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

Abstract

For a certain range of the value 𝑝 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.

1. Introduction

This paper is devoted to study the initial value problem for the nonlinear Schrödinger equation 𝑖𝑢_𝑡+(−Δ)^𝑚𝑢=𝜆|𝑢|^𝑝𝑢,𝑥∈𝑅^𝑛,𝑡∈𝑅⁺,𝑢(𝑥,0)=𝑓(𝑥),𝑥∈𝑅^𝑛, (1.1)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 Schrödinger equation of the second order:𝑖𝑢_𝑡−Δ𝑢=𝜆|𝑢|^𝑝𝑢. (1.2)

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 𝐿^𝑝 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 that 𝑝 satisfies𝑝₀<𝑝<4𝑚𝑛−2𝑚,𝑛>2𝑚,𝑝₀<𝑝<+∞,𝑛≤2𝑚, (1.3)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 to𝑝+2<𝑝+1𝑛𝑝2𝑚<𝑝+2. (1.4)For 𝑝 which satisfies (1.3) or (1.4), let𝜃=4𝑚−(𝑛−2𝑚)𝑝2𝑚𝑝(𝑝+2), (1.5)then we may introduce our work space 𝑋 as follows. Let 𝑋 be a space consisting of all Bochner measurable functions:𝑢(𝑡)∶(0,+∞)⟶𝐿^𝑝+2(𝑅^𝑛), (1.6)such that‖𝑢‖_𝑋=sup_𝑡>0𝑡^𝜃‖𝑢(𝑡)‖_𝑝+2<+∞. (1.7)

In order to prove our main result, we should transform the Cauchy problem (1.1) into the following equivalent integral equation:𝑢(𝑡)=𝑆(𝑡)𝑓(𝑥)−𝑖𝜆^𝑡₀𝑆(𝑡−𝑠)(|𝑢(𝑠)|^𝑝𝑢(𝑠))𝑑𝑠, (1.8)where 𝑆(𝑡)=𝑒^{𝑖(−Δ)𝑚𝑡}=ℱ⁻¹(𝑒^{𝑖|𝜉|2𝑚𝑡}ℱ⋅) is a free group produced by the free Schrödinger equation 𝑖𝑣_𝑡+(−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. 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, (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 𝜑(𝜉)=1, |𝜉|≤1, 𝜑(𝜉)=0, |𝜉|>2, then𝜑_𝑗(𝜉)=𝜑(2^−𝑗𝜉),𝜓_𝑗(𝜉)=𝜓(2^−𝑗𝜉)=𝜑(2^−𝑗𝜉)−𝜑(2^−𝑗+1𝜉),𝑗∈𝑍 (1.9)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, Δ_𝑗𝑓=ℱ⁻¹𝜓_𝑗ℱ𝑓=𝜓_𝑗∗𝑓,𝑆_𝑗𝑓=ℱ⁻¹𝜑_𝑗ℱ𝑓=𝜑_𝑗∗𝑓∀𝑗∈𝑍. (1.10)If 𝑠∈𝑅, 1≤𝑝≤+∞, 1≤𝑞<+∞, then𝐵̇_𝑝^𝑠,𝑞={𝑓∈𝒮^′∶‖𝑓‖_̇𝐵𝑝^𝑠,𝑞=[_𝑗∈𝑍2^𝑠𝑗𝑞‖Δ_𝑗𝑓‖^𝑞_𝑝]^1/𝑞<+∞} (1.11)is called a homogeneous Besov space and𝐵̇_𝑝^𝑠,∞={𝑓∈𝒮^′∶‖𝑓‖_̇𝐵𝑝^𝑠,∞=sup_𝑗∈𝑍2^𝑗𝑠‖Δ_𝑗𝑓‖_𝑝<+∞}. (1.12)

2. Lemmas and Main Results

The linear Schrödinger group 𝑆(𝑡)=𝑒^{𝑖(−Δ)𝑚𝑡} satisfies the following 𝐿^𝑞′−𝐿^𝑞 estimate [14, 15]:‖𝑆(𝑡)𝑓(𝑥)‖_𝑞=‖ℱ⁻¹(𝑒^{𝑖|𝜉|2𝑚𝑡}ℱ𝑓)‖_𝑞≤𝐶|𝑡|^{−(𝑛/𝑚)(1/2−1/𝑞)}‖𝜑‖_𝑞′,2≤𝑞≤+∞∀𝑡>0. (2.1)We first provide two lemmas that may be useful in in the following.

