内吸收多重非线性抛物组奇性解的渐近分析
|
摘要
本文研究几类具有内部吸收和耦合边界流的多重非线性抛物方程组奇性解的渐近行为，包括由不同非线性机制占优而导致的多重blow-up速率，非同时与同时blow-up等．首先讨论了一个具有混合型内部吸收与耦合边界流的多重非线性模型．通过引入特征代数方程组得到其blow-up临界指标的清晰刻画．特别引入包含两个新参数的另一类特征代数方程组用以刻画多重blow-up速率．非常有趣的是，这里得到了两个与吸收有关的blow-up速率，而现有文献中的所有blow-up速率结果都是与吸收无关的．为进一步分析这个现象到底来自混合型非线性还是耦合机制，又对比考虑了单一非线性耦合情形，同样得到与吸收有关的blow-up速率，从而确认耦合机制在这里所起的关键作用．此外，本文还分别讨论了(对称的)负-负源和(非对称的)正-负源的边界流耦合抛物方程组的非同时与同时blow-up问题，由此确定源的符号对引发非同时blow-up的影响．
本文得到的主要结果概述如下：
(Ⅰ)关于多重blow-up速率
在第二章中，对于具有混合型内部吸收的方程u_t=△u-a_1u~m，v_t=△v-a_2e~(nv)，(x，t)∈Ω×(0，T)，附加边界条件(?)=e~(pv)，(?)=u~q，(x，t)∈(?)Ω×(0，T)的问题，先根据比较原理建立blow-up临界指标，再利用Green表示公式和多重形式下的Scaling方法得到了上述问题在N=1(N表示维数)时的blow-up速率．需要提及的是，在现有文献中，吸收项会影响临界指标、blow-up时间以及blow-up解所需初值等等，但不影响blow-up速率，而这里得到的多重blow-up速率中有两种是与吸收有关的．
第三章研究了具有幂型内部吸收的方程u_t=u_(xx)-a_1u~m，v_t=v_(xx)-a_2v~n，(x，t)∈(0，1)×(0，T)经由边界条件u_x(1，t)=v~p(1，t)，v_x(1，t)=v~q(1，t)，u_x(0，t)=0，v_x(0，t)=0，t∈(0，T)耦合的初边值问题，得到基于非线性指标分类完全的多重blow-up速率，其中也有两种速率是与吸收有关的．这说明与吸收有关的blow-up速率是由耦合机制造成的．若p=q，m=n，初值u_0(x)=v_0(x)，则方程组化为单个方程，而此时恰属于blow-up速率与吸收项无关的指标区域，进一步说明存在与吸收项有关的blow-up速率是耦合方程组区别于单个方程问题所特有的现象．
(Ⅱ)关于非同时与同时blow-up
第四章考虑了方程u_t=u_(xx)-λ_1u~(α_1)，v_t=v_(xx)-λ_2~(β_1)，(x，t)∈(0，1)×(0，T)，附加边界条件u_x(1，t)=v~(α_2)v~p，v_x(1，t)=u~qv~(β_2)，u_x(0，t)=v_x(0，t)=0，t∈(0，T)的解的非同时blow-up．首先借助对辅助问题的研究并引入截断函数得到一个基本引理，继而，结合Green表示公式和Scaling方法最终得到了存在初值发生非同时blow-up的充分必要条件，以及所有blow-up必为非同时blow-up的充分条件．
第五章研究了u_t=u_(xx)+u~(α_1)，v_t=v_(xx)-v~(β_1)，(x，t)∈(0，1)×(0，T)经由边界条件u_x(1，t)=u~(α_2)v~p，v_x(1，t)=u~qv~(β_2)，u_x(0，t)=(0，t)=0，t∈(0，T)耦合的方程组的解的非同时blow-up，得到了与第四章相对应的结果．两个分量u和v的非对称性导致了讨论过程和所得结果都较前一模型更为复杂．与第四章结果的对比可见源的符号对引发非同时blow-up的作用．
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic systems with inner absorptions and coupled boundary fluxes, such as multiple blow-up rates under different dominations of nonlinearities, non-simultaneous versus simultaneous blow-up, and so on. Firstly, a multi-nonlinear model with inner absorptions and coupled boundary fluxes of mixed type nonlinearities is discussed. The critical exponent is obtained, and clearly described via a so called characteristic algebraic system. In particular, another characteristic algebraic system with two new parameters is introduced to simply show the multiple blow-up rates. It is interesting to observe that two of the multiple blow-up rates obtained here do depend on the absorption exponents, unlike all the known results that the blow-up rates in the current literatures, to our knowledge, are all absorption-independent. Furthermore, in order to explore the reason for the new phenomenon (i.e., it is due to the mixed type nonlinearities or the coupling mechanism), the same problem is considered for the case of single type nonlinearities, and the blowup rates of the same propeties are obtained also. This is to confirm that the coupling mechanism plays the key role for the absorption-relevant blow-up rates. In addition, the thesis discusses non-simultaneous blow-up of solutions for coupled parabolic systems with negative-negative sources and positive-negative sources, respectively. Thereby, the influence of the sign of sources to non-simultaneous blow-up is determined.
