半线性双温度热传导方程柯西问题
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摘要
本文研究以下半线性双温度热传导方程u_(t)—△u—△u_(t)=u=f(u),x∈R~n,＞0,u(x,0)=u_0(x),x∈R~n.的柯西问题(初值问题).半线性双温度热传导方程是在物理学中提出的一类非线性拟抛物方程.
本文采取的研究方法主要是位势井及位势井族理论.
首先,应用位势井族理论研究了解的不变集合,得到了解的真空隔离现象.
其次,研究了该问题的整体弱解的存在性与解的blow-up,得到了解的整体存在性与不存在性的门槛结果.
再次,运用Galerkin方法并且结合了位势井族理论研究了在临界条件J(u_0)=d下,问题的整体弱解的存在性.
最后,讨论了问题解的渐近性.
In this paper,we study the Cauchy problem for a class of semilinear double temperature heat equation u_t-△u-△u_t+u=f(u),x∈R~n,t＞0, u(x,0)=u_0(x),x∈R~n. Semilinear double temperature heat equation is one of nonlinear pseudoparabolic equations arised from physics.
This paper adopts the method of potential well and the family of potential wells.
Firstly,we study the invariant sets of solutions by using the family of potential wells and obtain vacuum isolating behavior of solutions.
Secondly,we study the the existence of the global weak solutions and blow-up of solutions.We obtain a threshold result of global existence and nonexistence of solutions.
Thirdly,we study the existence of the global weak solutions of the problem with critical initial condition J(u_0)=d by using the method of Galerkin and the family of potential wells.
Finally,we study the asymptotic behavior of solutions of the problem.
本文采取的研究方法主要是位势井及位势井族理论.
首先,应用位势井族理论研究了解的不变集合,得到了解的真空隔离现象.
其次,研究了该问题的整体弱解的存在性与解的blow-up,得到了解的整体存在性与不存在性的门槛结果.
再次,运用Galerkin方法并且结合了位势井族理论研究了在临界条件J(u_0)=d下,问题的整体弱解的存在性.
最后,讨论了问题解的渐近性.
In this paper,we study the Cauchy problem for a class of semilinear double temperature heat equation u_t-△u-△u_t+u=f(u),x∈R~n,t＞0, u(x,0)=u_0(x),x∈R~n. Semilinear double temperature heat equation is one of nonlinear pseudoparabolic equations arised from physics.
This paper adopts the method of potential well and the family of potential wells.
Firstly,we study the invariant sets of solutions by using the family of potential wells and obtain vacuum isolating behavior of solutions.
Secondly,we study the the existence of the global weak solutions and blow-up of solutions.We obtain a threshold result of global existence and nonexistence of solutions.
Thirdly,we study the existence of the global weak solutions of the problem with critical initial condition J(u_0)=d by using the method of Galerkin and the family of potential wells.
Finally,we study the asymptotic behavior of solutions of the problem.
引文
[1]Yacheng Liu.On potential wells and vacuum isolating of solutions for semilinear wave equations.Journal of Differential Equations,2003,192:155-169P
[2]Payne L E,Sattinger D H.Sadle points and instability of nonlinear hyperbolic equations.Israel J Math,1975,22:273-303P
[3]Nakao M.L~(p) -estimates of solutions of some nonlinear degenerate diffusion equations.J Math Soc Japan,1985,37(1):23-30P
[4]刘亚成,王峰.一类非线性多维Sobolev-Galpern方程.应用数学学报.1994,17(4):569-577页
[5]GUO Boling,Miao Changxing.On inhomogeneous GBBM equation.J Partial Diff Eqs,1995,8(3):193-204P
[6]刘亚成,于涛.神经传播型方程解的blow-up.应用数学学报.1995,18(4):264-272页
