非線性擾動下二維有界域半線性波方式爆炸解之穩定性研究 (II)

(1)

行政院國家科學委員會專題研究計畫成果報告

非線性擾動下二維有界域半線性波方式爆炸解之穩定性研

究 (II)

研究成果報告(精簡版)

計畫類別：個別型

計畫編號： NSC 98-2115-M-004-004-

執行期間： 98 年 08 月 01 日至 99 年 07 月 31 日

執行單位：國立政治大學應用數學學系

計畫主持人：李明融

共同主持人：謝宗翰、白仁德

報告附件：國外研究心得報告

處理方式：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中華民國 99 年 09 月 11 日

(2)

Stability of positive solutions for some semilinear wave

equations under nonlinear perturbation near blow-up

solutions in 2-space dimension

u

p₊

_u

q ₌

₀

_II

Meng-Rong Li

Department of Mathematical Sciences National Chengchi University

AbstractIn this research we treat the stability of positive solutions of some par-ticular semilinear wave equations under nonlinear perturbation in bounded domain near blow-up solutions in 2-space dimension.

1.

Introduction

Consider the initial value problem for the semilinear wave equation of the type u + g (u) = 0 in [0; T ) R2;

(0.1)

u (0; ) = u0; _u (0; ) = u1;

(0.2)

where g : R ! Ris a real valued function, the initial data are given su¢ ciently smooth functions and u := utt 4u; 4 is the Laplace operator. The linear

case g(u) = mu, where m is a constant, corresponds to the classical Klein Gordon equation in relativistic particle physics; the constant m is interpreted as the mass and is assumed to be nonnegative generally. To model also nonlinear phenomena like quantization, in the 1950s equations of (0:1) type with nonlinearities like g(u) = mu + u3_{; m} _{0; were proposed as models in relativistic quantum mechanics with}

local interaction. Solutions could be considered as real or complex valued functions. In the latter case it was assumed that the nonlinearity commutes with the phase; that is, g ei'u = "ei'_{g (u) for ' 2 R and that g(0) = 0. In this case, g may} be expressed g(u) = uf juj2 , which gives the study of equation (0:1) [J ]. In the

(3)

2

noncoercive case it is easy to construct solutions of (0:1) with smooth initial data that blow up in …nite time; for instance, for any > 0 the function u(t; x) = (1 t) 1=m _{solves the equation} _{u + (1 + )u juj}2m

= 0; m 2 N and blows up at t = 1. Modifying the initial data o¤ fx : jxj 2g, say, we even posses a singular solution with C1_{-data having compact support.}

In this study we want to deal with the stability of positive solutions for the semilinear wave equation

(1:1) u = up+ uq in [0; T ) ; R2

with boundary value null and initial values u (0; ) = u0( ) 2 H2( ) \ H01( ) and

_u (0; ) = u1( ) 2 H01( ) ; where p; q 2 (1; 1) and is a bounded domain in R2:

We will use the following notations: := @ @t; Du := ( _u; ru) ; ru := @2_u @2_x+ @2_u @2_y; a (t) := Z u2(t; x) dx; E (t) := Z jDuj2 _{p + 1}2 up+1 2 q + 1u q+1 _{(t; x) dx:}

For a Banach space X and 0 < T _{1 we set}

Ck(0; T; X) = Space of Ck _{functions : [0; T ) ! X;} H1 := C1 _{0; T; H}1

0( ) \ C2 0; T; L2( ) :

The existence result to the equation (1:1) is proved [Li 3] and the positive solu-tion blows-up in …nite time if 0 [Li 2] ; this means that the positive solutions for the semilinear wave equation

u = up in [0; T ) ; (1:2)

u (0; ) = u0( ) 2 H2( ) \ H01( ) ;

_u (0; ) = u1( ) 2 H01( ) ;

is stable under nonlinear perturbation uq _{providing p > 1; q > 1;} _{> 0; but it is}

not clearly whether it is also true for any p > 1; q > 1; < 0 ? If so, we would want to estimate the blow-up time and the blow-up rate under such a situation.