The main results obtained in this thesis can be summarized as follows:
(Ⅰ) Multiple blow-up rate
In Chapter 2, for the system with mixed type nonlinearities u_t =Δu - a_1u~m, v_t =Δv-a_2e~(nv) in (0,1)×(0,T), (?) =e~(pv), (?) = u~q on d (?)Ω×(0, T), the critical blow-up exponent is established by using the comparision principle. Furthermore, multiple simultaneous blow-up rates of solutions with N = 1 (N is the space dimension) are established by Green's identity and the Scaling method . It should be mentioned that in previous literatures, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates, while here some absorption-erlevant simultaneous blow-up rates are obtained.
In Chapter 3, the system u_t = u_(xx) - a_1u~m, v_t = v_(xx) - a_2v~n in (0,1)×(0, T) with coupled boundary fluxes u_x(1,t) = v~p, v_x(1,t) = u~q, u_x(0,t) = v_x(0,t) = 0 is considered. The multiple simultaneous blow-up rates obtained with a complete classification for all the nonlinear parameters of the model, where two absorption-relevant ones are observed also. This is to say that the absorption-relevant blow-up rates should be caused by the coupling mechanism. If p = q, m = n with u_0(x) = v_0(x), the system reduces to a scalar problem, which belongs to the class of absorption-independent blow-up rate. This shows a substantial difference between the coupled systems and the scalar equations with inner absorptions.
(Ⅱ) Non-simultaneous versus simultaneous blow-up
Chapter 4 deals with the initial-boundary problem for u_t = u_(xx) -λ_1u~(α_1), v_t = v_(xx) -λ_2v~(β_1) in (0,1)×(0, T) coupled via u_x(1, t) = u~(α_2)v~p, v_x(1, t) = u~qv~(β_2), u_x(0, t) = v_x(0, t) = 0, t∈(0, T). Firstly, by introducing an auxilary problem and a cut-off function, a basic lemma is proved. Then, combining with Green's identity and the Scaling method, the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data as well as the sufficient conditions under which any blow-up of solutions would be non-simultaneous are established.
Chapter 5 considers non-simultaneous blow-up of solutions for the system with positive-negative sources u_t = u_(xx) + u~(α_1) v_t = v_(xx) - v~(β_1) in (0,1)×(0,T), coupled via boundary conditions u_x(1,t) = u~(α_2)v~p, v_x(1,t) = u~qv~(β_2), u_x(0,t) = vx_(0,t) = 0, t∈(0,T). The non-symmetry of components u and v leads to a more complicated discussion. Comparing with the corresponding conclusions for the two models in Chapters 4 and 5, the contributions of the signs of sources to the non-simultaneous blow-up of solutions are shown here clearly.