[7]刘亚成,徐润章.半线性拟抛物方程的整体W~(1,2)解.哈尔滨工程大学学报.2004,25(2):251-253页
[8]刘亚成,徐润章.半线性拟抛物方程的整体W~(1,p)解.哈尔滨工程大学学报.2004,25(5):677-679页
[9]刘亚成,徐润章.半线性拟抛物方程的整体W~(2,p)(1＜p＜2)解.哈尔滨工程大学学报.2004,25(3):394-396页
[10]刘亚成,徐润章.半线性拟抛物方程的整体W~(k,p)(k≥3,1＜p＜∞)解与古典解.哈尔滨工程大学学报.2005,26(2):277-278页
[11]Liu Yacheng,Du Juan.Global W~(2,2) solutions of semilinear pseudoparabolic equations on unbounded domain.Journal of natural science of Heilongjiang university,2005,22(2):219-223P
[12]杜鹃,钱妍.半线性拟抛物方程古典解的Blow-up问题.哈尔滨师范大学自然科学学报.2006,22(3):17-18页
[13]杨海鸥,刘亚成.任意维数半线性拟抛物方程的初边值问题.哈尔滨工程大学学报.2003,24(4):460-463页
[14]徐润章,刘亚成.两类非线性拟抛物方程解的有限时间爆破和长时间行为.工程数学学报.2005,22(5):800-806页
[15]尚亚东,郭柏灵.非线性拟抛物型积分微分方程的初边值问题与初值问题.应用数学.2002,15(1):40-45页
[16]雷沛东.拟线性退缩抛物方程Cauchy问题广义解的稳定性和唯一性.吉林大学自然科学学报.1996,2:1-6页
[17]张新华.一类非自治系统的非线性抛物方程解的爆破.四川师范大学学报(自然科学版).2001,24(2):154-156页
[18]陈义安.一类拟线性抛物方程解的Blow-up.数学理论与应用.2004,24(1):1-3页
[19]Mizoguchi N,Yanagida E.Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation.Math Ann,1997,307(4):663-675P
[20]Kano T,Nishida T.A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves.Osaka J Math,1986,23:389-413P
[21]杨晗.一类非线性热传导方程混合问题整体解的存在唯一性.四川师范大学学报(自然科学版).1995,18(1):92-96页
[22]杨从军.一类拟线性退化抛物型方程的初边值问题.数学学报.1995,38(1):134-139页
[23]杨宏志,呼青英.一类非线性发展方程组整体解的存在性和不存在性.数学的实践与认识.2002,32(4):658-663页
[24]刘亚成.方程u_((tt)-△u_(t)=f(u)的整体解.数学物理学报.1989,9(2):155-166页
[25]刘亚成,徐润章.方程u_(tt)-△u-△u_(tt)=f(x)的整体广义解.哈尔滨工程大学学报.2005,3:406-408页
[26]王远弟.抛物型方程一类初值问题解的存在性和衰减.数学年刊.2006,27A(6):749-760页
[27]陶双平,崔尚斌.七阶非线性色散方程初值问题解的局部和整体存在性.数学物理学报.2005,25A(4):451-460页
[28]杨光俊.半线性抛物组Cauchy问题的局部解和整体解的存在性.云南大学学报(自然科学版).1997,19(3):241-248页
[29]肖应昆,梅茗.非线性四阶抛物型方程Cauchy问题整体经典解.江西师范大学学报(自然科学版).1994,18(3):197-204,235页
[30]谌德,向新民.全直线上非线性BBM方程Cauchy问题解的存在唯一性.上海师范大学学报(自然科学版).2005,34(3):7-10页
[31]田茂英.一类非线性抛物型方程的Cauchy问题与自由边界.宁夏大学学报(自然科学版).1996,17(1):25-26页
[32]幸冬梅.一类退化抛物方程初值问题的Blow-up.南昌大学学报(理科版).2001,25(3):326-329页
[33]Peter J C,Curtin M E.On a theory of heat conduction involving two temperatures.Z Angew Math Phys,1968,19:614-627P
[34]Peter J C,Curtin M E,Williams W O.On the thermodynamics of Non-simplelastic materials with two temperatures.Z Angew Math Phys,1969,20:107-112P
[35]刘亚成.半线性热方程整体解的存在性与非存在性.数学年刊.1997,18A(1):65-72页
[36]Fujita H.On some nonexistence and nonuniqueness theorem for nonlinear parabolic equations.Proc Sympos Pure Math,1969,18(1):105-113P
[37]Han Weiwei.Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for inhomogeneous quasilinear hyperbolic systems.J Partial Diff Eqs,2007,20:273-288P
[38]Li Shumin.Cauchy problem for general first order inhomogeneous quasilinear hyperbolic systems.J Partial Diff Eqs,2002,15:46-68P
[39]Chen Yunmei.Remark on the global existence for the GBBM equation in arbitrary dimensions.Appl Anal,30(1-3)(1988):1-15P
[40]Goldstein J A,Kajikiya R,Oharn S.On some nonlinear dispersive equations in several space variables.Differential and Integral Equation,1990,40(3):617-732P
[41]蹇素雯,杨凤藻.奇异半线性热方程的非奇性初值问题解的存在性与不存在性.云南工业大学学报.1997,13(2):83-86页
[42]蹇素雯.一类变系数半线性热方程初值问题解的Blow-up问题.云南师范大学学报.1997,17(2):1-9页
