具有指数型内部源及边界流多重耦合的抛物方程组的奇性分析
|
摘要
本文研究的问题是具有指数型源或边界流及它们之间相互耦合的非线性反应扩散方程(组)解的性质,例如解的整体存在性、有限Blow-up性及Blow-up速率等。通过考察所研究问题非线性机制与解的奇性之间的关系,引入了和系统参数有关的特征线性方程组,利用这个线性方程组的解(或特征参数)简洁明了地刻画了系统解的整体存在和有限时刻Blow-up的判断准则(临界Blow-up指标)和解的奇性传播;在只有源或边界流耦合时,给出了在适当条件下解的最大模Blow-up速率的估计,这个估计恰是特征参数为指数的幂的自然对数。另外,本文中还对具有局部化源的反应扩散方程组解的性质进行了研究,特别是对Blow-up解边界层的较为细致的分析。本文得到的主要结果概述如下:
(1)对于方程组u_t=Δu+λe~(p1u+q1υ),υ_t=Δυ+μe~(p2υ+q2υ)的齐次Dirichlet初边值问题,本文通过引进线性方程组给出了其临界Blow-up指标I(α,β);在径向对称的条件下,给出了解的Blow-up速率及逐点估计,其中解的Blow-up速率为supu(·,t)=O(log(T-t)~(-α),supυ(·,t)=O(log(T-t)~(-β)。
(2)对于方程ut=Δu,vt=Δv经由边界条件耦合的初边值问题(非线性边界流),本文通过线性方程组给出了其临界Blow-up指标I(α,β);在径向对称的条件下,给出了解的Blow-up速率及逐点估计,其中解的Blow-up速率为sup u(·,t)=O(log(T-t)~(-α),sup υ(·,t)=(log(T-t)~(-β)。
(3)对于方程组ut=Δu+λe~(p1u+q1v),vt=Δv+μe~(p2u+q2v)的齐次Neumann初边值问题,本文同样引入与(1)相同的线性方程组,给出了所研究问的临界Blow-up指标I(α,β)和Blow-up速率;在一维的情况下,讨论了解的逐点估计及初值对Blow-up集的影响。其中解的Blow-up速率为supu(·,t)=O(log(T-t)~(-α)),supv(·,t)=O(log(T-t)~(-β)),与齐次Dirichlet边值的情形类似。
(4)
边值杂
对于方程组叭
△。+入1。p‘,”+”‘,v,。‘=△v+入Zep,,”+p,2”具有Neumann
二如丽
=娜xc口11趾+,12,,,
二拼2。q2l”+卿”的问题,本文通过引入线性方程组
凡以之::)一(
k‘十(,一“‘)/2、,
kZ+(1一kZ)/2/
其中PIc,,、2=l无伊‘,+(1一凡‘)仇,]2义:为2x2的矩阵,无1,k:任{1,0},给出了所研究
问题的临界Blow一up指标I.
本文所讨论的模型中,非线性参数为任意实数,但要求藕合参数为非负实
数.
(句本文对具有局部化源的Dihchlet初边值抛物方程组问题进行了研究,讨
论了解的整体存在和Blow一uP的条件、Blow一uP速率、渐近估计以及边界层等.
