夏 濱

(四川建筑職業(yè)技術(shù)學(xué)院 教務(wù)處, 四川 德陽 618000)

帶逆平方勢的非線性Schr?dinger方程的阻尼影響

夏 濱

(四川建筑職業(yè)技術(shù)學(xué)院 教務(wù)處, 四川 德陽 618000)

研究一類帶逆平方勢的阻尼非線性Schr?dinger方程,此方程相對(duì)論分子物理中有磁性的粒子捕獲電子的現(xiàn)象.物理上,阻尼通常弱化系統(tǒng)爆破.從數(shù)學(xué)上探究分析阻尼對(duì)系統(tǒng)爆破的精確影響.對(duì)于臨界情形,建立一個(gè)產(chǎn)生爆破解的阻尼門檻.對(duì)于超臨界情形,導(dǎo)出一個(gè)產(chǎn)生爆破解的阻尼區(qū)間.

非線性Schr?dinger方程; 逆平方勢; 阻尼; 爆破

一類帶逆平方勢的阻尼非線性Schr?dinger方程[1-3]

(1)

物理上,阻尼通常能夠弱化系統(tǒng)爆破.本文旨在從理論上分析探究阻尼對(duì)系統(tǒng)爆破的精確影響,發(fā)展了M. Tsutsumi[11-12]、G. Fibich[13]、M. Ohta等[14]的方法,對(duì)于系統(tǒng)(1)的臨界情形,建立一個(gè)產(chǎn)生爆破解的阻尼門檻(見定理2.1);對(duì)于系統(tǒng)(1)的超臨界情形,導(dǎo)出一個(gè)產(chǎn)生爆破解的阻尼區(qū)間(見定理3.1).

1 預(yù)備知識(shí)

首先賦予方程(1)初值

φ(x,0)=φ0(x),x∈RD.

(2)

于是,(1)和(2)式的局部適定性如下.

-△u+u-|u|p-1u=0,u∈H1(RD),

(3)

那么Gagliardo-Nirenberg不等式

中的最佳常數(shù)C*gt;0滿足

注1.1方程(3)解的存在性見文獻(xiàn)[16],唯一性見文獻(xiàn)[17].

下面分析Cauchy問題(1)和(2)的特征.

M(φ(t))=e-2atM(φ0)

(4)

和能量方程

(5)

其中

(6)

和

(7)

其中

(8)

然后用|x|2乘以(8)式,并在RD上積分可得

于是可得(6)式.

2)

x.

(9)

注意到

(10)

和

那么由(9)～(11)式易得(7)式成立.命題得證.

下面給出一個(gè)變換

v(x,t)=eatφ(x,t),

(12)

其中φ(x,t)是方程(1)的解,那么v(x,t)滿足

(13)

和

v(x,0)=φ(x,0)=φ0(x).

(14)

eat‖φ(t)‖2=‖v(t)‖2=‖φ0‖2

(15)

和

H(t):=‖‖
e-a(p-1)t‖
[e-a(p-1)τ‖).

(16)

(17)

因此,由(17)和(12)式可得

eat‖φ(t)‖2=‖v(t)‖2=‖φ0‖2.

(18)

對(duì)(18)式關(guān)于t在[0,t]上積分得(16)式.命題得證.

由(15)式可得

‖‖
[e-a(p-1)t‖
a(p-1)[e-a(p-1)τ‖

(19)

其中C*是引理1.1中Gagliardo-Nirenberg不等式的最佳常數(shù).

證明由(16)式,引理1.1和(15)式可得

H(0)=H(t)=‖‖
e-a(p-1)t‖
[e-a(p-1)τ‖
‖‖
a(p-1)[e-a(p-1)τ‖
‖‖
‖
‖‖
‖‖
‖
‖‖
‖‖
{e-a(p-1)t‖
[e-a(p-1)τ‖

于是(19)式成立.命題得證.

2 臨界情形下的爆破

由物理可知,阻尼能夠弱化爆破解[13],阻尼越強(qiáng),解越容易整體存在,因此,假設(shè)(A):對(duì)任意給定的初值φ0,系統(tǒng)解的最大存在時(shí)間Ta(φ0)關(guān)于阻尼參數(shù)a單增.

下面給出本文的第一個(gè)主要結(jié)論.

1) 當(dāng)alt;a*,Cauchy問題(1)和(2)的解φ(t)在有限時(shí)間內(nèi)爆破;

2) 當(dāng)agt;a*,Cauchy問題(1)和(2)的解φ(t)在H1(RD)中整體存在.

