非線性擾動下三維有界域半線性波方式爆炸解之穩定性研究

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行政院國家科學委員會專題研究計畫成果報告

非線性擾動下三維有界域半線性波方式爆炸解之穩定性研

究

研究成果報告(精簡版)

計畫類別：個別型

計畫編號： NSC 99-2115-M-004-001-

執行期間： 99 年 08 月 01 日至 100 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 100 年 09 月 23 日

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Stability of positive solutions for some semilinear wave

equations under nonlinear perturbation near blow-up

solutions in 3-space dimension

u

p₊

_u

q ₌

₀

Meng-Rong Li

Department of Mathematical Sciences National Chengchi University

Abstract In this research we treat the stability of positive solutions of some particular semilinear wave equations under nonlinear perturbation in bounded do-main near blow-up solutions in 3-space dimension.

1.

Introduction

Consider the initial value problem for the semilinear wave equation of the type u + g (u) = 0 in [0; T ) R3;

(0.1)

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

(0.2)

where g : R ! R is 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

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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 ) ; R3

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 R3:

We will use the following notations: := @ @t; Du := ( _u; ru) ; 4u := @2_u @2_x 1 + @ 2_u @2_x 2 + @ 2_u @2_x 3 ; a (t) := Z u2(t; x) dx; E (t) := Z jDuj2 2 p + 1u p+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 C}k _{functions : [0; T ) ! X;}

H1 := C1 0; T; H₀1_{( ) \ C}2 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;

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

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3

In this study we hope that our ideals used in [Li 2]; [Li 4]; [Li 5],[Li 7] ; [Li8]; [LiLinShieh]; [ShiehLi] and [SLLLW ] 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+ uq; 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

a0(t) > 0 _{8t > t}0

and a0_(t 0) = 0:

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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 a0(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, x 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}

a0_{(t) > 0 8t > t}5; a0(t5) = 0

and

t5 t6:=

a0₍₀₎

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

where is the positive root of the equation 2

p + 1

p+1

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5

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],[Li 7] ; [Li8]; [LiLinShieh]; [ShiehLi] and [SLLLW ] 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 Größen 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.

[Li 4] Meng-Rong Li: Blow-up solutions to the nonlinear second order di¤erential equation u00= up(c1+ 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)

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[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, 2012 to appear.

[Li8 ] Meng-Rong Li; Chuen-Hsin Chang: Analysis on the Duration of Deep Earthquakes, Taiwanese Journal of Mathematics, Vol.12, No.7, Oct. 2008, pp.1667-1680.

[LiShiehYuLiLi] M.R. Li; T.H. Shieh; etc : Parabola Method in Ordinary Dif-ferential Equation, Taiwanese Journal of Mathematics, 2012 to appear.

[PaiLiChangChiu] J.T. Pai; Meng-Rong Li; Y.L. Chang; S.M. Chiu: A mathe-matical model of enterprise competitive abilitry and performance through a partic-ular Emden-Fowler equation ( to appear in Acta Mathematica scientia 2011,31B(5), SCIE.)

[LiLinShieh] Meng-Rong Li; Y.J. Lin; T.H. Shieh: The ‡ux model of the move-ment of tumor cells and health cells using a system of nonlinear heat equations. Journal of Computational Biology, January 2011, ahead of print.

[ShiehLi ] T.H. Shieh; Meng-Rong Li: Numeric treatment of contact disconti-nuity with multi-gases Journal of computational and applied mathematics (Journal of Computational and Applied Mathematics, 2009 August pp.656-673.

[SLLLW] T.H. Shieh; T.M. Liou; Meng-Rong Li; C.H. Liu; W.J. Wu: Analysis on numerical results for stage separation with di¤erent exhaust holes, International communications in heat and mass transfer, 2009 April pp.342-345.

[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.pp.378-406(1984).

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

[W2] von Wahl W.: Klassische Lösungen nichtlinearer gedämpfter Wellengle-ichungen im Großen. Manuscripta Math.3. pp7-33(1970).

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國科會補助計畫衍生研發成果推廣資料表

日期:2011/09/20

國科會補助計畫

計畫名稱: 非線性擾動下三維有界域半線性波方式爆炸解之穩定性研究計畫主持人: 李明融計畫編號: 99-2115-M-004-001- 學門領域: 微分方程

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99 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：

99-2115-M-004-001-計畫名稱：