关键词:非线性抛物方程(组),非线性边界流,非线性反应,整体存在,B10w一uP,
临界指标,渐近行为,Blow一uP速率,Blow一uP集·
The main problems studied in this thesis are the properties of the solutions to reaction-diffusion systems coupled via nonlinear sources or(and) nonlinear boundary flux with exponent type, such as global existence, non-global existence, and blow-up rate, etc. Based on observing the relation between the singularity of solutions and the nonlinear mechanism of the systems considered, the so callled linear charateristic algebraic systems are introduced. The solutions of the algebraic systems, or characteristic parameters, are very convenient to describe the critical exponents as well as the propagations of singularities of solutions for the reaction-diffusion systems. For the case with only inner sources or boundary flux coupling, the blow-up rates of solutions are obtained under reasonable conditions, which just are the logarithm of the powers of the characteristic parameters. In particular, the reaction-diffusion systems with localized nonlinear sources are discussed also with a precise analysis to evolution of the boundary layers in the blow-up solutions. The main results obtained in this thesis are summarized as follows:(1) For homogeneous Dirichlet initial-boundary problem about the system through introducing linear algebraic systemwe obtain the critical exponent I(α,β);Under the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·,t) = O(log(T-t)-α), sup v(·,t) = O(log(T-t)-β).(2) For initial-boundary problem ut = △u, vt = △v coupled via boundary conditions (nonlinear boundary flux), we obtain the critical exponent I(α,β) through introducing the linear algebraic systemUnder the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·, t) = O(log(T - t)-α), sup v(·, t) = O(log(T - t)-β).
(3) For the systemwith homogeneous Neumann boundary conditions, we obtain the critical exponent I(α,β) and blow-up rates through introducing the same linear algebraic system as in (1); At the one-dimensional case, we obtain the pointwise estimates of the solutions and discuss the relation between initail value and the blow-up set.(4) For the system with Neumann boundary conditions ,we introduce some linearalgebraic systemswhere is a matrix, Using the solutions of the linear algebraic systems, we obtain the critical exponent of the problem.The nonlinear parameters in the models considered in our thesis are any real numbers, while the coupling ones among them should be nonnegative.(5) For parabolic systems with localized nonlinear sources, we study the properties of the solutions, such as global existence, non-global existence, asymptotic behavior at blow-up time, and estimates of boundary layer.
(1)对于方程组u_t=Δu+λe~(p1u+q1υ),υ_t=Δυ+μe~(p2υ+q2υ)的齐次Dirichlet初边值问题,本文通过引进线性方程组给出了其临界Blow-up指标I(α,β);在径向对称的条件下,给出了解的Blow-up速率及逐点估计,其中解的Blow-up速率为supu(·,t)=O(log(T-t)~(-α),supυ(·,t)=O(log(T-t)~(-β)。
(2)对于方程ut=Δu,vt=Δv经由边界条件耦合的初边值问题(非线性边界流),本文通过线性方程组给出了其临界Blow-up指标I(α,β);在径向对称的条件下,给出了解的Blow-up速率及逐点估计,其中解的Blow-up速率为sup u(·,t)=O(log(T-t)~(-α),sup υ(·,t)=(log(T-t)~(-β)。
(3)对于方程组ut=Δu+λe~(p1u+q1v),vt=Δv+μe~(p2u+q2v)的齐次Neumann初边值问题,本文同样引入与(1)相同的线性方程组,给出了所研究问的临界Blow-up指标I(α,β)和Blow-up速率;在一维的情况下,讨论了解的逐点估计及初值对Blow-up集的影响。其中解的Blow-up速率为supu(·,t)=O(log(T-t)~(-α)),supv(·,t)=O(log(T-t)~(-β)),与齐次Dirichlet边值的情形类似。
(4)
边值杂
对于方程组叭
△。+入1。p‘,”+”‘,v,。‘=△v+入Zep,,”+p,2”具有Neumann
二如丽
=娜xc口11趾+,12,,,
二拼2。q2l”+卿”的问题,本文通过引入线性方程组
凡以之::)一(
k‘十(,一“‘)/2、,
kZ+(1一kZ)/2/
其中PIc,,、2=l无伊‘,+(1一凡‘)仇,]2义:为2x2的矩阵,无1,k:任{1,0},给出了所研究
问题的临界Blow一up指标I.
本文所讨论的模型中,非线性参数为任意实数,但要求藕合参数为非负实
数.
(句本文对具有局部化源的Dihchlet初边值抛物方程组问题进行了研究,讨
论了解的整体存在和Blow一uP的条件、Blow一uP速率、渐近估计以及边界层等.
