一类半线性抛物方程组的爆破临界指标
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摘要
抛物型微分方程反应了许多物理学、化学、生物学等现象,并且已经取得了非常丰富的成果[1]-[2].从1966年Fujita H.对非线性抛物方程展开研究以来,有Levine H A.Pinsky R G.Escobedo M.等一大批数学家开始关注抛物型方程的非线性研究,在行波解、方程组正平衡解分支与稳定性、解的爆破问题和渐近性质等等各方面取得了丰硕的成果,不但丰富了数学本身的研究方法,也为我们改造世界提供了依据[3]-[11]。
本文考虑半线性抛物方程组的Cauchy问题:其中:s是正整数,p_i＞0,m_i＞-2为实数,u_i~0(x),i=1,2…,s是定义在R~N上的非负连续函数。和
其中:m＞-2,n＞-2,p_i≥0,q_i≥0,i=1,2,u_0(x),v_0(x)是定义在R~N上的非负连续函数。
获得了以下主要结果:
1、方程组(Ⅰ)的正解的整体存在的爆破临界指标:
(1)设u_i~0(x)≥0,且u_i~0(x)≠0,则当1＜γ＜1＋2/N(1＋β)时,其中i=1,2…,s,方程组(Ⅰ)的解在有限时刻爆破。
(2)设p_i＞1,γ＞1＋2/N(1＋β),则当u_i~0(x)充分大时,其中i=1,2…,s,问题(Ⅰ)的解在有限时间爆破;而当甜u_i~0(x)充分小时(Ⅰ)具有整体解。
其中,β=(?),β_i=(?),i=1,2…,s。
2、方程组(Ⅱ)的正解的整体存在的爆破临界指标：
(1)当u_0(x)≥0,v_0(x)≥0,且δ≠0且max{α,β}＞N/2,则问题(Ⅱ)的解在有限时刻爆破。
(2)当u_0(x)≥0,v_0(x)≥0,且δ≠0且max{α,β}＜N/2时,则当u_0(x),v_0(x)充分大时,问题(Ⅱ)的解在有限时间爆破;当u_0(x),v_0(x)充分小且p_1＋q_1＞1时(Ⅱ)具有整体解。
其中:(?),δ=det(A－I)当δ≠0时,x=(α,β)~T表示方程(?)的解。
Parabolic differential equation responses to a number of physics, chemistry, biology, and has realized ample achievements [1] - [2]. Since 1966 Fujita H. began to study the non-linear equations, a number of mathematicians like Levine H A. Pinsky R G. Escobedo M. mathematicians have begun to pay attention to the study of parabolic equation of nonlinear, and they have made great achievements on good wave solutions, branches and stability of the positive equilibrium solution, and asymptotic nature of the explosion, which not only enriched the methods of mathematics study, but also provided basis to change the world[3] - [11].
This paper considered semi-linear parabolic equations of the Cauchy problem:
where s∈Z~+,p_i＞0,m_i∈(-2,+∞),u_i~0(x),i=1,2…,s is defined as non-negative continuous function in R~N.
Where m＞-2,n＞-2,p_i≥0,q_i≥0,i=1,2,u_0(x),v_0(x) are definedas non-negative continuous function in R~N.
The main results are as following:
1、The Blowing up critical exponent of equation (Ⅰ):
(1)Suppose u_0(x)≥0, v_0(x)≥0 and u_i~0(x)≠0, if 1＜γ＜1＋2/N(1＋β),i=1,2…,s,Then the solution of equaton (Ⅰ) is nonglobal.
(2)Suppose p_i＞1,γ＞1＋2/N(1＋β), Then the solutions ofequation (Ⅰ) exist globally for u_i~0(x) small enough and blowing-up in finite time for u_i~0(x) large enough.
Whereβ=(?),β_i=(?),i=1,2…,s.
2、The Blowing up critical exponent of equation (2)
(1)Suppose u_0(x)≥0, v_0(x)≥0 andδ≠0, if max{α,β}＞N/2,Thenthe solution of equation(Ⅱ) is nonglobal.