本文得到的主要结果概述如下：
(Ⅰ)关于多重blow-up速率
在第二章中，对于具有混合型内部吸收的方程u_t=△u-a_1u~m，v_t=△v-a_2e~(nv)，(x，t)∈Ω×(0，T)，附加边界条件(?)=e~(pv)，(?)=u~q，(x，t)∈(?)Ω×(0，T)的问题，先根据比较原理建立blow-up临界指标，再利用Green表示公式和多重形式下的Scaling方法得到了上述问题在N=1(N表示维数)时的blow-up速率．需要提及的是，在现有文献中，吸收项会影响临界指标、blow-up时间以及blow-up解所需初值等等，但不影响blow-up速率，而这里得到的多重blow-up速率中有两种是与吸收有关的．
第三章研究了具有幂型内部吸收的方程u_t=u_(xx)-a_1u~m，v_t=v_(xx)-a_2v~n，(x，t)∈(0，1)×(0，T)经由边界条件u_x(1，t)=v~p(1，t)，v_x(1，t)=v~q(1，t)，u_x(0，t)=0，v_x(0，t)=0，t∈(0，T)耦合的初边值问题，得到基于非线性指标分类完全的多重blow-up速率，其中也有两种速率是与吸收有关的．这说明与吸收有关的blow-up速率是由耦合机制造成的．若p=q，m=n，初值u_0(x)=v_0(x)，则方程组化为单个方程，而此时恰属于blow-up速率与吸收项无关的指标区域，进一步说明存在与吸收项有关的blow-up速率是耦合方程组区别于单个方程问题所特有的现象．
(Ⅱ)关于非同时与同时blow-up
第四章考虑了方程u_t=u_(xx)-λ_1u~(α_1)，v_t=v_(xx)-λ_2~(β_1)，(x，t)∈(0，1)×(0，T)，附加边界条件u_x(1，t)=v~(α_2)v~p，v_x(1，t)=u~qv~(β_2)，u_x(0，t)=v_x(0，t)=0，t∈(0，T)的解的非同时blow-up．首先借助对辅助问题的研究并引入截断函数得到一个基本引理，继而，结合Green表示公式和Scaling方法最终得到了存在初值发生非同时blow-up的充分必要条件，以及所有blow-up必为非同时blow-up的充分条件．
第五章研究了u_t=u_(xx)+u~(α_1)，v_t=v_(xx)-v~(β_1)，(x，t)∈(0，1)×(0，T)经由边界条件u_x(1，t)=u~(α_2)v~p，v_x(1，t)=u~qv~(β_2)，u_x(0，t)=(0，t)=0，t∈(0，T)耦合的方程组的解的非同时blow-up，得到了与第四章相对应的结果．两个分量u和v的非对称性导致了讨论过程和所得结果都较前一模型更为复杂．与第四章结果的对比可见源的符号对引发非同时blow-up的作用．
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic systems with inner absorptions and coupled boundary fluxes, such as multiple blow-up rates under different dominations of nonlinearities, non-simultaneous versus simultaneous blow-up, and so on. Firstly, a multi-nonlinear model with inner absorptions and coupled boundary fluxes of mixed type nonlinearities is discussed. The critical exponent is obtained, and clearly described via a so called characteristic algebraic system. In particular, another characteristic algebraic system with two new parameters is introduced to simply show the multiple blow-up rates. It is interesting to observe that two of the multiple blow-up rates obtained here do depend on the absorption exponents, unlike all the known results that the blow-up rates in the current literatures, to our knowledge, are all absorption-independent. Furthermore, in order to explore the reason for the new phenomenon (i.e., it is due to the mixed type nonlinearities or the coupling mechanism), the same problem is considered for the case of single type nonlinearities, and the blowup rates of the same propeties are obtained also. This is to confirm that the coupling mechanism plays the key role for the absorption-relevant blow-up rates. In addition, the thesis discusses non-simultaneous blow-up of solutions for coupled parabolic systems with negative-negative sources and positive-negative sources, respectively. Thereby, the influence of the sign of sources to non-simultaneous blow-up is determined.
The main results obtained in this thesis can be summarized as follows:
(Ⅰ) Multiple blow-up rate
In Chapter 2, for the system with mixed type nonlinearities u_t =Δu - a_1u~m, v_t =Δv-a_2e~(nv) in (0,1)×(0,T), (?) =e~(pv), (?) = u~q on d (?)Ω×(0, T), the critical blow-up exponent is established by using the comparision principle. Furthermore, multiple simultaneous blow-up rates of solutions with N = 1 (N is the space dimension) are established by Green's identity and the Scaling method . It should be mentioned that in previous literatures, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates, while here some absorption-erlevant simultaneous blow-up rates are obtained.
In Chapter 3, the system u_t = u_(xx) - a_1u~m, v_t = v_(xx) - a_2v~n in (0,1)×(0, T) with coupled boundary fluxes u_x(1,t) = v~p, v_x(1,t) = u~q, u_x(0,t) = v_x(0,t) = 0 is considered. The multiple simultaneous blow-up rates obtained with a complete classification for all the nonlinear parameters of the model, where two absorption-relevant ones are observed also. This is to say that the absorption-relevant blow-up rates should be caused by the coupling mechanism. If p = q, m = n with u_0(x) = v_0(x), the system reduces to a scalar problem, which belongs to the class of absorption-independent blow-up rate. This shows a substantial difference between the coupled systems and the scalar equations with inner absorptions.