关键词:非线性抛物方程(组),非线性边界流,非线性反应,整体存在,B10w一uP,
临界指标,渐近行为,Blow一uP速率,Blow一uP集·
The main problems studied in this thesis are the properties of the solutions to reaction-diffusion systems coupled via nonlinear sources or(and) nonlinear boundary flux with exponent type, such as global existence, non-global existence, and blow-up rate, etc. Based on observing the relation between the singularity of solutions and the nonlinear mechanism of the systems considered, the so callled linear charateristic algebraic systems are introduced. The solutions of the algebraic systems, or characteristic parameters, are very convenient to describe the critical exponents as well as the propagations of singularities of solutions for the reaction-diffusion systems. For the case with only inner sources or boundary flux coupling, the blow-up rates of solutions are obtained under reasonable conditions, which just are the logarithm of the powers of the characteristic parameters. In particular, the reaction-diffusion systems with localized nonlinear sources are discussed also with a precise analysis to evolution of the boundary layers in the blow-up solutions. The main results obtained in this thesis are summarized as follows:(1) For homogeneous Dirichlet initial-boundary problem about the system through introducing linear algebraic systemwe obtain the critical exponent I(α,β);Under the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·,t) = O(log(T-t)-α), sup v(·,t) = O(log(T-t)-β).(2) For initial-boundary problem ut = △u, vt = △v coupled via boundary conditions (nonlinear boundary flux), we obtain the critical exponent I(α,β) through introducing the linear algebraic systemUnder the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·, t) = O(log(T - t)-α), sup v(·, t) = O(log(T - t)-β).
(3) For the systemwith homogeneous Neumann boundary conditions, we obtain the critical exponent I(α,β) and blow-up rates through introducing the same linear algebraic system as in (1); At the one-dimensional case, we obtain the pointwise estimates of the solutions and discuss the relation between initail value and the blow-up set.(4) For the system with Neumann boundary conditions ,we introduce some linearalgebraic systemswhere is a matrix, Using the solutions of the linear algebraic systems, we obtain the critical exponent of the problem.The nonlinear parameters in the models considered in our thesis are any real numbers, while the coupling ones among them should be nonnegative.(5) For parabolic systems with localized nonlinear sources, we study the properties of the solutions, such as global existence, non-global existence, asymptotic behavior at blow-up time, and estimates of boundary layer.
引文
[1] H. Amann, Paxabolic evolutions equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.
[2] D. J. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math., 30 (1978), 33-76.
[3] C. Bandle, Blow up in exterior domains, Recent Advances in Nonlinear Elliptic and Parabolic Problem, P. Benilan, M. Chipot, L. Evans, and M. Pierre, eds., Pitman Notes,208 (1988), 15-27.
[4] C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equation in sectorial domains, Trans. Amer. Math. Soc, 655 (1989),595-642.
[5] J. Bebernes and D. Kassy, A mathematical abalysis of blow-up for thermal reactions,SUM J. Appl. Math., 40 (1981), 476-484.
[6] J. Bebernes, A. Berssan and D. Eberly, A description of blow-up for the solid fuel ignition model, Indiana Univ. Math. J., 36 (1987) 295-305.
[7] Bei Hu and Hong-Ming Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc, 346 (1994), 117-135.
[8] K. Bimpong-Boat, P. Ortoleva and J. Ross, Far-from equilibrium phenomena at local sites of recation, J. Chem. Phys., 60 (1974), 3124.
[9] J. M. Chadma, A. Peirce and Hong-Ming Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl, 169 (1992),313-328.
[10] K. Deng, Global existence and blow-up for a system of heat equations with nonlinear boundary conditions, Math. Methods Appl Sci., 18 (1995), 307-315.
[11] K. Deng and M. X. Xu, Remark on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions, Proc. Amer. Math. Soc, 127 (1999), 167-172.
[12] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.
[13] M. Fila and P. Quittner, The blow-up rate for the heat equation with a non-linear boundary condition, Math. Methods Appl. Sci., 14 (1991), 197-205.
