多重非线性抛物方程（组）奇性解的渐近分析
|
摘要
本文主要研究几类多重非线性抛物方程(组)奇性解的渐近行为．所讨论的问题包括确定非线性扩散方程组的blow-up临界指标、考查梯度项对非线性抛物方程解的blow-up性质的影响，以及研究带有奇异非线性反应项的抛物问题解的quenching现象等．首先讨论一个具有内部吸收项及耦合边界流的非线性扩散方程组．通过对模型中非线性机制之间相互作用的精确分析，我们确定了其blow-up临界指标．其次考虑含梯度项的非线性抛物模型，探究其中对流项是否并且以何种程度影响解的blow-up行为．最后我们研究有限时刻quenching问题．对于具有耦合吸收项并附加正Dirichlet边界条件的非线性抛物方程组，我们考虑了解的同时与非同时quenching；而对于具有加权非线性吸收的热方程，我们描述了解的quenching时间和quenching集的渐近行为．
第一章概述本文所研究问题的实际背景和国内外的发展现状，并简要介绍本文的主要工作．
第二章考虑内吸收非线性扩散方程(u~m)_t=△u-a_1u~(α_1)，(v~n)_t=△v-a_2v~(β_1)经由边界流(?)耦合的初边值问题．通过引入特征代数方程组以及对所有八个非线性指标的完全分类，我们得到对该问题blow-up临界指标的简明而清晰的刻画．由于所考虑模型的一般性，这一工作包含了关于blow-up临界指标的许多已有结果．与单个方程结果的比较可见耦合机制对blow-up临界指标的本质性影响．
第三章致力于含梯度项的非线性抛物方程解的blow-up分析，其目的在于研究梯度项对解的渐近行为的影响．对于具有内部吸收项及正性梯度项的半线性抛物方程u_t=△u+|▽u|~r-ae~(pu),附加边界条件(?)的初边值问题，我们证明当且仅当r≥2时，梯度项对blow-up的形成起本质作用．进一步，当r>2时，对流项还将显著影响blow-up速率，并且使得blow-up速率具有关于模型参数的不连续性(discontinuous)．然而，梯度项对空间blow-up profile并无本质性影响．对非线性扩散方程w_t=(e~((m-1)w))_(xx)-λe~((p-1)w)附加Neumann边界条件w_x(0，t)=0，w_x(1，t)=e~((q-m)w(1,t))的初边值问题，利用scaling方法，我们建立了解的blow-up速率估计．通过适当变换，该方程等价于一含对流项的多孔介质类型方程．虽然此模型中的对流项不改变解的blow-up速率，但它给问题的讨论带来一定的困难．
第四章研究有限时刻quenching问题，包括非线性抛物方程组u_t=△u-v~(-p)，v_t=△u-u~(-q)，(x，t)∈Ω×(0，T)具有正Dirichlet边界条件的初边值问题，以及带有加权非线性吸收的热方程u_t=u_(xx)-Mf(x)u~(-p),(x，t)∈(-1，1)×(0，T)附加边界条件u(-1，t)=u(1，t)=1和初值φ(x)的模型．对于前一问题，我们建立了Ω=B_R径向解的非同时quenching准则：当p，q≥1时quenching必为同时，当p<1≤q或q<1≤p时必为非同时；若p，q<1且R>(?)，则quenching依赖于初值既可能同时也可能非同时．需要提及的是，在对共存性结果的证明中，我们工作的新意在于恰当地定义一个初值集合以使quenching解具有某些一致下界估计．对于后一问题，我们利用局部能量估计方法，得到当M→+∞时解的quenching时间与quenching集的渐近性态．我们有，当M-→+∞时，quenching时间T～(m/(p+1))·1/M，其中m=(?)；并且当M充分大时，quenching点将集中于f／φ~(p+1)的最大值点．
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equations (systems). The topics include the critical exponent for a nonlinear diffusion system, the influences of the gradient perturbations on the blow-up properties of solutions for nonlinear parabolic equations, and the quenching behavior of solutions for parabolic problems with singular absorptions. Firstly, we consider a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary fluxes. A precise analysis on interactions among the multi-nonlinearities in the system is given to determine the critical exponent. Secondly, we concern nonlinear parabolic models with convection so as to explore whether and in what extent the gradient terms influence blow-up behavior of solutions. Finally, in the studying of quenching phenomena, we determine simultaneous versus non-simultaneous quenching for a nonlinear parabolic system with coupled absorptions subject to positive Dirichlet boundary conditions, and characterize the asymptotic behavior of quenching time and set of solutions for heat equations with weighted nonlinear absorptions.
Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.
Chapter 2 deals with the initial-boundary problem for (u~m)_t =△u -α_1u~(α_1), (v~n)_t =△v -α_2v~(β_1) coupled via boundary flux (?). We introduce a so called characteristic algebraic system together with a complete classification for all the eight nonlinear parameters to obtain a simple and clear description to the critical exponent of the problem. Due to the generality of the model considered, this covers many known results on critical blow-up exponents. Comparing with those for scalar cases, the substantial effects of the coupling mechanism on critical exponents can be observed.
Chapter 3 is devoted to the blow-up analysis for nonlinear parabolic equations with convection. The aim is to investigate the influences of gradient perturbations on the asymptotic behavior of solutions. For the semilinear parabolic equation u_t =△u + |▽u|~r - ae~(pu) subject to nonlinear boundary flux (?) = e~(qu), we obtain that the gradient term makes a substantial contribution to the formation of blow-up if and only if r≥2. In addition, the gradient term would significantly affect the blow-up rate as well whenever r > 2. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. However, the gradient perturbations have no essential effects on the spatial blow-up profile. For the nonlinear diffusion equation w_t = (e~((m-1)w))_(xx) -λe~((p-1)w) with Neumann boundary conditions w_x(0,t) = 0, w_x(1,t) = e~((q-m)w(1,t)) , using the scaling method, we establish the blow-up rate estimates for blow-up solutions. Under a transformation, this equation is equivalent to a porous medium type one with convection. We find that the gradient term just leads to a more complicated discussion without changing the blow-up rate of solutions.
Chapter 4 studies two quenching problems, namely, coupled nonlinear parabolic system u_t =△u-v~(-p), v_t =△v-u~(-q)inΩ×(0, T) with positive Dirichlet boundary conditions, and scalar heat equations with weighted nonlinear absorptions u_t = u_(xx) - Mf(x)u~(-p) subject to boundary conditions u(-1,t) = u(1,t) = 1 and initial dataφ(x). For the former problem, we characterize the non-simultaneous quenching criteria for radial quenching solutions withΩ= B_r: The quenching is simultaneous if p,q≥1, and non-simultaneous if p < 1≤q or q < 1≤p; If p,q < 1 with R > (?), then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. It should be mentioned that to get the coexistence result, we have to skillfully construct a set of initial data admitting required uniform lower estimates on quenching solutions. For the latter model, the asymptotic behavior of quenching time and set of solutions as M→+∞is established by local energy estimates. It is obtained that the quenching time T - (m/(p+1)). M~(-1) with (?) as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/φ~(p+1) for large M.