It is also important to study the asymptotic behavior of the solution u ; the velocity and the rate of the approximation for approaches to zero.

Such questions are also not easy to answer even under the case for the ordinary di¤erential equation

u00= up(c + u0(t)q) ; (1.3)

u (0) = u0; u0(0) = u1;

p > 1; q > 1; c > 0; > 0:

We have studied the blow-up behavior of the solution for problem (1:3) and got some estimates on blow-up time and blow-up rate [Li4] but it is di¢ cult to …nd the

(4)

3

real blow-up time (life-span). Further literature could be fund in [S], [R], [W1] and [W2].

In this study we hope that our ideals used in [Li 4], [Li 5] can do help us dealing such problem (1:1) on our topics.

2. De…nition and Fundamental Lemma

There are many de…nitions of the weak solutions of the initial-boundary problems of the wave equation, we use here as following.

De…nition 2.1_{: For p > 1; u 2 H1 is called a positive weakly solution of} equation (1:1), if Z t 0 Z ( _u (r; x) _' (r; x) + (up+ uq) (r; x) ' (r; x)) dxdr = Z t 0 Z ru (r; x) r' (r; x) dxdr 8' 2 H1 and Z t 0 Z u (r; x) (r; x) dxdr 0 for each positive 2 C1

0 ([0; T ) ).

Remark 2.2:

1) The de…nition 1.1 is resulted from the multiplying with ' to the equation (1:1) and integrating in from 0 to t.

2) From the local Lipschitz functions up_{+ u}q_{; p > 1; q > 1 the initial-boundary}

value problem (1:1) possesses a unique solution in H1 [Li1]. Hereafter we use the notations: 1 C := 1= sup kuk2= @u @x ₂: u 2 H 1 0( ) ; q = sup kukq= @u @x ₂: u 2 H 1 0( ) \ Lq( ) ; q > 1:

In this study we need the following lemmas

Lemma 2.3:

_{Suppose that u 2 H1 is a weakly positive solution of (1:1) with}

E (0) = 0 for p > 1; q > 1; then for a (0) > 0 we have: (i) a 2 C2(R+) and E (t) = E (0) 8t 2 [0; T ). (ii) a0_{(t) > 0} _{8t 2 [0; T ), provided a}0_{(0) > 0.}

(iii) a0_{(t) > 0} _{8t 2 (0; T ), if a}0_{(0) = 0.}

(iv) For a0(0) < 0; there exists a constant t0> 0 with

(5)

4

and a0_{(t) = 0:}

Lemma 2.4

:

Suppose that u is a positive weakly solution in H1 of equation (1:1) with u (0; ) = 0 = _u (0; ) in L2_{( ). For p > 1; q > 1;} _{> 0; we have u} ₀

in H1.

According to Lemma 2.4, we discuss the following theme (3) E (0) = 0; a (0) > 0 and a0₍₀₎ _{0 or a}0_{(0) < 0.}

(4) E (0) < 0; a (0) > 0 and a0₍₀₎ _{0 or a}0_{(0) < 0.}

3. Estimates for the Life-Span

3.1. Estimates for the Life-Span of the Solutions of (1.1) under Null-Energy. We study the case that E (0) = 0, p > 1; q > 1; > 0 and divide it into two parts

(i) a (0) > 0; a0₍₀₎ ₀

and

(ii) a (0) > 0; a0_{(0) < 0:}

Remark 3.. 1) The local existence and uniqueness of solutions of equation (1:1) in H1 are known [Li 2].