(Ⅱ) Non-simultaneous versus simultaneous blow-up
Chapter 4 deals with the initial-boundary problem for u_t = u_(xx) -λ_1u~(α_1), v_t = v_(xx) -λ_2v~(β_1) in (0,1)×(0, T) coupled via u_x(1, t) = u~(α_2)v~p, v_x(1, t) = u~qv~(β_2), u_x(0, t) = v_x(0, t) = 0, t∈(0, T). Firstly, by introducing an auxilary problem and a cut-off function, a basic lemma is proved. Then, combining with Green's identity and the Scaling method, the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data as well as the sufficient conditions under which any blow-up of solutions would be non-simultaneous are established.
Chapter 5 considers non-simultaneous blow-up of solutions for the system with positive-negative sources u_t = u_(xx) + u~(α_1) v_t = v_(xx) - v~(β_1) in (0,1)×(0,T), coupled via boundary conditions u_x(1,t) = u~(α_2)v~p, v_x(1,t) = u~qv~(β_2), u_x(0,t) = vx_(0,t) = 0, t∈(0,T). The non-symmetry of components u and v leads to a more complicated discussion. Comparing with the corresponding conclusions for the two models in Chapters 4 and 5, the contributions of the signs of sources to the non-simultaneous blow-up of solutions are shown here clearly.
引文
[1] Hu B, Yin H M. The profile near blow-up time for the solution of the heat equation with a nonlinear boundary condition. Trans. Amer. Math. Soc, 1994, 346:117-135.
[2] Rodriguez-Bernal A, Tajdine A. Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differential Equations, 2001, 169(2) :332-372.
[3] Levine H A. The role of critical exponents in blowup theorems. SIAM Rev., 1990, 32(2):262-288.
[4] Deng K, Levine H A. The role of crictical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl., 2000, 243(1).
[5] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J. Differential Equations, 1981, 89(1):76-202.
[6] Galaktionov V A, Kurdyumov S P, Samarskii A A. A parabolic system of quasi-linear equations i. Differ. Equ., 1983, 19:1558-1571.
[7] Galaktionov V A, Kurdyumov S P, Samarskii A A. A parabolic system of quasi-linear equations ii. Differ. Equ., 1985, 21:1049-1062.
[8] Zheng S N. Global existence and global nonexistence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39(3):327-340.
[9] Zheng S N. Global boundedness of solutions to a reaction-diffusion system. Math. Methods Appl. Sci., 1999, 22(1):43-54.
[10] Deng K. Global existence and blow-up for a system of heat equations with a nonlinear boundary condition. Math. Methods. Appl. Sci., 1995, 18(4).
[11] Wang M X. Parabolic systems with nonlinear boundary conditions. Chinese Sci. Bull., 1995, 40(17):1412-1414.
[12] Zheng S N, Song X F. Interactions among multi-nonliearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux. Nonlinear Anal., 2004, 57:519-530.
[13] Zheng S N, Wang W. Critical exponents for a nonlinear diffusion system. Nonlinear Anal., 2007, 67:1190-1210.
[14] Zheng S N, Li F J. Critical exponents for a reaction-diffusion model with absorptions and coupled boundary flux. Proc. Edinb. Math. Soc., 2005, 48:241-252.
[15] Rossi J D, Wolanski N. Blow-up versus global existence for a semilinear reaction-diffusion system in a bounded domain. Comm. Partial Differential Equations, 1995, 20:1991-2001.
[16] Mu C L, Su Y. Global existence and blow-up for a quasilinear degenerate parabolic system in a cylinder. Appl. Math. Lett., 2001, 14:715-723.
[17] Friedman A, Mcleod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J., 1995, 34(2):425-447.
[18] Fila M, Quittner P. The blow-up rate for the heat equation with a non-linear boundary condition. Math. Methods Appl. Sci., 1991, 14(3):197-205.
[19] Zheng S N. Nonexistence of positive solutions to a semi-linear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl., 1999, 232(2):293-311.
[20] Wang M X. Blow-up estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl., 2001, 257(1):46-51.
[21] Fila M, Levine H A. On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition. J. Math. Appl. Anal., 1996, 204(2):494-521.
[22] Lin Z G, Xie C H, Wang M X. On critical exponents for a semilinear heat equations with nonlinear boundary conditions. Appl. Math. J. Chinese Univ. Ser. B, 1998, 13(4):363-372.