Lemma 2.1. Let 𝑓(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 and 𝑓(𝑥)=𝜆^2𝑚/𝑝𝑓(𝜆𝑥), we get𝑆(𝑡)𝑓=𝜆^2𝑚/𝑝[𝑆(𝜆^2𝑚𝑡)]𝑓(𝜆𝑥)∀𝜆>0. (2.2)Let 𝜆=1/√^2𝑚𝑡, then𝑆(𝑡)𝑓=𝑡^−1/𝑝𝑥[𝑆(1)]𝑓(_2𝑚√𝑡). (2.3)Thus𝑡^𝜃‖𝑆(𝑡)𝑓‖_𝑝+2=𝑡^{𝜃−1/𝑝}𝑥‖[𝑆(1)𝑓](_2𝑚√𝑡)‖_𝑝+2=𝑡^{𝜃−1/𝑝+𝑛/2𝑚(𝑝+2)}‖𝑆(1)𝑓‖_𝑝+2. (2.4)Since1𝜃−𝑝+𝑛2𝑚(𝑝+2)=0, (2.5)It is easy to see that from (2.4) and (2.5),sup_𝑡>0𝑡^𝜃‖𝑆(𝑡)𝑓‖_𝑝+2=‖𝑆(1)𝑓‖_𝑝+2, (2.6)namely,‖𝑢₀‖_𝑋=‖𝑆(𝑡)𝑓‖_𝑋=‖𝑆(1)𝑓‖_𝑝+2. (2.7)

Lemma 2.2. Let Ω∈𝐶^𝑘(𝑆^𝑛−1), 𝑘≥0, 𝑓(𝑥)=Ω(𝑥/|𝑥|)|𝑥|^−𝑑, 0<𝑑<𝑛, then|Δ₀(𝑓)(𝑥)|≤𝐶‖Ω‖_𝐶^𝑘(1+|𝑥|)^{−𝑘−𝑑}. (2.8)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 (1.3) 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‖𝑢₀‖_𝑋=‖𝑆(𝑡)𝑓‖_𝑋≤𝜀, (2.9)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 ‖𝑢−𝑣‖_𝑋≤𝐶‖𝑆(𝑡)(𝑓−𝑔)‖_𝑋. (2.10) In addition, if sup_𝑡>0𝑡^𝜃(1+𝑡)^𝛿‖𝑆(𝑡)(𝑓−𝑔)‖_𝑝+2<+∞, (2.11) then ‖𝑢−𝑣‖_𝑝+2≤𝐶𝑡^−𝜃(1+𝑡)^−𝛿, (2.12) 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 that 𝑢(𝑡,𝑥) is a solution of the Cauchy problem (1.1), if𝑢(𝑡,𝑥)=𝑢_𝜆(𝑡,𝑥)=𝜆^2𝑚/𝑝𝑢(𝜆^2𝑚𝑡,𝜆𝑥)∀𝜆>0, (2.13) then 𝑢(𝑡,𝑥) is called the self-similar solution of the problem (1.1).
One easily knows from the above definition that 𝑢_𝜆(𝑡,𝑥)=𝜆^2𝑚/𝑝𝑢(𝜆^2𝑚𝑡,𝜆𝑥)∀𝜆>0 is a solution of the problem (1.1) which satisfies the initial value 𝜆^2𝑚/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. Let 𝑝 satisfy (1.3) or (1.4), 𝜃=(4𝑚−(𝑛−2𝑚)𝑝)/2𝑚𝑝(𝑝+2), Ω∈𝐶^𝑛(𝑆^𝑛−1), and 𝑓(𝑥)=Ω(𝑥/|𝑥|)|𝑥|^2𝑚/𝑝, (2.14)𝑢₀(𝑡,𝑥)=𝑆(𝑡)𝑓(𝑥), then‖𝑢₀‖_𝑋≤𝐶‖Ω‖_𝐶^𝑛. (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).

3. The Proof of Main Result

To prove Theorem 2.6, we should provide the following two propositions.

Proposition 3.1. Let 𝑥𝑓(𝑥)=Ω(|𝑥|)|𝑥|^{−2𝑚/𝑝},𝜃=4𝑚−(𝑛−2𝑚)𝑝2𝑚𝑝(𝑝+2), (3.1) then ‖𝑢₀‖_𝑋=‖𝑆(𝑡)𝑓‖_𝑋≤𝐶‖Δ₀(𝑓)‖_(𝑝+2)′. (3.2)