2) For _{= 0, p > 1 and E (0) = 0; the life-span of the positive solution u 2 H1} of equation (1.1) is bounded by T 1:= k21sin 1 k2 k1a p 1 4 (0) ! with k1:= p 1 4 a p 1 4 (0) q a0_{(0) a} 2_{(0) + 4C}2_{; k} 2:= p 1 2 C ; 1

C := 1= sup kuk2= kDuk2: u 2 H

1 0( ) :

3.2. Estimates for the Life-Span of the Solutions of equation (1.1) under Negativ-Energy. We use the following result and those argumentations of proof are not true for positive energy, so under positive energy we need another method to show the similar results.

Lemma 3

_{: Suppose that u 2 H1 is a positive weakly solution of equation (1.1)}

with a (0) > 0 and E (0) < 0 for = 0. Then (i) for a0₍₀₎ _{0, we have a}0_{(t) > 0 8t > 0:}

(ii) for a0_{(0) < 0; there exists a constant t}

5> 0 with

(6)

5

and

t5 t6:=

a0₍₀₎

(p 1) ( 2 _{E (0))};

where is the positive root of the equation 2

p + 1

p+1

p+1 rp+1 r2+ E (0) = 0:

4. Stability of positive solutions of equation (1.1) near blow-up solutions under Negativ-Energy

In this study we use our ideals used in [Li 2]; [Li 4]; [Li 5] and [Li 7]to deal such problem (1:1) on our topics under negative energy and obtain the following results:

Theorem 4.1:

E (0) 0 for p > 1; q > 1; then for a (0) > 0 we have:

The equation (1:1) is stable for _{! 0}+; this means that weakly positive solution u of (SL) blows up in …nite time for _{! 0}+:

Theorem 4.2:

The equation (1:1) is stable for p > q; _{! 0 ; this means that weakly positive} solution u of (1:1) blows up in …nite time for p > q; _{! 0 :}

Theorem 4.3:

The equation (1:1) is unstable for p < q; _{! 0 ; this means that some weakly} positive solution u of (1:1) blow up in …nite time for p < q; _{! 0 ; but also} there were some global weakly positive solution u of (1:1) for p < q; _{! 0 :}

Remark:

The decade rate of the di¤erence of life-spans T of u and T of u; can not be estimated very well for _{! 0; thus it will be a good topic on asymptotic behavior} near the blow-up solutions.

Reference:

[J] Jörgens,K.: Das Anfangswertproblem im Gröen fur eine Klasse nichtlinearer Wellengleichungen. M. Z.77. pp.295-307 (1961)

[Li1] Meng-Rong Li: On the Semi-Linear Wave Equations (I) : Taiwanese Journal of Math. Vol. 2, No. 3, pp. 329-345, Sept. 1998

[Li2] Meng-Rong Li: Estimates for the Life-Span of the Solutions of some Semi-linear Wave Equations. Communications on Pure and Applied Analysis vol.7,no. 2, pp.417-432. (2008).

[Li3] Meng-Rong Li: Nichtlineare Wellengleichungen 2. Ordnung auf beschränk-ten Gebiebeschränk-ten. PhD-Dissertation Tübingen 1994.

(7)

6

[Li 4] Meng-Rong Li: Blow-up solutions to the nonlinear second order di¤erential equation u00 _{= u}p_(c

1+ c2u0(t)q). Taiwanese Journal of Mathematics, vol.12,no.3,

pp.599-622, June 2008.

[Li5] Renjun Duan, Meng-Rong Li; Tong Yang: Propagation of Singularities in the Solutions to the Boltzmann Equation near Equilibrium, Mathematical Models and Methods in Applied Sciences (M3AS), vol.18, no.7, pp.1093-1114.(2008)

[Li6] Meng-Rong Li, Brain Pai: Quenching problem in some semilinear wave equations. Acta math. scientia vol.28,no.3, pp.523-529, July 2008.

[Li7] Meng-Rong Li: Estimates for the life-span of the solutions of some semilin-ear wave equation u up_{= 0 in one space dimension. Communications on Pure}

and Applied Analysis, 2010 to appear.

[R] Racke R.: Lectures on nonlinear Evolution Equations: Initial value problems. Aspects of Math. Braunschweig Wiesbaden Vieweg(1992).