[23] Fu S C, Guo J S. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Appl. Anal., 2002, 276(1 ):458-475.
[24] Wang M X. Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions. Appl. Math. Lett., 2003, 16(4):543-549.
[25] Rossi J D. The blow up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math. Univ. Comenian. (N.S.), 1998, 67:343-350.
[26] Zheng S N, Li F J, Liu B C. Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux. Appl. Math. Lett., 2006, 19(9):942-948.
[27] Song X F. Blow-up analysis for a system of heat equations coupled via nonlinear boundary conditions. Math. Methods Appl. Sci., 2007, 30(10):1135-1146.
[28] Song X F. Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws. In press, Nonlinear Anal.
[29] Samarskii A A, Galaktionov V A, Kurdyumov S P, Mikhailov A P. Blow-up in problems for quarsilinear parabolic equations. Berlin: Walter de Gruyter, 1995.
[30] Pinasco J P, Rossi J D. Simultaneous vs. non-simultaneous blow-up. New Zealand J. Math., 2000, 29:55-59.
[31] Quir6s F, Rossi J D. Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys., 2001, 52:342-346.
[32] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal., 2005, 4(3):523-536.
[33] Brandle C, Quir6s F, Rossi J D. The role of nonlinear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl., 2005, 308(1):92-104.
[34] Souplet P, Tayachi S. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan., 2004, 56(2):571-584.
[35] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18(4):405-418.
[36] Zheng S N, Liu B C, Li F J. Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system. J. Math. Anal. Appl., 2007, 326:414-431.
[37] Zheng S N, Liu B C, Li F J. Non-simultaneous blow-up for a multi-coupeld reaction-diffusion system. Nonlinear Anal., 2006, 64:1189-1202.
[38] Li F J, Liu B C, Zheng S N. Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary flux. Z. Angew. Math. Phys., 2007, 58:717-735.
[39] Amann H. Parabolic evolution equations and nonlinear boundary conditions. J. Differential Equations, 1988, 72:201-269.
[40] Bebernes J, Eberly D. Mathematical problem from combustion theory, Appl. Math. Sci. 83. Berlin: Springer-Verlag, 1989.
[41] Bebernes J, Bressan A, Eberly D. A description of blowup for the solid fuel ignition model. Indiana Univ. Math. J., 1987, 36:295-305.
[42] Bebernes J, Kassy D. A mathematical analysis of blow-up for thermal reactions. SIAM J. Appl. Math., 1981, 40:476-484.
[43] Caristi G, Mitidieri E. Blow-up estimates of positive solutions of a parabolic system. J. Differential Equations, 1994, 113:265-271.
[44] Giga Y, Kohn R V. Asymptotic self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 1985, 38:297-319.
[45] Lin Z G, Xie C H. The blow-up rate for a system of heat equations with nonlinear boundary conditions. Nonlinear Anal., 1998, 34:767-778.
[46] Weissler F B. Single point blow-up of semi-linear initial value problems. J. Differential Equations, 1984, 55:204-224.
[47] Ferreira R, Groisman P, Rossi J D. Numerical blow-up for the porous medium equation with a source. Numer. Methods Partial Differential Equations, 2004, 20:552-575.
[48] Galaktionov V A, Posashkov S A. On so some investigation method for unbounded solutions of quasilinear parabolic equations. USSR Comput. Math. Phys., 1988, 28:148-156.
[49] Galaktionov V A. On asymptotic self-similar behaviour for a quasilinear heat equation: single point blow-up. Math. Methods Appl. Sci., 1995, 26:675-693.
[50] Souplet P. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J. Differential Equations, 1999, 153:374-406.
[51] 姜朝欣，郑斯宁．具有吸收和非线性边界流的非线性扩散方程的爆破估计(英文)．数 学进展，2004，33(5)：615-620．
[52] Dibenedetto E. Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J., 1983, 32:83-118.
[53] Ladyzhenskaya O A, Solonnikov V A, Ural'ceva N N. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, Transl. Math. Mono. 23. Povidence RI: AMS, 1968.
[54] Ziemer W P. Interior and bounary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc, 1982, 271(2):733-748.
[55] Liberman G M. Second order parabolic differential equations. River Edge: World scientific, 1996.
[56] Friedman A. Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall, 1964.
[57] Quir(?)s F, Rossi J D. Blow-up sets and fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ. Math. J., 2001, 50:629-654.