[14] M. Fila and H. A. Levine, On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition, J. Math. Anal. Appl., 204 (1996), 494-512.
[15] J. Filo, Diffusivity versus absorption through the boundary, J. Differential Equations, 99 (1992), 281-305.
[16] A. Friedman, "Partial Differential Equations of Parabolic Type", Prentice-Hall, Engle-wood Cliffs, NJ, 1964.
[17] A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations,Indiana Univ. Math. J., 34 (1985), 425-447.
[18] F. C. Li and C. H. Xie, Global existence and Blow-up for a nonlinear porous medium equation, Applied Mathematics Letters, 16 (2003), 185-192.
[19] H. Fujjita, On the blowing up of solutions of the Cauchy problem for ut-△u+ u1+α,J. Fac. Sci. Univ. Tokyo Sect. A Math., 16 (1966), 105-113.
[20] I. Fukuda and A. Okada, Behavior of solutions of nonlinear parabolic equations with localized reactions, Nonlinear Anal., 47 (2001), 3215-3221.
[21] V. A. Galaktionov, Boundary value problems for the nonlinear parabolic equation ut-△uσ+1+uβ, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 17 (1981),551-555. (In English.), 17 (1981), 836-842.
[22] V. A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations, U.S.S.R. Comput. Math. and Math. Phys., 23 (1983),35-44.
[23] V. A. Galaktionov, A parabolic system of quasilinear equations II, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 21 (1985), 1544-1559. (In English), 21 (1985), 1049-1062.
[24] V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii, A parabolic system of quasi-linear equations I, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 19 (1983), 2123-2140. (In English), 19 (1983), 1558-1571.
[25] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,Comm. Pure Appl. Math., 38 (1985), 297-319.
[26] Jong-Sheng Guo and Bei Hu, Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary, J. Math. Anal. Appl., 269 (2002), 28-49.
[27] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,Proc. Japan Acad., 49 (1973), 503-525.
[28] D. Kassoy and J. Poland, The thermal explosion confined by a constant temperature boundary , SIAM J. Appl. Math., 39 (1980), 412-430.
[29] K. Kobayashi, T. Siaro and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.
[30] A. A. Lacey, Mathematical abalysis of thermal runway for spatially inhomogeneous re- actions, SIAM J. Appl Math., 43 (1983), 1350-1366.
[31] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = -Au + F(u), Arch. Rational. Mech. anal., 51 (1973), 371-386.
[32] H. A. Levine and P. Meier, The value of the criticle exponent for reaction-diffusion equations in cones, Arch. Rational. Mech. anal., 109 (1990), 73-80.
[33] H. A. Levine and L. E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl, 55 (1976), 329-334.
[34] H. A. Levine and R. A. Smith, A potential well theory for the heat equation with a nonlinear boundary condition, Math. Methods Appl Sci., 9 (1987), 127-136.
[35] G. M. Lieberman, "Second Order Differential Equations", World Scientific, Singgapore,1996.
[36] Lin Zhigui, Xie Chunhong and Wang Mingxin, On critical exponents for a semilinear heat equations with nonlinear boundary conditions, Appl. Math. -JCU, 13B (1998), 363-372.
[37] Lin Zhigui, Xie Chunhong and Wang Mingxin, The blow-up rate of positive solution of a parabolic system, Northeast. Math. J., 13 (1997) 372-378.
[38] Lin Zhigui and Xie Chunhong, The blow-up for a system of heat equations with Neumann boundary conditions, Acta Mathematica Sinica, English Series, 15 (1999), 549-554.
[39] Liwen Wang and Qingyi Chen, The asymptoyic behavior of blowup solution of localized nonlinear equation, J. Math. Anal. AppL, 200 (1996), 315-321.
[40] W. Liu, The blow-up rate of solutions of semilinear heat equations, J. Differential Equations, 77 (1989), 104-122.
[41] P. Meier, Existence et non-existence de solutions globales d'une equation de la chaleur semi-lineaire: extension d'un theoreme de Fujita, C. R. Acad. Sci. Paris Serie I, 303 (1986), 635-637.