Proof. By Lemma 2.1, we only illustrate that the following inequality is valid:‖𝑆(1)𝑓‖_𝑝+2≤𝐶‖Δ₀(𝑓)‖_(𝑝+2)′. (3.3)It follows that from the embedding 𝐵̇^0,1_𝑝+2↪𝐻̇⁰_𝑝+2=𝐿^𝑝+2, it is necessary to prove‖𝑆(1)𝑓‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀(𝑓)‖_(𝑝+2)′. (3.4)
Denote 𝐹=𝑆(1)𝑓, and then 𝐹 can be decomposed as follows:𝐹=𝐹₁+𝐹₂,𝐹₁=𝑆(1)(𝜑∗𝑓),𝐹₂=𝑆(1)(ℱ⁻¹(1−𝜑)∗𝑓), (3.5)where 𝜑 is referred in the introduction.
Making use of the estimate (2.1) and noting that Δ_𝑗ℱ⁻¹(1−𝜑)=0∀𝑗≤−1, then we have‖𝐹₂‖_̇𝐵^0,1_𝑝+2=_𝑗∈𝑍‖Δ_𝑗𝐹₂‖_𝑝+2=_𝑗∈𝑍‖𝑆(1)Δ_𝑗(ℱ⁻¹((1−𝜑)∗𝑓))‖_𝑝+2≤_𝑗∈𝑍‖Δ_𝑗(ℱ⁻¹((1−𝜑)∗𝑓))‖_(𝑝+2)′=_𝑗≥0‖Δ_𝑗(ℱ⁻¹((1−𝜑)∗𝑓))‖_(𝑝+2)′=_𝑗≥0‖Δ^{0𝑥002𝑑𝑐}_𝑗Δ_𝑗(ℱ⁻¹((1−𝜑)∗𝑓))‖_(𝑝+2)′, (3.6)where ^0𝑥0Δ^02𝑑𝑐_𝑗=∑^𝑙=1_𝑙=−1Δ_𝑗+𝑙.
For 𝑙=±1,0, we have‖Δ_𝑗+𝑙ℱ⁻¹(1−𝜑)‖₁=‖ℱ⁻¹(𝜓_𝑗+𝑙(1−𝜑))‖₁≤‖𝜓_𝑗+𝑙‖₁+‖𝜓_𝑗+𝑙∗𝜑‖₁. (3.7)Since 𝜓_𝑗+𝑙(𝑥)=2^{(𝑗+𝑙)𝑛}𝜓₀(2^𝑗+𝑙𝑥), then ‖𝜓_𝑗+𝑙‖₁=‖𝜓₀‖₁. Thus, it follows that from the Young inequality‖Δ_𝑗+𝑙ℱ⁻¹(1−𝜑)‖₁≤𝐶‖𝜓_𝑗+𝑙‖₁(1+‖𝜑‖₁)≤𝐶. (3.8)Besides, as 𝑓(𝜆𝑥)=𝜆^{−2𝑚/𝑝}𝑓(𝑥), so thatΔ_𝑗𝑓(𝑥)=𝜓_𝑗∗𝑓(𝑥)=_𝑅𝑛𝜓_𝑗(𝑥−𝑦)𝑓(𝑦)𝑑𝑦=2^𝑗𝑛_𝑅𝑛𝜓₀(2^𝑗𝑥−2^𝑗=𝑦)𝑓(𝑦)𝑑𝑦_𝑅𝑛𝜓₀(2^𝑗𝑥−𝑧)𝑓(2^−𝑗𝑧)𝑑𝑧=2^{𝑗(2𝑚/𝑝)}(𝜓₀∗𝑓)(2^𝑗𝑥)=2^{𝑗(2𝑚/𝑝)}Δ₀𝑓(2^𝑗𝑥). (3.9)Therefore,‖Δ_𝑗𝑓‖_(𝑝+2)′=2^{𝑗(2𝑚/𝑝)}‖Δ₀𝑓(2^𝑗⋅)‖_(𝑝+2)′=2^{𝑗(2𝑚/𝑝−𝑛/(𝑝+2)′)}‖Δ₀𝑓‖_(𝑝+2)′. (3.10)By (3.6) together with the Young inequality, we obtain‖𝐹₂‖_̇𝐵^0,1_𝑝+2≤_𝑗≥0‖Δ^{0𝑥002𝑑𝑐}_𝑗(ℱ⁻¹(1−𝜑))‖₁‖Δ_𝑗𝑓‖_(𝑝+2)′≤𝐶‖Δ₀𝑓‖_{(𝑝+2)′𝑗≥0}2^{𝑗(2𝑚/𝑝−𝑛/(𝑝+2)′)}. (3.11)We know that from the left side of the inequality (1.4),2𝑚𝑝−𝑛(𝑝+2)^′<0. (3.12)It yields from (3.11) that‖𝐹₂‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀𝑓‖_(𝑝+2)′. (3.13)On the other hand, Δ_𝑗𝜑=0 for 𝑗≥2, thus‖𝐹₁‖_̇𝐵^0,1_𝑝+2=_𝑗∈𝑍‖Δ_𝑗𝐹₁‖_𝑝+2=_𝑗≤1‖𝑆(1)Δ_𝑗(𝜑∗𝑓)‖_𝑝+2. (3.14)It follows that by the Young inequality,‖𝑆(1)Δ_𝑗(𝜑∗𝑓)‖_𝑝+2≤‖𝑆(1)𝜑‖₁‖Δ_𝑗𝑓‖_𝑝+2. (3.15)We get that from (3.15) and ‖Δ_𝑗𝑓‖_𝑝+2=2^{𝑗(2𝑚/𝑝−𝑛/(𝑝+2))}‖Δ₀𝑓‖_𝑝+2,
‖𝑆(1)Δ_𝑗(𝜑∗𝑓)‖_𝑝+2≤𝐶2^{𝑗(2𝑚/𝑝−𝑛/(𝑝+2))}‖Δ₀𝑓‖_𝑝+2. (3.16)Correspondingly,‖𝐹₁‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀𝑓‖_{𝑝+2𝑗≤1}2^{𝑗(2𝑚/𝑝−𝑛/(𝑝+2))}. (3.17)The right side of (1.4) shows that 2𝑚/𝑝−𝑛/(𝑝+2)>0, consequently‖𝐹₁‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀𝑓‖_𝑝+2. (3.18)From (𝑝+2)^′≤𝑝+2 and the Bernstein inequality, we get‖𝐹₁‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀𝑓‖_𝑝+2≤𝐶‖Δ₀𝑓‖_(𝑝+2)′, (3.19)Combining (3.13) with (3.19), we have‖𝐹‖_̇𝐵^0,1_𝑝+2≤𝐶‖Δ₀𝑓‖_(𝑝+2)′. (3.20) The proof of Proposition 3.1 is finished.