[58] Chen Y P. Blow-up for a system of heat equations with nonlocal sources and absorptions. Comput. Math. Appl., 2004, 48:361-372.
[59] Li H L, Wang M X. Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system. Nonlinear Anal., 2005, 63(9):1083-1093.
[60] 刘其林，李玉样，谢春红．带局部非线性反应项的退化抛物方程解的爆破性质．数学 学报，2003，46(6)：1135-1142．
[61] Souplet P. Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Math. Methods Appl. Sci., 2004, 27:1819-1829.
[62] Zheng S N, Ji R H. Asymptotic estimates to non-global solutions of a multi-coupled parabolic system. Preprint.
[63] Zheng S N, Su H. A quasilinear reaction-diffusion system coupled via nonlocal sources. Appl. Math. Comput., 2006, 180:295-308.
[64] Zheng S N, Wang W. Blow-up rate for a nonlinear diffusion equation. Appl. Math. Lett., 2006, 19(12): 1385-1389.
[65] Brandle C, Quiros F, Rossi J D. A complete classification of simultaneous blow-up rates. Appl. Math. Lett., 2006, 19:607-611.
[66] Deng K. Blow-up rates for parabolic systems. Comput. Math. Appl., 1995, 46:110-118.
[67] Pao C V. Nonlinear parabolic and elliptic equations. New york: Plenum Press, 1992.
[68] Quir6s F, Rossi J D. Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud., 2003, 3:397-418.
[69] Chipot M, Fila M, Quittner P. Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. Univ. Comenian. (N. s.), 1991, 60(1).
[70] Levine H A, Payne L E. Nonexistence theorems for the heat equation with a nonlinear boundary conditions for the porous medium eqution backward in time. J. Differential Equations, 1974, 16:319-334.
[71] Lopez J, Marquez V, Wolanski N. Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Un. Mat. Argentina, 1993, 38:196-209.
[72] Walter W. On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition. SIAM J. Math. Anal., 1975, 6:85-90.
[73] Wu Y H, Wang M X. Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary condition. Chinese Ann. Math. Ser. B, 1995, 16(3):371-378.
[74] Wang M X, Wu Y H. Global solutions and blow-up problems for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J. Math. Anal., 1993, 24(6):1515-1521.
[75] Deng K, Fila M, Levine H A. On critical exponents for a system of heat equations coupled in the boundary conditions. Acta Math. Univ. Comenian. (N. S.), 1994, 63(2).
[76] Pederson M, Lin Z G. Blow up estimates of the positive solutions of a parabolic system. J. Math. Anal. Appl., 2001, 255:551-563.
[77] Zheng S N, Liang W M, Song X F. Critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux. Appl. Math. Comput., 2004, 154:567-581.
[78] Lin Z G, Wang M X. The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions. Z. Angew. Math. Phys., 1999, 50:361-374.
[79] Du L L. Note on non-simultaneous blow-up for a reaction-diffusion system. Appl. Math. Lett., 2000, 20:81-85.
[2] Rodriguez-Bernal A, Tajdine A. Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differential Equations, 2001, 169(2) :332-372.
[3] Levine H A. The role of critical exponents in blowup theorems. SIAM Rev., 1990, 32(2):262-288.
[4] Deng K, Levine H A. The role of crictical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl., 2000, 243(1).
[5] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J. Differential Equations, 1981, 89(1):76-202.
[6] Galaktionov V A, Kurdyumov S P, Samarskii A A. A parabolic system of quasi-linear equations i. Differ. Equ., 1983, 19:1558-1571.
[7] Galaktionov V A, Kurdyumov S P, Samarskii A A. A parabolic system of quasi-linear equations ii. Differ. Equ., 1985, 21:1049-1062.
[8] Zheng S N. Global existence and global nonexistence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39(3):327-340.
[9] Zheng S N. Global boundedness of solutions to a reaction-diffusion system. Math. Methods Appl. Sci., 1999, 22(1):43-54.
[10] Deng K. Global existence and blow-up for a system of heat equations with a nonlinear boundary condition. Math. Methods. Appl. Sci., 1995, 18(4).
[11] Wang M X. Parabolic systems with nonlinear boundary conditions. Chinese Sci. Bull., 1995, 40(17):1412-1414.
[12] Zheng S N, Song X F. Interactions among multi-nonliearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux. Nonlinear Anal., 2004, 57:519-530.