[42] P. Meier, Blow up of solutions of semilinear parabolic differential equations, Z. Angew. Math. Phys., 39 (1988), 135-149.
[43] P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63-71.
[44] P. Ortoleva and J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys., 56 (1972), 4397.
[45] C. V. Pao, "Nonlinear Parabolic and Elliptic Equations", Plenum Press, New York and London, 1992.
[46] M. Pedersen and Zhigui Lin, Coupled diffusion systems with localized nonlinear reactions,Comput. Math. AppL, 42 (2001), 807-816.
[47] J. Poland, Transient phenomena in thermal explosions in confined gases, Ph. D. Thesis,University of Colorrado, Boulder, CO., 1978.
[48] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,Prentice-Hall, New York, 1967.
[49] A. Rodriguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up, J. Differential Equations, 169 (2001), 332-372.
[50] Julio D. Rossi, The blow-up rate for a system of heat equations with non-trivial coupling at the boundary, Math. Methods Appl. Sci., 20 (1997), 1-11.
[51] A. A. Samanskii, V. A. Galaktionov, S. P. Kordiumov and A. P. Mikhaylov, Blowup of solutions in problems for quasilinear parabolic equations, Nauka Pub., Moscow, 1987.(In Russian).
[52] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.
[53] Sheng-Chen Fu and Jong-Shenq Guo, Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Appl, 276 (2002),458-475.
[54] P. Souplet, Uniform blow-up profile and boundary behavior for diffusion equation with Nonlinear source, J. Differential Equations, 153 (1999), 374-406.
[55] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., Kyoto University, 8 (1972), 221-229.
[56] F. B. Walter, Differential and integral inequalities, Ergibnesse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York, (1970).
[57] M. X. Wang, Global existence and finite time blow-up for a reaction-diffusion system, Z. Angew. Math. Phys., 51 (2000), 160-167.
[58] M. X. Wang, Blowup estimates for a semilinear reaction diffusion system, J. Math. Anal.Appl, 257 (2001), No. 1, 292-295.
[59] M. X. Wang, Blow-up rate for a semilinear reaction system, Comput. Math. Appl, 44 (2002), 573-585.
[60] M. X. Wang, Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions, Applied Mathematics Letters, 16 (2003), 543-540.
[61] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.
[62] F. B. Weissler, An L∞ blow-up estimate for a semilinear heat equation, Comm. Pure Appl. Math., 38 (1985), 292-295.
[63] 叶其孝,李正元,《反应扩散方程引论》,科学出版社,1994.
[64] S. N. Zheng, Global boundedness of solutions to a reaction-diffusion system, Math. Methods Appl Sci., 22 (1999), 43-54.
[65] S. N. Zheng, Nonexistence of positive solutions to a semilinear elliptic system and blow-up
estimates for a reaction-diffusion system, J. Math. Anal. Appl, 232 (1999), 293-311.
[66] S. N. Zheng, Global existence and global non-existence of solutions to a reaction-diffusion system, Nonlinear Anal., 39 (2000), 327-340.
[2] D. J. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math., 30 (1978), 33-76.
[3] C. Bandle, Blow up in exterior domains, Recent Advances in Nonlinear Elliptic and Parabolic Problem, P. Benilan, M. Chipot, L. Evans, and M. Pierre, eds., Pitman Notes,208 (1988), 15-27.
[4] C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equation in sectorial domains, Trans. Amer. Math. Soc, 655 (1989),595-642.
[5] J. Bebernes and D. Kassy, A mathematical abalysis of blow-up for thermal reactions,SUM J. Appl. Math., 40 (1981), 476-484.
[6] J. Bebernes, A. Berssan and D. Eberly, A description of blow-up for the solid fuel ignition model, Indiana Univ. Math. J., 36 (1987) 295-305.
[7] Bei Hu and Hong-Ming Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc, 346 (1994), 117-135.