(2)Suppose u_0(x)≥0,v_0(x)≥0 andδ≠0 and max{α,β}＜N/2,Then the solution of equation (Ⅱ) exist globally for u_0(x),v_0(x)small enough and p_1＋q_1＞1,blowing-up in finite time foru_0(x),v_0(x) large enough.
Where A=(?),δ=det(A－I),ifδ≠0, let x=(α,β)~T be the unique solution of (?).
本文考虑半线性抛物方程组的Cauchy问题:其中:s是正整数,p_i＞0,m_i＞-2为实数,u_i~0(x),i=1,2…,s是定义在R~N上的非负连续函数。和
其中:m＞-2,n＞-2,p_i≥0,q_i≥0,i=1,2,u_0(x),v_0(x)是定义在R~N上的非负连续函数。
获得了以下主要结果:
1、方程组(Ⅰ)的正解的整体存在的爆破临界指标:
(1)设u_i~0(x)≥0,且u_i~0(x)≠0,则当1＜γ＜1＋2/N(1＋β)时,其中i=1,2…,s,方程组(Ⅰ)的解在有限时刻爆破。
(2)设p_i＞1,γ＞1＋2/N(1＋β),则当u_i~0(x)充分大时,其中i=1,2…,s,问题(Ⅰ)的解在有限时间爆破;而当甜u_i~0(x)充分小时(Ⅰ)具有整体解。
其中,β=(?),β_i=(?),i=1,2…,s。
2、方程组(Ⅱ)的正解的整体存在的爆破临界指标：
(1)当u_0(x)≥0,v_0(x)≥0,且δ≠0且max{α,β}＞N/2,则问题(Ⅱ)的解在有限时刻爆破。
(2)当u_0(x)≥0,v_0(x)≥0,且δ≠0且max{α,β}＜N/2时,则当u_0(x),v_0(x)充分大时,问题(Ⅱ)的解在有限时间爆破;当u_0(x),v_0(x)充分小且p_1＋q_1＞1时(Ⅱ)具有整体解。
其中:(?),δ=det(A－I)当δ≠0时,x=(α,β)~T表示方程(?)的解。
Parabolic differential equation responses to a number of physics, chemistry, biology, and has realized ample achievements [1] - [2]. Since 1966 Fujita H. began to study the non-linear equations, a number of mathematicians like Levine H A. Pinsky R G. Escobedo M. mathematicians have begun to pay attention to the study of parabolic equation of nonlinear, and they have made great achievements on good wave solutions, branches and stability of the positive equilibrium solution, and asymptotic nature of the explosion, which not only enriched the methods of mathematics study, but also provided basis to change the world[3] - [11].
This paper considered semi-linear parabolic equations of the Cauchy problem:
where s∈Z~+,p_i＞0,m_i∈(-2,+∞),u_i~0(x),i=1,2…,s is defined as non-negative continuous function in R~N.
Where m＞-2,n＞-2,p_i≥0,q_i≥0,i=1,2,u_0(x),v_0(x) are definedas non-negative continuous function in R~N.
The main results are as following:
1、The Blowing up critical exponent of equation (Ⅰ):
(1)Suppose u_0(x)≥0, v_0(x)≥0 and u_i~0(x)≠0, if 1＜γ＜1＋2/N(1＋β),i=1,2…,s,Then the solution of equaton (Ⅰ) is nonglobal.
(2)Suppose p_i＞1,γ＞1＋2/N(1＋β), Then the solutions ofequation (Ⅰ) exist globally for u_i~0(x) small enough and blowing-up in finite time for u_i~0(x) large enough.
Whereβ=(?),β_i=(?),i=1,2…,s.
2、The Blowing up critical exponent of equation (2)
(1)Suppose u_0(x)≥0, v_0(x)≥0 andδ≠0, if max{α,β}＞N/2,Thenthe solution of equation(Ⅱ) is nonglobal.
(2)Suppose u_0(x)≥0,v_0(x)≥0 andδ≠0 and max{α,β}＜N/2,Then the solution of equation (Ⅱ) exist globally for u_0(x),v_0(x)small enough and p_1＋q_1＞1,blowing-up in finite time foru_0(x),v_0(x) large enough.