[43]Mizoguchi N,Yanagida E.Blowup and life span of solutions for a semilinear parabolic equation.SIAM J Math Anal,1998,29(6):1434-1446P
[44]Bandle C,Levine HA,Zhang Q S.Critical exponents of Fujita type for inhomogeneous parabolic equations and system.J Math Anal Appl,2000,251(2):624-648P
[45]Weissler F B.Existence and non-existence of global solutions for a semilinear heat equations.Israel J Math,1981,28(1-2):29-40P
[46]杨志坚,陈国旺.关于一类拟线性抛物方程解的Blow-up问题.高校应用数学学报.1994,9A(3):228-236页
[47]Yacheng Liu,Junsheng Zhao.Nonlinear parabolic equations with critical initial conditions J(u_0)=d or I(u_0)=0Nonlinear Anal,2004,58:873-883P
[48]Yacheng Liu,Junsheng Zhao.Multidimensional viscoelasticity equationtions with nonlinear damping and source terms.Nonlinear Anal,2004,56:851-865P
[49]Yacheng Liu,Weiming Wan.Gtobal solutions of GBBM equations in arbitrary dimensions.J Partial Diff Eqs,1999,12(1):63-70P
[50]Guo Boling.Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equation.Chin Ann Math,1987,8:226-238P
[51]王光烈,廉松哲.一个来自投资理论的抛物型Monge-Ampere方程初值问题.数学年刊.2005,26A(3):435-440页
[52]查中伟.非线性发展方程混合问题解的Blow-up.三峡大学学报.2002,24(3):276-279页
[53]曾文平.二维抛物型方程精细积分法与差分法比较.华侨大学学报(自然科学版).2002,23(1):5-11页
[54]谭忠.具有特殊扩散过程的反应扩散方程.数学年刊.2001,22A(5):597-606页
[2]Payne L E,Sattinger D H.Sadle points and instability of nonlinear hyperbolic equations.Israel J Math,1975,22:273-303P
[3]Nakao M.L~(p) -estimates of solutions of some nonlinear degenerate diffusion equations.J Math Soc Japan,1985,37(1):23-30P
[4]刘亚成,王峰.一类非线性多维Sobolev-Galpern方程.应用数学学报.1994,17(4):569-577页
[5]GUO Boling,Miao Changxing.On inhomogeneous GBBM equation.J Partial Diff Eqs,1995,8(3):193-204P
[6]刘亚成,于涛.神经传播型方程解的blow-up.应用数学学报.1995,18(4):264-272页
[7]刘亚成,徐润章.半线性拟抛物方程的整体W~(1,2)解.哈尔滨工程大学学报.2004,25(2):251-253页
[8]刘亚成,徐润章.半线性拟抛物方程的整体W~(1,p)解.哈尔滨工程大学学报.2004,25(5):677-679页
[9]刘亚成,徐润章.半线性拟抛物方程的整体W~(2,p)(1＜p＜2)解.哈尔滨工程大学学报.2004,25(3):394-396页
[10]刘亚成,徐润章.半线性拟抛物方程的整体W~(k,p)(k≥3,1＜p＜∞)解与古典解.哈尔滨工程大学学报.2005,26(2):277-278页
[11]Liu Yacheng,Du Juan.Global W~(2,2) solutions of semilinear pseudoparabolic equations on unbounded domain.Journal of natural science of Heilongjiang university,2005,22(2):219-223P
[12]杜鹃,钱妍.半线性拟抛物方程古典解的Blow-up问题.哈尔滨师范大学自然科学学报.2006,22(3):17-18页
[13]杨海鸥,刘亚成.任意维数半线性拟抛物方程的初边值问题.哈尔滨工程大学学报.2003,24(4):460-463页
[14]徐润章,刘亚成.两类非线性拟抛物方程解的有限时间爆破和长时间行为.工程数学学报.2005,22(5):800-806页
[15]尚亚东,郭柏灵.非线性拟抛物型积分微分方程的初边值问题与初值问题.应用数学.2002,15(1):40-45页
[16]雷沛东.拟线性退缩抛物方程Cauchy问题广义解的稳定性和唯一性.吉林大学自然科学学报.1996,2:1-6页
[17]张新华.一类非自治系统的非线性抛物方程解的爆破.四川师范大学学报(自然科学版).2001,24(2):154-156页
[18]陈义安.一类拟线性抛物方程解的Blow-up.数学理论与应用.2004,24(1):1-3页
[19]Mizoguchi N,Yanagida E.Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation.Math Ann,1997,307(4):663-675P
[20]Kano T,Nishida T.A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves.Osaka J Math,1986,23:389-413P
[21]杨晗.一类非线性热传导方程混合问题整体解的存在唯一性.四川师范大学学报(自然科学版).1995,18(1):92-96页
[22]杨从军.一类拟线性退化抛物型方程的初边值问题.数学学报.1995,38(1):134-139页
[23]杨宏志,呼青英.一类非线性发展方程组整体解的存在性和不存在性.数学的实践与认识.2002,32(4):658-663页