[S] Strauss W.A.: Nonlinear Wave Equations. A.M.S. Providence(1989). Di-mensions. J. Di¤erential Equations52.p.378-406(1984)

[W1] von Wahl W.: Klassische Lösungen nichtlinearer Wellengleichungen im Großen.M.Z.112.p.241-279(1969).

[W2] von Wahl W.: Klassische Lösungen nichtlinearer gedämpfter Wellengle-ichungen im Großen. Manuscripta.Math.3.p7-33(1970).

(8)

訪香港城市大學數學系

楊彤教授(講座教授理工學院副院長)

報告

十月十五日討論波茲曼方程在實際上應用於其它領域的可行性

1. 建立個體經濟新模型之可行性

2. 建立總體經濟新模型之可行性

3. 建立人口模型之可行性

4. 稀薄流新的思考方向建立新模型之可行性

十月十六日討論波茲曼方程中碰撞項於實際應用在其它領域時

進行實質分類的可行性

1. 經濟行為如個經總經或上市公司間之交互持股其效益

問題建模時其碰撞項的分類明顯重要

2. 人口問題亦應可利用波茲曼方程的概念來建模於建模時

其碰撞項的分類亦格外重要

3. 若可建立稀薄流新模型時其碰撞項的分類格外重要尤其

在經費的支出上

十月十七日因楊彤教授赴廣州大學參加學術研討會所以無法共同

討論細節相約于十一月份再進行細節上的討論併討論楊彤

教授訪問政大應數系的可能性與時間

十月十八日整理行裝準備返台

Report on visiting Chair Professor Tong Yang

(Professor of Institute of Technology Vice-President)

Department of Mathematics, City University of Hong Kong

October 15 Discussion about the feasibility of the applications of Boltzmann equations

practically, to other areas:

1. To establish the feasibility of a new model of Microeconomics

2. The establishment of the feasibility of a new model for Macroeconomics

3. To establish the feasibility of the population model

4. Possibilities to explore a new direction on Rarefied flow and the feasibility

establishing a new model

(9)

October 16 to discuss of the substantive classification of Boltzmann equations in the

collision term of the practical application in other research fields

1. Economic behavior, such as the listed company or by the interaction between the

holdings of its benefits

Problem modeling the classification of items of its obvious importance collisions

2. The population problem should be able to use the concept of Boltzmann equation to

model in the modeling

Its impact is particularly important to the classification of items

3. If it is able to establish the thin stream of new models of its impact is particularly

important, particularly the classification of items

The expenditure of funds,

October 17 by Professor Yang Tong went to the Guangzhou University Symposium on

it is not possible to participate in the common

Meet in November to discuss the details further discussion of the details and

discuss the Tong Yang

Visit the Department of National Chengchi University professor to be a few

possibilities and the time

(10)

訪香港城市大學數學系

楊彤教授(講座教授理工學院副院長)

報告

十一月十九日討論波茲曼方程在實際上應用於其它相關領域的可行性

1. 建立個體經濟之分子運動非線性模型之可行性

2.建立總體經濟之分子運動非線性模型之可行性

3.建立非線性人口模型之可行性

4.稀薄流新的思考方向建立二次非線性模型之可行性

十一月二十日討論波茲曼方程中碰撞項於實際應用在其它領域時

進行二次非線型逼近的可行性

1. 經濟行為之效益問題建模時以二次非線型分子運動逼近明顯重要

2. 人口問題亦可以波茲曼方程概念建模其碰撞項之二次非線型逼近亦格外

重要

3. 建立稀薄流時碰撞項二次非線性模型

十一月二十一日相約于二 O 一 O 年六月楊彤教授訪問政大應數系

十一月二十二日整理行裝準備返台

(11)

(12)

98 年度專題研究計畫研究成果彙整表

計畫主持人：

李明融

計畫編號：

98-2115-M-004-004-計畫名稱：