一類交叉耦合拋物方程組解的整體存在及爆破

薛應(yīng)珍

(西安外事學(xué)院 商學(xué)院, 陜西 西安 710077)

一類交叉耦合拋物方程組解的整體存在及爆破

薛應(yīng)珍

(西安外事學(xué)院 商學(xué)院, 陜西 西安 710077)

為了更好描述3種混合物質(zhì)燃燒的熱傳導(dǎo)過程，或者化學(xué)反應(yīng)中3種反應(yīng)物的反應(yīng)情況，研究了一類具有3個變量交叉耦合且?guī)в蟹蔷植吭醇胺蔷植窟吔缌鲯佄镄头匠探M解的整體存在及有限時刻爆破問題,打破常用的第一特征值等構(gòu)造上下解的方法，而采用常微分方程方法構(gòu)造了該方程組的上、下解，引用比較定理，證明得到了由冪函數(shù)局部源和指數(shù)函數(shù)非局部源交叉耦合的退化拋物型方程組解的整體存在及解在有限時刻爆破的充分條件，為熱傳導(dǎo)和化學(xué)反應(yīng)問題提供更好的理論支持.

交叉拋物型方程組；比較原理；整體存在；爆破

針對交叉耦合的拋物方程組解的漸近性態(tài)問題，許多學(xué)者已做了大量研究，文獻[1]研究了具有3個變量交叉耦合的局部源和非局部邊界拋物型方程組解的漸近性態(tài)，得到了解整體存在及有限時刻爆破的充分條件.文獻[2]研究了具有2個冪函數(shù)作為局部源耦合的拋物型方程組解局部存在、整體存在和全局爆破的充分條件.文獻[3]將文獻[2]的結(jié)論進行了推廣.文獻[4-6]研究了具有冪函數(shù)耦合拋物型方程組解的漸近性態(tài).文獻[7]研究了一類擬線性拋物型方程組解在有限時刻爆破的充分條件及同時爆破的充分必要條件. 文獻[8-19]研究了其他如具有非局部吸收源等交叉耦合的拋物型方程組解的漸近性態(tài)等.

基于以上工作，本文研究了如下由冪函數(shù)和對數(shù)函數(shù)的非局部源交叉耦合，且具有3個變量交叉耦合退化拋物型方程組解的整體存在及解在有限時刻爆破的充分條件.

(1)

具有非局部邊界流

(2)

及連續(xù)有界初值

(3)

1 預(yù)備知識

(4)

具有非局部邊界流

(5)

及連續(xù)有界初值

(6)

由文獻[20],有如下的比較引理:

2 解的整體存在

定理1 如果m1m2m3>p1p2p3+q1q2q3+3，對于小初值u0(x)、v0(x)、w0(x)，方程組(1)-(3)的解整體存在.

證明：設(shè)φ(x)滿足

(7)

ap3bq3K1(p3m2+q3m1)/m1m2Ω.

綜上可知，只要存在a、b、c，使得

(8)

下證這樣的a、b、c存在.令bp1=am1c-q1K1-(p1m3+q1m2)/m2m3Ω-1ε0，將此式代入式(8)中，可得關(guān)于a的不等式

(9)

由定理1條件知m1m2m3>p1p2p3+q1q2q3+3，(m1m2-p1q2)(m3p1+q1q3)>(p1q2+m2q1)·(p1p3+m1q3)，只要取a充分大時，可使得式(9)成立.另只要a、b、c充分大，又對于小初值u0(x)、v0(x)、w0(x)，就可以保證式(8)的后3個式子成立.定理1證畢.

3 解的有限時刻爆破

討論解的整體存在問題時，引入以下2個引理:

引理3 設(shè)θ>λ>1，k、l>0,h(t)是問題

(10)

的正解，則當(dāng)h0充分大時，h(t)>2在有限時刻爆破.

引理4 設(shè)λ2>λ1>1,Q2>Q1>1，則引理3的(h(t))滿足

引理3及引理4的證明見文獻[5].

定理2 如果m1m2m3

證明：設(shè)φ(x)是滿足方程

(11)

的解，則存在C>2,使得0≤φ(x)≤C.令

其中：l1、l2、l3均大于1，h(t)待定，由式(10)-(11)可知，hliφl2(x)>2(i=1,2,3),則對于任意正實數(shù)α，利用拉格朗日中值定理證明可知lnα[hli(t)φli(x)]>h-αli(t)φ-αli(x)，記

∫Ωhl2p1(t)φl2p1(x)lnq1(hl3(t)φl3(x))dx≤l1hl1-1(t)φl1(x)h'(t)-

l1m1(l1m1-1)hl1m1(t)φl1m1-2(x)Δφ(x)-∫Ωhl2p1(t)φl2p1(x)h-l3q1(t)φ-l3q1(x)dx=

l1hl1-1(t)φl1(x)[h'(t)+m1(l1m1-1)hl1m1-l1+1(t)φl1m1-l1-2(x)-

綜上由引理3的條件可知，只要存在l1、l2、l3，使得

(12)

成立.則由引理4知，存在滿足引理3的h(t)使得

(13)

由定理2條件m1m2m3

(q1q2+m1p2)(p1p3-m1q3)>(p1q2+m1m2)(m1m2+p3q1).