[8] K. Bimpong-Boat, P. Ortoleva and J. Ross, Far-from equilibrium phenomena at local sites of recation, J. Chem. Phys., 60 (1974), 3124.
[9] J. M. Chadma, A. Peirce and Hong-Ming Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl, 169 (1992),313-328.
[10] K. Deng, Global existence and blow-up for a system of heat equations with nonlinear boundary conditions, Math. Methods Appl Sci., 18 (1995), 307-315.
[11] K. Deng and M. X. Xu, Remark on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions, Proc. Amer. Math. Soc, 127 (1999), 167-172.
[12] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.
[13] M. Fila and P. Quittner, The blow-up rate for the heat equation with a non-linear boundary condition, Math. Methods Appl. Sci., 14 (1991), 197-205.
[14] M. Fila and H. A. Levine, On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition, J. Math. Anal. Appl., 204 (1996), 494-512.
[15] J. Filo, Diffusivity versus absorption through the boundary, J. Differential Equations, 99 (1992), 281-305.
[16] A. Friedman, "Partial Differential Equations of Parabolic Type", Prentice-Hall, Engle-wood Cliffs, NJ, 1964.
[17] A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations,Indiana Univ. Math. J., 34 (1985), 425-447.
[18] F. C. Li and C. H. Xie, Global existence and Blow-up for a nonlinear porous medium equation, Applied Mathematics Letters, 16 (2003), 185-192.
[19] H. Fujjita, On the blowing up of solutions of the Cauchy problem for ut-△u+ u1+α,J. Fac. Sci. Univ. Tokyo Sect. A Math., 16 (1966), 105-113.
[20] I. Fukuda and A. Okada, Behavior of solutions of nonlinear parabolic equations with localized reactions, Nonlinear Anal., 47 (2001), 3215-3221.
[21] V. A. Galaktionov, Boundary value problems for the nonlinear parabolic equation ut-△uσ+1+uβ, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 17 (1981),551-555. (In English.), 17 (1981), 836-842.
[22] V. A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations, U.S.S.R. Comput. Math. and Math. Phys., 23 (1983),35-44.
[23] V. A. Galaktionov, A parabolic system of quasilinear equations II, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 21 (1985), 1544-1559. (In English), 21 (1985), 1049-1062.
[24] V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii, A parabolic system of quasi-linear equations I, Differentsial'nye Uravneniya(In Russian.) Differential Equations, 19 (1983), 2123-2140. (In English), 19 (1983), 1558-1571.
[25] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,Comm. Pure Appl. Math., 38 (1985), 297-319.
[26] Jong-Sheng Guo and Bei Hu, Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary, J. Math. Anal. Appl., 269 (2002), 28-49.
[27] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,Proc. Japan Acad., 49 (1973), 503-525.
[28] D. Kassoy and J. Poland, The thermal explosion confined by a constant temperature boundary , SIAM J. Appl. Math., 39 (1980), 412-430.
[29] K. Kobayashi, T. Siaro and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.
[30] A. A. Lacey, Mathematical abalysis of thermal runway for spatially inhomogeneous re- actions, SIAM J. Appl Math., 43 (1983), 1350-1366.
[31] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = -Au + F(u), Arch. Rational. Mech. anal., 51 (1973), 371-386.
[32] H. A. Levine and P. Meier, The value of the criticle exponent for reaction-diffusion equations in cones, Arch. Rational. Mech. anal., 109 (1990), 73-80.
[33] H. A. Levine and L. E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl, 55 (1976), 329-334.
[34] H. A. Levine and R. A. Smith, A potential well theory for the heat equation with a nonlinear boundary condition, Math. Methods Appl Sci., 9 (1987), 127-136.
[35] G. M. Lieberman, "Second Order Differential Equations", World Scientific, Singgapore,1996.
[36] Lin Zhigui, Xie Chunhong and Wang Mingxin, On critical exponents for a semilinear heat equations with nonlinear boundary conditions, Appl. Math. -JCU, 13B (1998), 363-372.