第一章概述本文所研究问题的实际背景和国内外的发展现状，并简要介绍本文的主要工作．
第二章考虑内吸收非线性扩散方程(u~m)_t=△u-a_1u~(α_1)，(v~n)_t=△v-a_2v~(β_1)经由边界流(?)耦合的初边值问题．通过引入特征代数方程组以及对所有八个非线性指标的完全分类，我们得到对该问题blow-up临界指标的简明而清晰的刻画．由于所考虑模型的一般性，这一工作包含了关于blow-up临界指标的许多已有结果．与单个方程结果的比较可见耦合机制对blow-up临界指标的本质性影响．
第三章致力于含梯度项的非线性抛物方程解的blow-up分析，其目的在于研究梯度项对解的渐近行为的影响．对于具有内部吸收项及正性梯度项的半线性抛物方程u_t=△u+|▽u|~r-ae~(pu),附加边界条件(?)的初边值问题，我们证明当且仅当r≥2时，梯度项对blow-up的形成起本质作用．进一步，当r>2时，对流项还将显著影响blow-up速率，并且使得blow-up速率具有关于模型参数的不连续性(discontinuous)．然而，梯度项对空间blow-up profile并无本质性影响．对非线性扩散方程w_t=(e~((m-1)w))_(xx)-λe~((p-1)w)附加Neumann边界条件w_x(0，t)=0，w_x(1，t)=e~((q-m)w(1,t))的初边值问题，利用scaling方法，我们建立了解的blow-up速率估计．通过适当变换，该方程等价于一含对流项的多孔介质类型方程．虽然此模型中的对流项不改变解的blow-up速率，但它给问题的讨论带来一定的困难．
第四章研究有限时刻quenching问题，包括非线性抛物方程组u_t=△u-v~(-p)，v_t=△u-u~(-q)，(x，t)∈Ω×(0，T)具有正Dirichlet边界条件的初边值问题，以及带有加权非线性吸收的热方程u_t=u_(xx)-Mf(x)u~(-p),(x，t)∈(-1，1)×(0，T)附加边界条件u(-1，t)=u(1，t)=1和初值φ(x)的模型．对于前一问题，我们建立了Ω=B_R径向解的非同时quenching准则：当p，q≥1时quenching必为同时，当p<1≤q或q<1≤p时必为非同时；若p，q<1且R>(?)，则quenching依赖于初值既可能同时也可能非同时．需要提及的是，在对共存性结果的证明中，我们工作的新意在于恰当地定义一个初值集合以使quenching解具有某些一致下界估计．对于后一问题，我们利用局部能量估计方法，得到当M→+∞时解的quenching时间与quenching集的渐近性态．我们有，当M-→+∞时，quenching时间T～(m/(p+1))·1/M，其中m=(?)；并且当M充分大时，quenching点将集中于f／φ~(p+1)的最大值点．
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equations (systems). The topics include the critical exponent for a nonlinear diffusion system, the influences of the gradient perturbations on the blow-up properties of solutions for nonlinear parabolic equations, and the quenching behavior of solutions for parabolic problems with singular absorptions. Firstly, we consider a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary fluxes. A precise analysis on interactions among the multi-nonlinearities in the system is given to determine the critical exponent. Secondly, we concern nonlinear parabolic models with convection so as to explore whether and in what extent the gradient terms influence blow-up behavior of solutions. Finally, in the studying of quenching phenomena, we determine simultaneous versus non-simultaneous quenching for a nonlinear parabolic system with coupled absorptions subject to positive Dirichlet boundary conditions, and characterize the asymptotic behavior of quenching time and set of solutions for heat equations with weighted nonlinear absorptions.
Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.
Chapter 2 deals with the initial-boundary problem for (u~m)_t =△u -α_1u~(α_1), (v~n)_t =△v -α_2v~(β_1) coupled via boundary flux (?). We introduce a so called characteristic algebraic system together with a complete classification for all the eight nonlinear parameters to obtain a simple and clear description to the critical exponent of the problem. Due to the generality of the model considered, this covers many known results on critical blow-up exponents. Comparing with those for scalar cases, the substantial effects of the coupling mechanism on critical exponents can be observed.
Chapter 3 is devoted to the blow-up analysis for nonlinear parabolic equations with convection. The aim is to investigate the influences of gradient perturbations on the asymptotic behavior of solutions. For the semilinear parabolic equation u_t =△u + |▽u|~r - ae~(pu) subject to nonlinear boundary flux (?) = e~(qu), we obtain that the gradient term makes a substantial contribution to the formation of blow-up if and only if r≥2. In addition, the gradient term would significantly affect the blow-up rate as well whenever r > 2. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. However, the gradient perturbations have no essential effects on the spatial blow-up profile. For the nonlinear diffusion equation w_t = (e~((m-1)w))_(xx) -λe~((p-1)w) with Neumann boundary conditions w_x(0,t) = 0, w_x(1,t) = e~((q-m)w(1,t)) , using the scaling method, we establish the blow-up rate estimates for blow-up solutions. Under a transformation, this equation is equivalent to a porous medium type one with convection. We find that the gradient term just leads to a more complicated discussion without changing the blow-up rate of solutions.
