非線性擾動下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 97-2115-M-004-004-

執行期間： 97 年 08 月 01 日至 98 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

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 98 年 08 月 05 日

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MENG-RONG LI

Stability of positive solutions for some semilinear wave equations

under nonlinear perturbation near blow-up solutions in 1-space

dimension

u

p₊

_u

q ₌

₀

_I

Abstract

In this research we treat the stability of positive solutions of some particular semilinear wave equations under nonlinear perturbation in bounded domain near blow-up solutions in 1-space dimension.

1.Introduction

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

(0:1) u = up+ uq in [0; T ) (R1; R2)

with boundary value null and initial values u (0; x) = u0(x) 2 H2(R1; R2) \

H₀1(R1; R2) and _u (0; x) = u1(x) 2 H01(R1; R2) ; where p; q 2 (1; 1) and (R1; R2)

R:

We will use the following notations: := @ @t; Du := _u; @u @x ; u := @2_u @t2 @2_u @x2; a (t) := Z R2 R1 u2(t; x) dx; E (t) := Z R2 R1 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; H}1

0(R1; R2) \ C2 0; T; L2(R1; R2) : 1

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2 M EN G -RO N G LI

The existence result to the equation (0:1) 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 ) (R1; R2) ;

(0:2)

u (0; x) = u0(x) 2 H2(R1; R2) \ H01(R1; R2) ;

_u (0; x) = u1(x) 2 H01(R1; R2) ;

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) ; (0.3)

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 (0:3) 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].

In this study we hope that our ideals used in [Li 4], [Li 5] can do help us dealing such problem (0:1) on our topics.

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1. 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 1.1_{: For p > 1; u 2 H1 is called a positive weakly solution of} equation (0:1), if Z t 0 Z R2 R1 _u (r; x) _' (r; x) @ @xu (r; x) @ @x' (r; x) + (up_{+ u}q_{) (r; x) ' (r; x)} dxdr = 0 8' 2 H1 and _Z t 0 Z R2 R1 u (r; x) (r; x) dxdr 0 for each positive 2 C01([0; T ) (R1; R2)).

Remark 1.2:

1) Our de…nition 1.1 is resulted from the multiplication with ' to the equation (0:1) and integration in (R1; R2) from 0 to t.

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

value problem (0:1) possesses a unique solution in H1 [Li1]. Hereafter we use the notations: 1 C(R1;R2) := 1= sup kuk₂= @u @x ₂: u 2 H 1 0(R1; R2) ; q = sup kukq= @u @x ₂: u 2 H 1 0( ) \ Lq( ) ; q > 1:

In this study we need the following lemmas

Lemma 1.3:

_{Suppose that u 2 H1 is a weakly positive solution of (SL) 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 t}₀_{> 0 with}

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

and a0_{(t) = 0:}

Lemma 1.4

:

Suppose that u is a positive weakly solution in H1 of equation (0.1) with u (0; ) = 0 = _u (0; ) in L2_(R

1; R2). For p > 1; q > 1; > 0; we have

u 0 in H1.

According to Lemma 1.4, 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.}

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We should study the problem (0:1) under the following situations: 2. Estimates for the Life-Span of the Solutions of (0.1) 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 2.1. 1) The local existence and uniqueness of solutions of equation (0:1) in H1 are known [Li2].

2) For higher space dimensional special cases under general bounded domain Rn_; _{= 0:}

i) If n = 2; p > 1 and E (0) = 0; the life-span of the positive solution u 2 H1 of equation (0.1) 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( ) :

ii) For n = 3; p = 2 and E (0) = 0; the life-span of the positive solution u 2 H1 of equation (0.1) is bounded

T 2:= 2 1sin 1 2C a0(0)2a (0) 2+ 4C2

1 2

for some constant C : If T = 2; then a (t) ! 1; t ! T:

iii) For n = 3 , p = 3; E (0) = 0 the life-span of the positive solution u 2 H1 of equation (0.1) is bounded T 3:= 1sin 1 2C a0(0)2a (0) 2+ 4C2 1 2 : If T = 3; then a (t) ! 1; t ! T:

iv) For a0(0) = 0, we have

1=

p 1C : v) For j j ! 1; we have also 1!

1 p 1 a (0) a0₍₀₎: As j j ! 0; then 1! 2 p 1sin 1 1 4C 1 _:

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3. Estimates for the Life-Span of the Solutions of equation (0.1) 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 (0.1)}

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

a0_{(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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6 M EN G -RO N G LI

4. Stability of positive solutions of equation (0.1) near blow-up solutions under Negativ-Energy

In this study we use our ideals used in [Li4]; [Li5] and [Li 7]to deal such problem (0:1) on our topics under negative energy and obtain the following results:

Theorem 4.1: Suppose that u 2 H1 is a weakly positive solution of (SL) with E (0) 0 for p > 1; q > 1; then for a (0) > 0 we have:

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

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

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

Remarks:

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.

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Reference:

[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 _{= 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, 2009 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).

非線性擾動下有界域半線性波方式爆炸解之穩定性研究

行政院國家科學委員會專題研究計畫 成果報告