即式 (13)成立，定理2證畢.

[1] 吳春晨.一類非局部邊值條件拋物型方程組解的性質(zhì)[J].江南大學(xué)學(xué)報(自然科學(xué)版)，2015，14(2):222-225. DOI:10.3969/j.issn.1671-7147.2015.02.018. WU C C. A nonlinear parabolic system with nonlocal boundary conditions[J]. Journal of Jiangnan University( Natural Science Edition), 2015，14(2):222-225. DOI:10.3969/j.issn.1671-7147.2015.02.018.

[2] 宋慧，曾有棟.具有非局部邊界和局部化源拋物方程組解的全局存在與爆破性[J].生物數(shù)學(xué)學(xué)報，2014,29(4)：711-717. SONG H, ZENG Y D. Global blow-up for a localized nonlinear parabolic system with a nonlocal boundary condition[J]. Journal of Biomathematics, 2014,29(4)：711-717.

[3] 吳春晨.一類非局部邊值條件拋物型方程組解的性質(zhì)研究[J] .鄭州大學(xué)學(xué)報( 理學(xué)版),2014,46(4):18-22。 WU C C.The properties of solutions for a nonlinear parabolic system with nonlocal boundary conditions[J] .Zhengzhou Univ ( Nat Sci Ed ) ,2014,46(4):18-22.

[4] 樊彩虹，容躍堂，房春梅，等.退化反應(yīng)擴散方程組解的整體存在和有限爆破[J].紡織高校基礎(chǔ)科學(xué)學(xué)報，2016,29(1): 35-38. DOI:10.13338/j.issn.1006-8341.2016.01.006. FAN C H,RONG Y T,FANG C M, etc. Global existence and finite time blow-up for a degenerate reaction-diffusion system [J]. Basic Sciences Journal of Textile Universities, 2016,29(1):35-38. DOI:10.13338/j.issn.1006-8341.2016.01.006.

[5] 王文海.具非局部源和非局部邊界條件拋物方程組解的性質(zhì)[J].中北大學(xué)學(xué)報(自然科學(xué)版)，2012，33(4): 372-375. DOI:10.3969/j.issn.1673-3193.2012.04.005. WANG W H. Properties of solution to a parabolic system with nonlocal source and nonlocal boundary conditions[J].Journal of North University of China(Natural Science Edition) ，2012，33(4):372-375. DOI:10.3969/j.issn.1673-3193.2012.04.005.

[6] 樊彩虹,李平，郭曉霞，等.半線性反應(yīng)擴散耦合系統(tǒng)解的整體存在與爆破[J].內(nèi)蒙古師范大學(xué)學(xué)報(自然科學(xué)漢文版),2015,44(6):735-737,742. DOI:10.3969/j.issn.1001-8735.2015.06.004. FAN C H,LI P,GUO X X,et al.The Global existence and blow-up for the solutions of the system of semilinear reaction diffusion couling[J].Journal of Inner Mongolia Normal University(Natural Science tdition),2014,43(6): 680-688. DOI:10.3969/j.issn.1001-8735.2015.06.004.

[7] 王玉蘭，穆春來.一類具有非局部源的擬線性拋物型方程組的一致爆破模式[J].四川大學(xué)學(xué)報(自然科學(xué)版)，2008，45(5):7001-3101. DOI:10.3969/j.issn.0490-6756.2008.05.002. WANG Y L, MU C L. Uniform blow up profiles for a quasilinear parabolic system with nonlocal sources[J].Journal of Sichuan University(Natural Science Edition), 2008，45(5):7001-3101. DOI:10.3969/j.issn.0490-6756.2008.05.002.

[8] 周澤文, 凌征球.源項耦合的退化拋物型方程組解的爆破和整體存在[J].應(yīng)用數(shù)學(xué)，2015, 28(3): 540-548. ZHOU Z W, LING Z Q. Blow-up and global existence of solutions to a degenerate parabolic equations coupled via nonlinear sources[J]. Mathematica Applicata, 2015, 28(3): 540-548.

[9] 薛應(yīng)珍.一類具有非線性吸收項和邊界流的拋物型方程組解的整體存在及爆破問題[J].紡織高校基礎(chǔ)科學(xué)學(xué)報，2013，26(2)：214-219. DOI:10.3969/j.issn.1006-8341.2013.02.017. XUE Y Z. Global existence and blow up problem for a parabolic equations with nonlinear absorption term and boundary flux[J]. Basic Sciences Journal of Textile Universities, 2013，26(2)：214-219. DOI:10.3969/j.issn.1006-8341.2013.02.017.