[13] Zheng S N, Wang W. Critical exponents for a nonlinear diffusion system. Nonlinear Anal., 2007, 67:1190-1210.
[14] Zheng S N, Li F J. Critical exponents for a reaction-diffusion model with absorptions and coupled boundary flux. Proc. Edinb. Math. Soc., 2005, 48:241-252.
[15] Rossi J D, Wolanski N. Blow-up versus global existence for a semilinear reaction-diffusion system in a bounded domain. Comm. Partial Differential Equations, 1995, 20:1991-2001.
[16] Mu C L, Su Y. Global existence and blow-up for a quasilinear degenerate parabolic system in a cylinder. Appl. Math. Lett., 2001, 14:715-723.
[17] Friedman A, Mcleod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J., 1995, 34(2):425-447.
[18] Fila M, Quittner P. The blow-up rate for the heat equation with a non-linear boundary condition. Math. Methods Appl. Sci., 1991, 14(3):197-205.
[19] Zheng S N. Nonexistence of positive solutions to a semi-linear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl., 1999, 232(2):293-311.
[20] Wang M X. Blow-up estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl., 2001, 257(1):46-51.
[21] Fila M, Levine H A. On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition. J. Math. Appl. Anal., 1996, 204(2):494-521.
[22] Lin Z G, Xie C H, Wang M X. On critical exponents for a semilinear heat equations with nonlinear boundary conditions. Appl. Math. J. Chinese Univ. Ser. B, 1998, 13(4):363-372.
[23] Fu S C, Guo J S. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Appl. Anal., 2002, 276(1 ):458-475.
[24] Wang M X. Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions. Appl. Math. Lett., 2003, 16(4):543-549.
[25] Rossi J D. The blow up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math. Univ. Comenian. (N.S.), 1998, 67:343-350.
[26] Zheng S N, Li F J, Liu B C. Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux. Appl. Math. Lett., 2006, 19(9):942-948.
[27] Song X F. Blow-up analysis for a system of heat equations coupled via nonlinear boundary conditions. Math. Methods Appl. Sci., 2007, 30(10):1135-1146.
[28] Song X F. Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws. In press, Nonlinear Anal.
[29] Samarskii A A, Galaktionov V A, Kurdyumov S P, Mikhailov A P. Blow-up in problems for quarsilinear parabolic equations. Berlin: Walter de Gruyter, 1995.
[30] Pinasco J P, Rossi J D. Simultaneous vs. non-simultaneous blow-up. New Zealand J. Math., 2000, 29:55-59.
[31] Quir6s F, Rossi J D. Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys., 2001, 52:342-346.
[32] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal., 2005, 4(3):523-536.
[33] Brandle C, Quir6s F, Rossi J D. The role of nonlinear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl., 2005, 308(1):92-104.
[34] Souplet P, Tayachi S. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan., 2004, 56(2):571-584.
[35] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18(4):405-418.
[36] Zheng S N, Liu B C, Li F J. Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system. J. Math. Anal. Appl., 2007, 326:414-431.
[37] Zheng S N, Liu B C, Li F J. Non-simultaneous blow-up for a multi-coupeld reaction-diffusion system. Nonlinear Anal., 2006, 64:1189-1202.
[38] Li F J, Liu B C, Zheng S N. Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary flux. Z. Angew. Math. Phys., 2007, 58:717-735.
[39] Amann H. Parabolic evolution equations and nonlinear boundary conditions. J. Differential Equations, 1988, 72:201-269.
[40] Bebernes J, Eberly D. Mathematical problem from combustion theory, Appl. Math. Sci. 83. Berlin: Springer-Verlag, 1989.
[41] Bebernes J, Bressan A, Eberly D. A description of blowup for the solid fuel ignition model. Indiana Univ. Math. J., 1987, 36:295-305.
[42] Bebernes J, Kassy D. A mathematical analysis of blow-up for thermal reactions. SIAM J. Appl. Math., 1981, 40:476-484.
[43] Caristi G, Mitidieri E. Blow-up estimates of positive solutions of a parabolic system. J. Differential Equations, 1994, 113:265-271.
[44] Giga Y, Kohn R V. Asymptotic self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 1985, 38:297-319.
[45] Lin Z G, Xie C H. The blow-up rate for a system of heat equations with nonlinear boundary conditions. Nonlinear Anal., 1998, 34:767-778.
[46] Weissler F B. Single point blow-up of semi-linear initial value problems. J. Differential Equations, 1984, 55:204-224.