Where A=(?),δ=det(A－I),ifδ≠0, let x=(α,β)~T be the unique solution of (?).
引文
[1] 叶其孝,李正元.反应扩散方程引论[M].北京:科学出版社,1999.
[2] 王明新.非线性抛物行型方程[M].北京:科学出版社,1993.
[3] Fujita H. On the blowing up of solutions of the Cauchy problem for u_t=Δu + u~(1+α). J. Fac. Sci.. Univ. Tokyo Sect.Ⅰ,1966,13: 109-124.
[4] Bandle C, Levine H A. On the existence and nonexistence of global solution of reaction-diffusion equations in sectorial domains. Trans. Amer. Math. Soc., 1989, 316: 595-622.
[5] Levine H. The role of critical exponents in blow-up theorems. SIAM Rew., 1990,32: 262-288.
[6] Pinsky R G. Existence and nonexistence of global solutions for u_t=Δu + α(x)u~p in R~d. J.Diff. Equ., 1997, 133: 152-177.
[7] Escobedo M, Herrero M A . Boundness and blow-up for a semilinear reaction-diffusion systems. J. Diff. Equ. , 1991,89:176-202.
[8] 戴求亿,一类半线性抛物方程组的爆破临界指标。应用数学学报,2002,25:339-346.
[9] Renclawowice J, Global existence and blow-up of solutions for a completely coupled Fujita type system of reaction-diffusion equations.Applicationes Mathemeticae 1998, 25(3): 313-326.
[10] Escobedo M, Levine H A, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch Rat Mech Anal, 1995, 129:47-100.
[11] Escobedo M, Herrero M A . A semilinear paraolic system in a bounded domain. Annali di Math. Pura ed Appl. (Ⅳ), 1993,115:315-336.
[12] D. J. Aronson and H.F.Weinberger, Multidimensional nonlinear diffusion arising in propulation genetics, Adv. in Math. 1978, 30:33-76.
[13] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equation-diffusion equations Quart, J. Math. Oxford 28 (1977) : 473-486.
[14] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear farabolic equations, Proc. Sympos. Pure Math.18(1969): 105-113.
[15] K.Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan. Acad 49(1973):503-525.
[16] K. Kobayashi, T. Siaro, and H. Tanaka, On the blowing up problem for semilinear heat equations,J. Math. Soc. Japan 29(1977): 407-422.
[17]H.Levine,The role of critical exponents in blowup theorems, SIAM Rev. 32(1990) :262-288.
[18]H. Levine and P. Meier, The value of the critical exponent for reaction-diffusion equations in cones, Arch. Rational Mech. Anal. 109 (1990), 73-80.
[19]A. Pazy, " Senugriyos of linear operators and applications to partial differential equations," Springer-Verlag,Berlin/New York, 1983.
[20]F.B. Weissler,Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math.38(1981): 29-40.
[21]Kawarada H., On solutions ofinitial-oundary walue problem for (?),Publ. Res. Inst. Math. Sci., Kyoto Univ., 1975, 10:729-736.
[22]Acker A.,Walter W., The quenching problem for nonlinear parabolic differential equation, Lecture Note in Math., NO. 564:1-12, New York, Spring-Verlag, 1976.
[23]Levine H. A.,Advances on quenching,Progress in Nonlinear Differential Equations and Its Applications, Vol 7:319-346,Birkhauser Boston, 1992.
[24]Levine H. A. , The phenomenon of quenching , A Survey, in Proc.Vith. Int. Conf. On Trends in the Theory and Practice of Nonlinear Analysis, New York, North Holland, 1985.
[25]Dai Q.Y.,Gu Y.G., Ashort note on quenching phenomena for semilienar parabolic equations, J. Diff. Equ., 1997, 137(2) :240-250.
[26]Fila M., Kawohl B., Levine H. A., Quenching for quasilinear equations , Comm. Partial Diff. Equ., 1992, 17(3/4) :593-614.
[27]Fila M. , Kawahl B. , Asymptotic analsysis of quenching problems, Rocky Mount. J. Math., 1992, 22:563-577.
[28]Deng K., Quenching for solutions of a plasma type equations, Nonlinear Anal., 1992, 18(8) :731-742.
