一类交叉耦合抛物型方程组解的渐近性态
2017-04-14薛应珍
薛应珍
(西安外事学院商学院,陕西西安 710077)
一类交叉耦合抛物型方程组解的渐近性态
薛应珍
(西安外事学院商学院,陕西西安 710077)
为了更好地描述3种混合物质燃烧的热传导过程,即3种化学反应中反应物的反应情况,研究了一类具有3个变量交叉耦合且带有非局部源及非局部边界流抛物型方程组解的整体存在和有限时刻爆破问题,打破常用的第一特征值的构造上下解的方法,采用常微分方程方法构造了该方程组的上下解,引用比较定理,证明得到了由幂函数局部源和指数函数非局部源交叉耦合的退化抛物型方程组解的整体存在及解在有限时刻爆破的充分条件,为热传导和化学反应问题提供了理论支持。
抛物型偏微分方程;比较原理;整体存在;爆破;热传导
针对交叉耦合抛物方程组解的渐近性态问题,文献[1]研究了如下具有3个变量交叉耦合的局部源和非局部边界抛物型方程组解的渐近性态,得到了解整体存在及有限时刻爆破的充分条件。
u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω。
文献[2]研究了具有2个幂函数作为局部源耦合抛物型方程组解的性质,得到了方程组解局部存在,整体存在和全局爆破的充分条件。文献[3]将文献[2]的结论推广到了3个变量的情形。文献[4—6]研究了具有幂函数耦合非局部源和边界流抛物型方程组解的整体存在及有限时刻爆破的充分条件。文献[7]研究了如下具有幂函数和指数函数交叉耦合的非局部源拟线性抛物型方程组的一致爆破模式,得到了解在有限时刻爆破的充分条件及同时爆破的一个必要条件。
u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω。
对具有非局部吸收源等交叉耦合抛物型方程组解的渐近性态研究,参见文献[8—19]。
基于以上工作,本文研究了如下由幂函数局部源和指数函数非局部源交叉耦合,且具有3个变量交叉耦合退化抛物型方程组解的整体存在及解在有限时刻爆破的充分条件。
(1)
具有边界流:
(2)
及初值:
u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω,
(3)
1 基本理论
首先给出2种情况下上、下解的定义。
(4)
具有如下边界流:
(5)
及初值:
(6)
定义2 若定义1中式(4)变为
(7)
可得m,n,h>1,αi>0,βi<0(i=1,2,3),情形下,方程组(1)-方程组(3)的上解,若改变不等号方向,可定义其下解。
由文献[20],有如下比较引理。
(8)
2 解的整体存在
讨论方程组(1)-方程组(3)解的整体存在问题时,设αi>0,βi<0(i=1,2,3),引用定义2来证明方程组(1)-方程组(3)解整体存在的充分条件。
定理1 若mnh>α1α2α3,当初值u0(x),v0(x),w0(x)充分小时,方程组(1)-方程组(3)的解整体存在。
证明 设φ(x)满足:
(9)
联立式(7)-式(9),得到:
类似地有:
在边界上:
类似地有:
在初值上:
综上,只要证明存在a,b,c,使得:
(10)
(11)
成立,由引理1知,方程组(1)—方程组(3)的解整体存在。
(12)
由定理1条件mnh>α1α2α3可知,当a充分大时式(12)成立。另当a,b,c充分大,存在小初值u0(x),v0(x),w0(x),使得式(11)成立,定理1证毕。
3 解的有限时刻爆破
讨论解的有限时刻问题时,设αi,βi>0,i=1,2,3,引用定义1来证明方程组(1)—方程组(3)解有限时刻爆破的充分条件。首先引入以下引理。
引理3 设θ>λ>1,k,l>0,h(t)是问题
(13)
的正解,则当h0充分大时,h(t)在有限时刻爆破。
引理4 设λ2>λ1>1,θ2>θ1>1,则存在如引理3的h(t)是满足:
引理3及引理4证明见文献[5]。
定理2 如果mnh<α1α2α3,则当初值u0(x),v0(x),w0(x)充分大时,方程组(1)—方程组(3)的解在有限时刻爆破。
证明 设φ(x)是满足方程:
(14)
的解,则存在C>0,使得0≤φ(x)≤C。令
其中,l1,l2,l3>1,h(t)待定,由式(13)及式(14)可知,h(t)φ(x)>1,显然,hli(t)φli(x)>1,i=1,2,3,则对于任意正实数η,利用拉格朗日中值定理证明可知,
eη hli(t)φli(x)>ηehli(t)φli(x)>ηelnli[1+h(t)φ(x)]。
记k=max{m(l1m-1)Cl1(m-1)-2,n(l2n-1)Cl2(n-1)-2,h(l3h-1)Cl3(h-1)-2},
则
l1hl1-1(t)φl1(x)h′(t)-l1m(l1m-1)hl1m(t)φl1m-2(x)Δφ(x)-hl2α1(t)φl2α1(x)∫Ωeβ1hl3(t)φl3(t)dx≤
l1hl1-1(t)φl1(x)h′(t)+l1m(l1m-1)hl1m(t)φl1m-2(x)-hl2α1(t)φl2α1(x)∫Ωeβ1hl3(t)φl3(t)dx=
l1hl1-1(t)φl1(x)[h′(t)+khl1(m-1)+1(t)-lhl2α1+l3-l1+1(t)]。
同理,
综上由引理3的条件可知,只要存在l1,l2,l3使得:
(15)
成立,则由引理4知,存在满足引理3的h(t)使得:
h′(t)≤-khl1(m-1)+1(t)+lhl2α1+l3-l1+1(t),
h′(t)≤-khl2(n-1)+1(t)+lhl3α2+l1-l2+1(t),
h′(t)≤-khl3(h-1)+1(t)+lhl1α3+l2-l3+1(t),
在边界上:
由式(14)知,φ(x)=0,x∈∂Ω,有:
在初值上:
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Asymptotic behavior for cross coupled parabolic equations
XUE Yingzhen
(Business School, Xi’an International University, Xi’an, Shaanxi 710077, China)
In order to better describe the heat transfer process of three kinds of mixed substances, namely 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. The structure of the equations of the upper and lower solutions by using the method of ordinary differential equation reference is broken, with comparison theorem, the proof shows that obtained by local source power function and exponential function of parabolic equations is broken, with the sufficient conditions for global existence of clegerate purubolic equations solutions cross coupled by local source power function and non local sources exponential function are proved, as soon as the solution of blowing up in finite time degradation of non local sources of cross coupling, providing better support for the theory of heat transfer and chemical reaction problem.
the parabolic partial differential equations; comparison principle; global existence; blow-up question; conduction of heat
1008-1542(2017)02-0137-06
10.7535/hbkd.2017yx02006
2016-11-18;
2016-12-27;责任编辑:张 军
陕西省自然科学基础研究计划项目(2016JM1036);陕西省教育厅科学研究计划项目(15JK2134);西安外事学院高等教育教学改革研究项目(2015B04)
薛应珍(1980—),男,甘肃庆阳人,副教授,硕士,主要从事偏微分方程理论及应用方面的研究。
E-mail:xueyingzhen@126.com
O175.26 MSC(2010)主题分类:35K46
A
薛应珍.一类交叉耦合抛物型方程组解的渐近性态[J].河北科技大学学报,2017,38(2):137-142.
XUE Yingzhen.Asymptotic behavior for cross coupled parabolic equations[J].Journal of Hebei University of Science and Technology,2017,38(2):137-142.