Proposition 3.2. Let Ω∈𝐶^𝑛(𝑆^𝑛−1), 𝑓(𝑥)=Ω(𝑥/|𝑥|)|𝑥|^{−2𝑚/𝑝}, then ‖Δ₀𝑓‖_(𝑝+2)′≤𝐶‖Ω‖_𝐶^𝑛. (3.21)

Proof. Since (𝑝+2)^′≥1, then (𝑛+2𝑚/𝑝)(𝑝+2)^′>𝑛. Accordingly, we obtain by Lemma 2.2 that‖Δ₀𝑓‖^(𝑝+2)′_(𝑝+2)′=_𝑅𝑛|Δ₀𝑓(𝑥)|^(𝑝+2)′𝑑𝑥≤𝐶_𝑅𝑛‖Ω‖^(𝑝+2)′_𝐶𝑛(1+|𝑥|)^{−(𝑛+2𝑚/𝑝)(𝑝+2)′}𝑑𝑥≤𝐶‖Ω‖^(𝑝+2)′_𝐶𝑛0^+∞(1+𝑟)^{−(𝑛+2𝑚/𝑝)(𝑝+2)′+𝑛−1}𝑑𝑟≤𝐶‖Ω‖^(𝑝+2)′_𝐶𝑛, (3.22) which implies that‖Δ₀𝑓‖_(𝑝+2)′≤𝐶‖Ω‖_𝐶^𝑛. (3.23)The proof is concluded.

Now, we are ready to prove Theorem 2.6.

Proof. For 𝑥𝑓(𝑥)=Ω(|𝑥|)|𝑥|^{−2𝑚/𝑝}, (3.24)we have from Proposition 3.1‖𝑆(𝑡)𝑓‖_𝑋≤𝐶‖Δ₀(𝑓)‖_(𝑝+2)′. (3.25)However, noting that Ω∈𝐶^𝑛(𝑆^𝑛−1) as well as Proposition 3.2, we get‖Δ₀(𝑓)‖_(𝑝+2)′≤𝐶‖Ω‖_𝐶^𝑛. (3.26)Then, it follows from (3.25) and (3.26) that‖𝑢₀‖_𝑋=‖𝑆(𝑡)𝑓‖_𝑋≤𝐶‖Ω‖_𝐶^𝑛. (3.27)Choosing 𝜀^′=𝜀/𝐶>0, then we have ‖𝑢₀‖_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,𝜆^2𝑚/𝑝𝑥𝑓(𝜆𝑥)=Ω(|𝑥|)|𝑥|^{−2𝑚/𝑝}=𝑓(𝑥), (3.28)which gives that by uniqueness𝑢(𝑥,𝑡)=𝜆^2𝑚/𝑝𝑢(𝜆𝑥,𝜆^2𝑚𝑡). (3.29)Thus, 𝑢(𝑥,𝑡) is just a self-similar solution of the problem (1.1).
This completes the proof of Theorem 2.6.

Acknowledgment

This work is supported by Natural Science Foundation of Henan Province Education Commission (no. 2007110013) and the Program for Outstanding Young Teacher in Henan Province (2004–2006).