[29]Dai Q. Y., Uniformly asymptotic global solution for singular semi linear parabolic equation, J. xiangtan Min. Inst., 1999,14(4) :85-91.
[30]Dai Q. Y., Zeng X. Z., The quenching phenomena for the cauchy problem of semilinear parabolic equation, J. Diff. Equ., 2001,175(1) :163-174.
[31]Dai Q. Y. , Gu Y. G., Asymptotic analsysis of global solutions for singular semilinear parabolic equation, Acta. Math. Appl.Sinica, 1998,21(1):57-65.
[32]Souplet P., Decay of heat semigroups in L~∞ and applications to nonlinear parabolic problems in unbounded domains, J. Funct.Anal., 2000, 173 (2) : 343-360.
[33]Ambrosetti A. ,Brezis H., Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J.Funct. Anal., 1994,122:519-543.
[34]Jin Z. R., Solvability of dirichlet problems for semilinear elliptic equations of certain domains,Pacific J. Math., 1996,176(1) :117-128
[35]Kazdan J. L., Kramer R. J., Invarint criteria for existence of solutions to second-order quasilinear elliptic equations,Comm. Pure Appl. Math., 1978, XXXI:619-645.
[36]Chizhonkov E. ,Vishik M., On a system of equations of magnetohydrodynamic type, Soviet Math. Dokl., 1984, 30:542-545.
[37]He C., The Cauchy problem for Navier-Stokes equations,J. Math. Anal. Appl. 1997, 209:228-242.
[38]Qu c., Song S., Wang P. , On the equations for the flow and the magnetic field within the Earth,J.Math. Anal. appl., 1994, 187:1003-1018.
[39]Solonnikov V. A., Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations,Am. Math. Soc. Trans. Ser. 2, 1968, 75:1-116.
[40]Rojas-Medar M. ,Boldrini J., Global strong solutions of equations of magnetohydrodynamic type,J.Austral.Math.Soc.Ser.B,1997,38:291-306.
[41]Sermange M.,Temam R.,Some mathematical questions related to the MHD equations.Comm.Pure Appl.Math.1983,36:635-664.
[2] 王明新.非线性抛物行型方程[M].北京:科学出版社,1993.
[3] Fujita H. On the blowing up of solutions of the Cauchy problem for u_t=Δu + u~(1+α). J. Fac. Sci.. Univ. Tokyo Sect.Ⅰ,1966,13: 109-124.
[4] Bandle C, Levine H A. On the existence and nonexistence of global solution of reaction-diffusion equations in sectorial domains. Trans. Amer. Math. Soc., 1989, 316: 595-622.
[5] Levine H. The role of critical exponents in blow-up theorems. SIAM Rew., 1990,32: 262-288.
[6] Pinsky R G. Existence and nonexistence of global solutions for u_t=Δu + α(x)u~p in R~d. J.Diff. Equ., 1997, 133: 152-177.
[7] Escobedo M, Herrero M A . Boundness and blow-up for a semilinear reaction-diffusion systems. J. Diff. Equ. , 1991,89:176-202.
[8] 戴求亿,一类半线性抛物方程组的爆破临界指标。应用数学学报,2002,25:339-346.
[9] Renclawowice J, Global existence and blow-up of solutions for a completely coupled Fujita type system of reaction-diffusion equations.Applicationes Mathemeticae 1998, 25(3): 313-326.
[10] Escobedo M, Levine H A, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch Rat Mech Anal, 1995, 129:47-100.
[11] Escobedo M, Herrero M A . A semilinear paraolic system in a bounded domain. Annali di Math. Pura ed Appl. (Ⅳ), 1993,115:315-336.
[12] D. J. Aronson and H.F.Weinberger, Multidimensional nonlinear diffusion arising in propulation genetics, Adv. in Math. 1978, 30:33-76.
[13] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equation-diffusion equations Quart, J. Math. Oxford 28 (1977) : 473-486.
[14] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear farabolic equations, Proc. Sympos. Pure Math.18(1969): 105-113.
[15] K.Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan. Acad 49(1973):503-525.
