+2≤C∥Δ0(f)∥(p+2)′.It follows that from the
embedding B˙p+20,1↪H˙p+20=Lp+2, it is necessary to prove(3.4)∥S(1)f∥B˙p+20,1≤C∥Δ0(f)∥(p+2)′.
DenoteF=S(1)f, and then F can be
decomposed as follows:(3.5)F=F1+F2,F1=S(1)(φ∗f),F2=S(1)(ℱ−1(1−φ^)∗f),where φ is referred in
the introduction.
Making use of the estimate (2.1) and noting that Δjℱ−1(1−φ^)=0∀j≤−1, then we have(3.6)∥F2∥B˙p+20,1=∑j∈Z∥ΔjF2∥p+2=∑j∈Z∥S(1)Δj(ℱ−1((1−φ^)∗f))∥p+2≤∑j∈Z∥Δj(ℱ−1((1−φ^)∗f))∥(p+2)′=∑j≥0∥Δj(ℱ−1((1−φ^)∗f))∥(p+2)′=∑j≥0∥Δ˜jΔj(ℱ−1((1−φ^)∗f))∥(p+2)′,where Δ˜j=∑l=−1l=1Δj+l.
Forl=±1,0, we have(3.7)∥Δj+lℱ−1(1−φ^)∥1=∥ℱ−1(ψ^j+l(1−φ^))∥1≤∥ψj+l∥1+∥ψj+l∗φ∥1.Since ψj+l(x)=2(j+l)nψ0(2j+lx), then ∥ψj+l∥1=∥ψ0∥1. Thus, it follows that from the Young inequality(3.8)∥Δj+lℱ−1(1−φ^)∥1≤C∥ψj+l∥1(1+∥φ∥1)≤C.Besides, as f(λx)=λ−2m/pf(x), so that(3.9)Δjf(x)=ψj∗f(x)=∫Rnψj(x−y)f(y)dy=2jn∫Rnψ0(2jx−2jy)f(y)dy=∫Rnψ0(2jx−z)f(2−jz)dz=2j(2m/p)(ψ0∗f)(2jx)=2j(2m/p)Δ0f(2jx).Therefore,(3.10)∥Δjf∥(p+2)′=2j(2m/p)∥Δ0f(2j⋅)∥(p+2)′=2j(2m/p−n/(p+2)′)∥Δ0f∥(p+2)′.By (3.6) together with the Young
inequality, we obtain(3.11)∥F2∥B˙p+20,1≤∑j≥0∥Δ˜j(ℱ−1(1−φ^))∥1∥Δjf∥(p+2)′≤C∥Δ0f∥(p+2)′∑j≥02j(2m/p−n/(p+2)′).We know that from the
left  side of the inequality
(1.4),(3.12)2mp−n(p+2)′<0.It yields from (3.11) that(3.13)∥F2∥B˙p+20,1≤C∥Δ0f∥(p+2)′.On the other hand, Δjφ=0 for j≥2, thus(3.14)∥F1∥B˙p+20,1=∑j∈Z∥ΔjF1∥p+2=∑j≤1∥S(1)Δj(φ∗f)∥p+2.It follows that by the Young
inequality,(3.15)∥S(1)Δj(φ∗f)∥p+2≤∥S(1)φ∥1∥Δjf∥p+2.We get that from (3.15) and ∥Δjf∥p+2=2j(2m/p−n/(p+2))∥Δ0f∥p+2,
(3.16)∥S(1)Δj(φ∗f)∥p+2≤C2j(2m/p−n/(p+2))∥Δ0f∥p+2.Correspondingly,(3.17)∥F1∥B˙p+20,1≤C∥Δ0f∥p+2∑j≤12j(2m/p−n/(p+2)).The
right  side of (1.4) shows that 2m/p−n/(p+2)>0, consequently(3.18)∥F1∥B˙p+20,1≤C∥Δ0f∥p+2.From (p+2)′≤p+2 and the
Bernstein inequality, we get(3.19)∥F1∥B˙p+20,1≤C∥Δ0f∥p+2≤C∥Δ0f∥(p+2)′,Combining (3.13) with (3.19), we
have(3.20)∥F∥B˙p+20,1≤C∥Δ0f∥(p+2)′.
The proof of Proposition 3.1 is finished.Proposition 3.2.
LetΩ∈Cn(Sn−1), f(x)=Ω(x/|x|)|x|−2m/p, then
(3.21)∥Δ0f∥(p+2)′≤C∥Ω∥Cn.Proof.
Since(p+2)′≥1, then (n+2m/p)(p+2)′>n. Accordingly, we obtain by Lemma 2.2 that(3.22)∥Δ0f∥(p+2)′(p+2)′=∫Rn|Δ0f(x)|(p+2)′dx≤C∫Rn∥Ω∥Cn(p+2)′(1+|x|)−(n+2m/p)(p+2)′dx≤C∥Ω∥Cn(p+2)′∫0+∞(1+r)−(n+2m/p)(p+2)′+n−1dr≤C∥Ω∥Cn(p+2)′,
which implies that(3.23)∥Δ0f∥(p+2)′≤C∥Ω∥Cn.The proof is concluded.Now, we are ready to prove Theorem2.6.Proof.
For(3.24)f(x)=Ω(x|x|)|x|−2m/p,we have from Proposition 3.1(3.25)∥S(t)f∥X≤C∥Δ0(f)∥(p+2)′.However, noting that Ω∈Cn(Sn−1) as well as
Proposition 3.2, we get(3.26)∥Δ0(f)∥(p+2)′≤C∥Ω∥Cn.Then, it follows from (3.25) and
(3.26) that(3.27)∥u0∥X=∥S(t)f∥X≤C∥Ω∥Cn.Choosing ε′=ε/C>0, then we have ∥u0∥X≤ε for any ∥Ω∥Cn≤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,(3.28)λ2m/pf(λx)=Ω(x|x|)|x|−2m/p=f(x),which gives that by uniqueness(3.29)u(x,t)=λ2m/pu(λx,λ2mt).Thus, u(x,t)
is just a
self-similar solution of the problem (1.1).
This completes the proof of Theorem2.6.



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*Source: 836124-2008-04-07.xml*