[37] Lin Zhigui, Xie Chunhong and Wang Mingxin, The blow-up rate of positive solution of a parabolic system, Northeast. Math. J., 13 (1997) 372-378.
[38] Lin Zhigui and Xie Chunhong, The blow-up for a system of heat equations with Neumann boundary conditions, Acta Mathematica Sinica, English Series, 15 (1999), 549-554.
[39] Liwen Wang and Qingyi Chen, The asymptoyic behavior of blowup solution of localized nonlinear equation, J. Math. Anal. AppL, 200 (1996), 315-321.
[40] W. Liu, The blow-up rate of solutions of semilinear heat equations, J. Differential Equations, 77 (1989), 104-122.
[41] P. Meier, Existence et non-existence de solutions globales d'une equation de la chaleur semi-lineaire: extension d'un theoreme de Fujita, C. R. Acad. Sci. Paris Serie I, 303 (1986), 635-637.
[42] P. Meier, Blow up of solutions of semilinear parabolic differential equations, Z. Angew. Math. Phys., 39 (1988), 135-149.
[43] P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63-71.
[44] P. Ortoleva and J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys., 56 (1972), 4397.
[45] C. V. Pao, "Nonlinear Parabolic and Elliptic Equations", Plenum Press, New York and London, 1992.
[46] M. Pedersen and Zhigui Lin, Coupled diffusion systems with localized nonlinear reactions,Comput. Math. AppL, 42 (2001), 807-816.
[47] J. Poland, Transient phenomena in thermal explosions in confined gases, Ph. D. Thesis,University of Colorrado, Boulder, CO., 1978.
[48] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,Prentice-Hall, New York, 1967.
[49] A. Rodriguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up, J. Differential Equations, 169 (2001), 332-372.
[50] Julio D. Rossi, The blow-up rate for a system of heat equations with non-trivial coupling at the boundary, Math. Methods Appl. Sci., 20 (1997), 1-11.
[51] A. A. Samanskii, V. A. Galaktionov, S. P. Kordiumov and A. P. Mikhaylov, Blowup of solutions in problems for quasilinear parabolic equations, Nauka Pub., Moscow, 1987.(In Russian).
[52] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.
[53] Sheng-Chen Fu and Jong-Shenq Guo, Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Appl, 276 (2002),458-475.
[54] P. Souplet, Uniform blow-up profile and boundary behavior for diffusion equation with Nonlinear source, J. Differential Equations, 153 (1999), 374-406.
[55] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., Kyoto University, 8 (1972), 221-229.
[56] F. B. Walter, Differential and integral inequalities, Ergibnesse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York, (1970).
[57] M. X. Wang, Global existence and finite time blow-up for a reaction-diffusion system, Z. Angew. Math. Phys., 51 (2000), 160-167.
[58] M. X. Wang, Blowup estimates for a semilinear reaction diffusion system, J. Math. Anal.Appl, 257 (2001), No. 1, 292-295.
[59] M. X. Wang, Blow-up rate for a semilinear reaction system, Comput. Math. Appl, 44 (2002), 573-585.
[60] M. X. Wang, Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions, Applied Mathematics Letters, 16 (2003), 543-540.
[61] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.
[62] F. B. Weissler, An L∞ blow-up estimate for a semilinear heat equation, Comm. Pure Appl. Math., 38 (1985), 292-295.
[63] 叶其孝,李正元,《反应扩散方程引论》,科学出版社,1994.
[64] S. N. Zheng, Global boundedness of solutions to a reaction-diffusion system, Math. Methods Appl Sci., 22 (1999), 43-54.
[65] S. N. Zheng, Nonexistence of positive solutions to a semilinear elliptic system and blow-up
estimates for a reaction-diffusion system, J. Math. Anal. Appl, 232 (1999), 293-311.
[66] S. N. Zheng, Global existence and global non-existence of solutions to a reaction-diffusion system, Nonlinear Anal., 39 (2000), 327-340.