Chapter 4 studies two quenching problems, namely, coupled nonlinear parabolic system u_t =△u-v~(-p), v_t =△v-u~(-q)inΩ×(0, T) with positive Dirichlet boundary conditions, and scalar heat equations with weighted nonlinear absorptions u_t = u_(xx) - Mf(x)u~(-p) subject to boundary conditions u(-1,t) = u(1,t) = 1 and initial dataφ(x). For the former problem, we characterize the non-simultaneous quenching criteria for radial quenching solutions withΩ= B_r: The quenching is simultaneous if p,q≥1, and non-simultaneous if p < 1≤q or q < 1≤p; If p,q < 1 with R > (?), then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. It should be mentioned that to get the coexistence result, we have to skillfully construct a set of initial data admitting required uniform lower estimates on quenching solutions. For the latter model, the asymptotic behavior of quenching time and set of solutions as M→+∞is established by local energy estimates. It is obtained that the quenching time T - (m/(p+1)). M~(-1) with (?) as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/φ~(p+1) for large M.
引文
[1] 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.
[2] Aronson D, Weinberger H F. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math., 1978, 30:33-76.
[3] Hayakawa K. On the nonexistence of global solutions of some semilinear parabolic equations. Proc. Japan Acad., 1973, 49:503-525.
[4] Kobayashi K, Sirao T, Tanaka H. On the blowing up problem for semilinear heat equations. J. Math. Soc. Japan, 1977, 29:407-424.
[5] Weissler F B. Existence and non-existence of global solutions for semilinear equations. Israel J. Math., 1981, 6:29-40.
[6] Galaktionov V A. Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc. Roy. Soc. Edinburgh Sect. A, 1994, 124:517-525.
[7] Galaktionov V A, Kurdjumov S P, Mikhailov A P, Samarskii A A. On unbounded solutions of the Cauchy problem for the parabolic equation u_t = ▽(u~σ▽u) + u~β. Dokl. Akad. Nauk SSSR, 1980, 252:1362-1364 (in Russian); English translation Sov. Phys. Dokl., 1980, 25:458-459.
[8] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J. Differential Equations, 1981, 89:176-202.
[9] Qi Y W, Levine H A. The critical exponent of degenerate parabolic systems. Z. Angew. Math. Phys., 1993, 44:249-265.
[10] Huang Q, Mochizuki K. A note on the global solutions of a degenerate parabolic system. Tokyo J. Math., 1997, 20:63-66.
[11] Levine H A. The role of critical exponents in blow-up theorems. SIAM Reviews, 1990, 32:262-288.
[12] Deng K, Levine H A. The role of crictical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl., 2000, 243:85-126.
[13] Friedman A. Blow-up of solutions of nonlinear parabolic equations. in: W.M. Ni, P.L. Peletier, J. Serrin (Eds.), Nonlinear Diffusion Equations and Their Equilibrium States I, Springer, New York, 1988.
[14] Friedman A, McLeod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J., 1985, 34:425-447.
[15] Giga Y, Kohn R V. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 1985, 38:297-319.
[16] Chipot M, Weissler F B. Some blow-up results for a nonlinear parabolic equation with a gradient term. SIAM J. Math. Anal., 1989, 20:886-907.
[17] Kawohl B, Peletier L A. Observations on blow-up and dead cores for nonlinear parabolic equations. Math. Z., 1989, 202:207-217.
[18] Fila M. Remarks on blow-up for a nonlinear parabolic equation with a gradient term. Proc. Amer. Math. Soc., 1991, 111:795-801.
[19] Quittner P. Blow-up for semilinear parabolic equations with a gradient term. Math. Methods Appl. Sci., 1991, 14:413-417.
[20] Quittner P. On global existence and stationary solutions for two classes of semilinear parabolic problems. Comment. Math. Univ. Carolinae, 1993, 34:105-124.
[21] Souplet P. Resultats d'explosion en temps fini pour une equation de la chaleur non lineaire. C. R. Acad. Sci. Paris Ser. I, 1995, 321:721-726.