在證明定理之前,先給一些準(zhǔn)備結(jié)論.

‖φ0‖2≥eaTa(φ0)‖R‖2,

(20)

其中R是方程(3)的正徑對(duì)稱解.

證明用反證法證明.如果(20)式不成立有

‖φ0‖2lt;eaTa(φ0)‖R‖2.

(21)

記g(t):=‖v(t)‖2.由于Cauchy問題(1)和(2)的解φ(t)在有限時(shí)間Ta(φ0)爆破,由命題1.5知Cauchy問題(13)和(14)的解v(t)也在Ta(φ0)爆破.進(jìn)一步有

(22)

因此可以聲稱：存在單增時(shí)間序列{tn}使得tn→Ta(φ0)且

(23)

下面證明(23)式成立.

(24)

從而一定存在tn→Ta(φ0)使得(23)式成立.

當(dāng)n→∞得

(25)

由(21)和(25)式知g(tn)是有界的,這與(22)式矛盾.命題得證.

注意到命題2.1僅僅是爆破的一個(gè)必要條件,不是充分條件,因?yàn)楸茣r(shí)間Ta(φ0)隨阻尼參數(shù)a變化而變化.為了獲得爆破解的阻尼門檻條件,將運(yùn)用前面的假設(shè)(A).

‖φ0‖2≥eaT0(φ0)‖R‖2,

(26)

其中R是方程(3)的正徑對(duì)稱解.

命題2.2直接由假設(shè)(A)和命題2.1可推得.

證明由

知

‖φ0‖2lt;eaT0(φ0)‖R‖2.

結(jié)合到假設(shè)(A),于是由命題2.1可立得推論成立.

定理得證.

3 超臨界情形下的爆破

1)E(φ0)lt;0;

2)E(φ0)=0和V(φ0)lt;0;

那么一定存在a*(φ0)使得對(duì)任意阻尼參數(shù)a∈(0,a*(φ0)),Cauchy問題(1)和(2)的解φ(t)在有限時(shí)間內(nèi)爆破.這里泛函E(φ)、V(φ)和J(φ)如命題1.2和1.3中所定義.

證明設(shè)初值φ0∈H1(RD)滿足|x|φ0∈L2(RD)和定理3.1的條件之一.假設(shè)定理3.1的結(jié)論不成立,則Cauchy問題(1)和(2)的解φ(t)在t∈[0,+∞)整體存在.

(27)

顯然,bgt;0.由命題1.2和1.3知

(28)

(29)

和

(30)

這里

(31)

于是由(30)式可得

(32)

(33)

為了簡單起見,記

以及k:=2a+b.

由(27)式有

(34)

(33)式變成

(35)

從而

(36)

由(29)式有

(37)

因此,由(36)和(37)式可知

(38)

另一方面,由(28)式和J(φ(t))≥0知

(39)

從而

(40)

由(38)和(40)式知

(41)

記

對(duì)于f(k,t)關(guān)于k運(yùn)用Taylor展式,對(duì)任意的tgt;0有

f(k,t)=f(0,t)+o(k)=
J(φ(0))+4V(φ(0))t+
8E(φ(0))t2+o(k),k→0.

(42)

由于初值φ0滿足定理3.1的條件之一,故一定存在Tlt;∞使得

J(φ(0))+4V(φ(0))T+8E(φ(0))T2lt;0,

(43)

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MSC2010：78A60; 35Q55

(編輯 鄭月蓉)

Effect of Damping for Nonlinear Schr?dinger Equation with Inverse Square Potential

XIA Bin

(AcademicAffairsOffice,SichuanCollegeofArchitecturalTechnology,Deyang618000,Sichuan)

This work is to concern the damped nonlinear Schr?dinger equation with inverse square potential, which models the process of an electron being captured by polar molecules in non-relativistic molecular physics. We are interested in the effect of the damping in this system. In physics, damping usually weakens the blowup of systems. We analyze and explore the exact effect of the damping in the view of mathematics. For the critical case, a threshold of the damping is established to derive the blowup solution. For the supercritical case, an interval of the damping is constructed to yield the blowup solution.

nonlinear Schr?dinger equation; inverse square potential; damping; blowup

O175.29

A

1001-8395(2017)06-0802-07

10.3969/j.issn.1001-8395.2017.06.016

2016-09-10

國家自然科學(xué)基金(11571245)和四川省教育廳重點(diǎn)科研項(xiàng)目(15ZA0031)

夏 濱(1969—),男,副教授,主要從事應(yīng)用非線性分析的研究,E-mail:xiabin690215@163.com