[16] K. Kobayashi, T. Siaro, and H. Tanaka, On the blowing up problem for semilinear heat equations,J. Math. Soc. Japan 29(1977): 407-422.
[17]H.Levine,The role of critical exponents in blowup theorems, SIAM Rev. 32(1990) :262-288.
[18]H. Levine and P. Meier, The value of the critical exponent for reaction-diffusion equations in cones, Arch. Rational Mech. Anal. 109 (1990), 73-80.
[19]A. Pazy, " Senugriyos of linear operators and applications to partial differential equations," Springer-Verlag,Berlin/New York, 1983.
[20]F.B. Weissler,Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math.38(1981): 29-40.
[21]Kawarada H., On solutions ofinitial-oundary walue problem for (?),Publ. Res. Inst. Math. Sci., Kyoto Univ., 1975, 10:729-736.
[22]Acker A.,Walter W., The quenching problem for nonlinear parabolic differential equation, Lecture Note in Math., NO. 564:1-12, New York, Spring-Verlag, 1976.
[23]Levine H. A.,Advances on quenching,Progress in Nonlinear Differential Equations and Its Applications, Vol 7:319-346,Birkhauser Boston, 1992.
[24]Levine H. A. , The phenomenon of quenching , A Survey, in Proc.Vith. Int. Conf. On Trends in the Theory and Practice of Nonlinear Analysis, New York, North Holland, 1985.
[25]Dai Q.Y.,Gu Y.G., Ashort note on quenching phenomena for semilienar parabolic equations, J. Diff. Equ., 1997, 137(2) :240-250.
[26]Fila M., Kawohl B., Levine H. A., Quenching for quasilinear equations , Comm. Partial Diff. Equ., 1992, 17(3/4) :593-614.
[27]Fila M. , Kawahl B. , Asymptotic analsysis of quenching problems, Rocky Mount. J. Math., 1992, 22:563-577.
[28]Deng K., Quenching for solutions of a plasma type equations, Nonlinear Anal., 1992, 18(8) :731-742.
[29]Dai Q. Y., Uniformly asymptotic global solution for singular semi linear parabolic equation, J. xiangtan Min. Inst., 1999,14(4) :85-91.
[30]Dai Q. Y., Zeng X. Z., The quenching phenomena for the cauchy problem of semilinear parabolic equation, J. Diff. Equ., 2001,175(1) :163-174.
[31]Dai Q. Y. , Gu Y. G., Asymptotic analsysis of global solutions for singular semilinear parabolic equation, Acta. Math. Appl.Sinica, 1998,21(1):57-65.
[32]Souplet P., Decay of heat semigroups in L~∞ and applications to nonlinear parabolic problems in unbounded domains, J. Funct.Anal., 2000, 173 (2) : 343-360.
[33]Ambrosetti A. ,Brezis H., Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J.Funct. Anal., 1994,122:519-543.
[34]Jin Z. R., Solvability of dirichlet problems for semilinear elliptic equations of certain domains,Pacific J. Math., 1996,176(1) :117-128
[35]Kazdan J. L., Kramer R. J., Invarint criteria for existence of solutions to second-order quasilinear elliptic equations,Comm. Pure Appl. Math., 1978, XXXI:619-645.
[36]Chizhonkov E. ,Vishik M., On a system of equations of magnetohydrodynamic type, Soviet Math. Dokl., 1984, 30:542-545.
[37]He C., The Cauchy problem for Navier-Stokes equations,J. Math. Anal. Appl. 1997, 209:228-242.
[38]Qu c., Song S., Wang P. , On the equations for the flow and the magnetic field within the Earth,J.Math. Anal. appl., 1994, 187:1003-1018.
[39]Solonnikov V. A., Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations,Am. Math. Soc. Trans. Ser. 2, 1968, 75:1-116.
[40]Rojas-Medar M. ,Boldrini J., Global strong solutions of equations of magnetohydrodynamic type,J.Austral.Math.Soc.Ser.B,1997,38:291-306.
[41]Sermange M.,Temam R.,Some mathematical questions related to the MHD equations.Comm.Pure Appl.Math.1983,36:635-664.