[47] Ferreira R, Groisman P, Rossi J D. Numerical blow-up for the porous medium equation with a source. Numer. Methods Partial Differential Equations, 2004, 20:552-575.
[48] Galaktionov V A, Posashkov S A. On so some investigation method for unbounded solutions of quasilinear parabolic equations. USSR Comput. Math. Phys., 1988, 28:148-156.
[49] Galaktionov V A. On asymptotic self-similar behaviour for a quasilinear heat equation: single point blow-up. Math. Methods Appl. Sci., 1995, 26:675-693.
[50] Souplet P. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J. Differential Equations, 1999, 153:374-406.
[51] 姜朝欣，郑斯宁．具有吸收和非线性边界流的非线性扩散方程的爆破估计(英文)．数 学进展，2004，33(5)：615-620．
[52] Dibenedetto E. Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J., 1983, 32:83-118.
[53] Ladyzhenskaya O A, Solonnikov V A, Ural'ceva N N. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, Transl. Math. Mono. 23. Povidence RI: AMS, 1968.
[54] Ziemer W P. Interior and bounary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc, 1982, 271(2):733-748.
[55] Liberman G M. Second order parabolic differential equations. River Edge: World scientific, 1996.
[56] Friedman A. Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall, 1964.
[57] Quir(?)s F, Rossi J D. Blow-up sets and fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ. Math. J., 2001, 50:629-654.
[58] Chen Y P. Blow-up for a system of heat equations with nonlocal sources and absorptions. Comput. Math. Appl., 2004, 48:361-372.
[59] Li H L, Wang M X. Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system. Nonlinear Anal., 2005, 63(9):1083-1093.
[60] 刘其林，李玉样，谢春红．带局部非线性反应项的退化抛物方程解的爆破性质．数学 学报，2003，46(6)：1135-1142．
[61] Souplet P. Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Math. Methods Appl. Sci., 2004, 27:1819-1829.
[62] Zheng S N, Ji R H. Asymptotic estimates to non-global solutions of a multi-coupled parabolic system. Preprint.
[63] Zheng S N, Su H. A quasilinear reaction-diffusion system coupled via nonlocal sources. Appl. Math. Comput., 2006, 180:295-308.
[64] Zheng S N, Wang W. Blow-up rate for a nonlinear diffusion equation. Appl. Math. Lett., 2006, 19(12): 1385-1389.
[65] Brandle C, Quiros F, Rossi J D. A complete classification of simultaneous blow-up rates. Appl. Math. Lett., 2006, 19:607-611.
[66] Deng K. Blow-up rates for parabolic systems. Comput. Math. Appl., 1995, 46:110-118.
[67] Pao C V. Nonlinear parabolic and elliptic equations. New york: Plenum Press, 1992.
[68] Quir6s F, Rossi J D. Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud., 2003, 3:397-418.
[69] Chipot M, Fila M, Quittner P. Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. Univ. Comenian. (N. s.), 1991, 60(1).
[70] Levine H A, Payne L E. Nonexistence theorems for the heat equation with a nonlinear boundary conditions for the porous medium eqution backward in time. J. Differential Equations, 1974, 16:319-334.
[71] Lopez J, Marquez V, Wolanski N. Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Un. Mat. Argentina, 1993, 38:196-209.
[72] Walter W. On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition. SIAM J. Math. Anal., 1975, 6:85-90.
[73] Wu Y H, Wang M X. Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary condition. Chinese Ann. Math. Ser. B, 1995, 16(3):371-378.
[74] Wang M X, Wu Y H. Global solutions and blow-up problems for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J. Math. Anal., 1993, 24(6):1515-1521.
[75] Deng K, Fila M, Levine H A. On critical exponents for a system of heat equations coupled in the boundary conditions. Acta Math. Univ. Comenian. (N. S.), 1994, 63(2).
[76] Pederson M, Lin Z G. Blow up estimates of the positive solutions of a parabolic system. J. Math. Anal. Appl., 2001, 255:551-563.
[77] Zheng S N, Liang W M, Song X F. Critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux. Appl. Math. Comput., 2004, 154:567-581.
[78] Lin Z G, Wang M X. The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions. Z. Angew. Math. Phys., 1999, 50:361-374.
[79] Du L L. Note on non-simultaneous blow-up for a reaction-diffusion system. Appl. Math. Lett., 2000, 20:81-85.