[24]刘亚成.方程u_((tt)-△u_(t)=f(u)的整体解.数学物理学报.1989,9(2):155-166页
[25]刘亚成,徐润章.方程u_(tt)-△u-△u_(tt)=f(x)的整体广义解.哈尔滨工程大学学报.2005,3:406-408页
[26]王远弟.抛物型方程一类初值问题解的存在性和衰减.数学年刊.2006,27A(6):749-760页
[27]陶双平,崔尚斌.七阶非线性色散方程初值问题解的局部和整体存在性.数学物理学报.2005,25A(4):451-460页
[28]杨光俊.半线性抛物组Cauchy问题的局部解和整体解的存在性.云南大学学报(自然科学版).1997,19(3):241-248页
[29]肖应昆,梅茗.非线性四阶抛物型方程Cauchy问题整体经典解.江西师范大学学报(自然科学版).1994,18(3):197-204,235页
[30]谌德,向新民.全直线上非线性BBM方程Cauchy问题解的存在唯一性.上海师范大学学报(自然科学版).2005,34(3):7-10页
[31]田茂英.一类非线性抛物型方程的Cauchy问题与自由边界.宁夏大学学报(自然科学版).1996,17(1):25-26页
[32]幸冬梅.一类退化抛物方程初值问题的Blow-up.南昌大学学报(理科版).2001,25(3):326-329页
[33]Peter J C,Curtin M E.On a theory of heat conduction involving two temperatures.Z Angew Math Phys,1968,19:614-627P
[34]Peter J C,Curtin M E,Williams W O.On the thermodynamics of Non-simplelastic materials with two temperatures.Z Angew Math Phys,1969,20:107-112P
[35]刘亚成.半线性热方程整体解的存在性与非存在性.数学年刊.1997,18A(1):65-72页
[36]Fujita H.On some nonexistence and nonuniqueness theorem for nonlinear parabolic equations.Proc Sympos Pure Math,1969,18(1):105-113P
[37]Han Weiwei.Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for inhomogeneous quasilinear hyperbolic systems.J Partial Diff Eqs,2007,20:273-288P
[38]Li Shumin.Cauchy problem for general first order inhomogeneous quasilinear hyperbolic systems.J Partial Diff Eqs,2002,15:46-68P
[39]Chen Yunmei.Remark on the global existence for the GBBM equation in arbitrary dimensions.Appl Anal,30(1-3)(1988):1-15P
[40]Goldstein J A,Kajikiya R,Oharn S.On some nonlinear dispersive equations in several space variables.Differential and Integral Equation,1990,40(3):617-732P
[41]蹇素雯,杨凤藻.奇异半线性热方程的非奇性初值问题解的存在性与不存在性.云南工业大学学报.1997,13(2):83-86页
[42]蹇素雯.一类变系数半线性热方程初值问题解的Blow-up问题.云南师范大学学报.1997,17(2):1-9页
[43]Mizoguchi N,Yanagida E.Blowup and life span of solutions for a semilinear parabolic equation.SIAM J Math Anal,1998,29(6):1434-1446P
[44]Bandle C,Levine HA,Zhang Q S.Critical exponents of Fujita type for inhomogeneous parabolic equations and system.J Math Anal Appl,2000,251(2):624-648P
[45]Weissler F B.Existence and non-existence of global solutions for a semilinear heat equations.Israel J Math,1981,28(1-2):29-40P
[46]杨志坚,陈国旺.关于一类拟线性抛物方程解的Blow-up问题.高校应用数学学报.1994,9A(3):228-236页
[47]Yacheng Liu,Junsheng Zhao.Nonlinear parabolic equations with critical initial conditions J(u_0)=d or I(u_0)=0Nonlinear Anal,2004,58:873-883P
[48]Yacheng Liu,Junsheng Zhao.Multidimensional viscoelasticity equationtions with nonlinear damping and source terms.Nonlinear Anal,2004,56:851-865P
[49]Yacheng Liu,Weiming Wan.Gtobal solutions of GBBM equations in arbitrary dimensions.J Partial Diff Eqs,1999,12(1):63-70P
[50]Guo Boling.Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equation.Chin Ann Math,1987,8:226-238P
[51]王光烈,廉松哲.一个来自投资理论的抛物型Monge-Ampere方程初值问题.数学年刊.2005,26A(3):435-440页
[52]查中伟.非线性发展方程混合问题解的Blow-up.三峡大学学报.2002,24(3):276-279页
[53]曾文平.二维抛物型方程精细积分法与差分法比较.华侨大学学报(自然科学版).2002,23(1):5-11页
[54]谭忠.具有特殊扩散过程的反应扩散方程.数学年刊.2001,22A(5):597-606页