[10] 龐鳳琴，王玉蘭，李慧芳. 一類帶吸引項的拋物型方程在記憶邊界條件下解的性質(zhì)[J]. 西華大學(xué)學(xué)報( 自然科學(xué)版),2016,35(2):82-87. DOI:10.3969/j.issn.1673-159X.2016.02.016. PANG F Q，WANG Y L，LI H F.The properties of a parabolic equation with absorb term and memoryboundary condition[J].Journal of Xihua University( Natural Science)2016,35(2):82-87. DOI:10.3969/j.issn.1673-159X.2016.02.016.

[11] ZHANG H，KONG L H, ZHANG S N. Propagations of singularities in a parabolic system with coupling nonlocal sources[J].Sci in China: Ser A，2009，52(1):181-194. DOI:10.1007/s11425-008-0135-7.

[12] LI Y X,DENG W B,XIE C H.Global existence and nonexistence for degenerate parabolic systems [J]. Proc Amer Math Soc,2002,130(12):3661-3670.

[13] DENG W B,LI Y X,XIE C H.Existence and nonexistence of global solution of some nonlocal degenerate parabolic system[J].Proc Amer Math Soc,2003,131(5):1573-1582. DOI:10.1016/s0893-9659(03)80118-0.

[14] ZHANG Y,SONG X J.Global existence and blow-up for a degenerate parabolic systems with nonlocal source [J].Journal of Southwest University(Natural Science Edition),2008,30(9):80-84.

[15] BUDDAL C J,DOLD J W,GALAKTIONOV V A.Global blow-up for a semilinear heat equation on a subspace [J].Proceedings of the Royal Society of Edinburgh:Section A Mathematics,2015,145(5):893-923. DOI:10.1017/s0308210515000256.

[16] 黨蘇娟,容躍堂,張航國.非局部反應(yīng)擴散方程組解的整體存在與爆破[J].紡織高校基礎(chǔ)科學(xué),2013,26(4): 481-485. DANG S J,RONG Y T,ZHANG H G.Global existence and blow-up of solution for nonlocal reaction diffusion equtions[J].Basic Sciences Journal of Textile Universities,2013,26(4):481-485.

[17] 樊彩虹,李平,郭曉霞，等.具非局部源退化拋物系統(tǒng)解的整體存在與爆破[J].內(nèi)蒙古師范大學(xué)學(xué)報(自然科學(xué)漢文版),2014,43(6):680-688. FAN C H,LI P ,GUO X X,et al.Global existence and blow-up for the solutions of a degenerate parabolic system with nonlocal sources[J].Journal of Inner Mongolia Normal University(Natural Science Edition),2014,43(6): 680-688.

[18] 樊彩虹,容躍堂.一類帶非局部源的反應(yīng)擴散方程組解的整體存在[J].紡織高校基礎(chǔ)科學(xué)學(xué)報,2009,22(2): 172-176. DOI:10.3969/j.issn.1006-8341.2009.02.011. FAN C H,RONG Y T.Global existence for a degenerate reaction-diffusion system with nonlocal sources[J]. Basic Sciences Journal of Textile Universities,2009,22(2):172-176. DOI:10.3969/j.issn.1006-8341.2009.02.011.

[19] ZHANG H，KONG L H, ZHENG S N.Propagations of singularities in a parabolic system with coupling nonlocal sources[J].Sci in China:Ser A，2009，52(1):181-194. DOI:10.3969/j.issn.1006-8341.2009.02.009.

[20] PAO C V. Nonlinear parabolic and elliptic equations[M].London:Plenum Press,1992.

(責(zé)任編輯：王蘭英)

Global existence and blow up problem for a parabolic equations cross coupled terms

XUE Yingzhen

(College of Business, Xi'an International University, Xi'an 710077, China)

In order to better describe the heat transfer process of three kinds of mixed substances, or the reaction of the reactants in the three chemical reactions,a class of three variable cross coupling with non parabolic equations of the whole existence of local source and non local boundary flow and the finite time blow up problem with breaking method for the solution of the first commonly used feature value structure are studied, and the structure of the equations of the upper and lower solutions by using the method of ordinary differential equation reference, comparison theorem, the proof obtained by local source power function and exponential function of parabolic equations and the sufficient conditions for global existence of solutions blow up in finite time degradation of non local sources of cross coupling, provide better support for the theory of heat transfer and chemical reaction problem.

parabolic equations cross coupled terms; the comparison principle; global solution; blow up

10.3969/j.issn.1000-1565.2017.04.002

2017-02-21

陜西省自然科學(xué)基礎(chǔ)研究計劃項目(2016JM1036)；陜西省教育科學(xué)十三五規(guī)劃課題(SGH16H292)

薛應(yīng)珍(1980—)，男，甘肅慶陽人，西安外事學(xué)院副教授，主要從事偏微分方程理論及應(yīng)用的研究. E-mail：xueyingzhen@126.com

O175.26

A

1000-1565(2017)04-0343-06