[22] Souplet P. Finite time blow-up for a nonlinear parabolic equation with a gradient term and applications. Math. Methods Appl. Sci., 1996, 19:1317-1333.
[23] Souplet P, Weissler F B. Self-similar subsolutions and blow-up for nonlinear parabolic equations. J. Math. Anal. Appl., 1997, 212:60-74.
[24] Chlebik M, Fila M. From critical exponents to blow-up rates for parabolic problems. Rend. Mat. Appl., Ser. VII, 1999, 19:449-470.
[25] Souplet P, Tayachi S. Blow-up rates for nonlinear heat equations with gradient terms and for parabolic inequalities. Colloq. Math., 2001, 88:135-154.
[26] Chlebik M, Fila M, Quittner P. Blow-up of positive solutions of a semilinear parabolic equation with a gradient term. Dyn. Contin. Discrete Impulsive Syst. Ser. A Math. Anal., 2003, 10:525-537.
[27] Fila M, Souplet P. The blow-up rate for semilinear parabolic problems on general domains. Nonlinear Differ. Equ. Appl., 2001, 8:473-480.
[28] Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Ann. Mat. Pura Appl., 1993, 165:315-336.
[29] Zheng S N. Global existence and global non-existence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39:327-340.
[30] Zheng S N. Global boundedness of solutions to a reaction-diffusion system. Math. Methods Appl. Sci., 1999, 22:43-54.
[31] Zheng S N. Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl., 1999, 232:293-311.
[32] Wang M X. Blow-up estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl., 2001, 257:46-51.
[33] Wang MX. Blow-up rate estimates for semilinear parabolic systems. J. Differential Equations, 2001, 170:317-324.
[34] Galaktionov V A, Kurdjumov S P, Samarskii A A. A parabolic system of quasilinear equations I. Differential Equations, 1983, 19:1558-1571.
[35] Galaktionov V A, Kurdjumov S P, Samarskii A A. A parabolic system of quasilinear equations II. Differential Equations, 1985, 21:1049-1062.
[36] Deng W B. Global existence and finite time blow-up for a degenerate reaction-diffusion system. Nonlinear Anal., 2005, 60:977-991.
[37] Lei P D, Zheng S N. Global and nonglobal weak solutions to a degenerate parabolic system. J. Math. Anal. Appl., 2006, 324:177-198.
[38] 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.
[39] Lopez-Gomez J, Marquez V, Wolanski N. Blow-up results and localization of blowup points for the heat equation with a nonlinear boundary condition. J. Differential Equations, 1991, 92:384-401.
[40] Fila M. Boundedness of global solutions for the heat equation with nonlinear boundary conditions. Comment. Math. Univ. Carolinae, 1989, 30:479-484.
[41] Levine H A, Payne L E. Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time. J. Differential Equations, 1974, 16:319-334.
[42] Levine H A, Smith R A. A potential well theory for the heat equation with a nonlinear boundary condition. Math. Methods Appl. Sci., 1987, 9:127-136.
[43] Hu B, Yin H M. The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition. Trans. Amer. Math. Soc., 1994, 346(1):117-135.
[44] Hu B. Remarks on the blow-up estimate for solution of the heat equation with a nonlinear boundary condition. Differential Integral Equations, 1996, 9:891-901.
[45] 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.
[46] 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. Comenianae (N.S.), 1991, 60:35-103.
[47] Andreu F, Mazon J M, Toledo J, Rossi J D. Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal., 2002, 49:541-563.
[48] Rossi J D. The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math. Univ. Comenianae (N.S.), 1998, 67:343-350.
[49] 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, 19:942-948.
[50] Filo J. Diffusivity versus absorption through the boundary. J. Differential Equations, 1992, 99:281-305.
[51] Deng K, Xu M X. Remark on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions. Proc. Amer. Math. Soc., 1999, 127:167-172.
[52] Song X F, Zheng S N. Blow-up and blow-up rate for a reaction-diffusion model with multiple nonlinearities. Nonlinear Anal., 2003, 54:279-289.
[53] Jiang Z X, Zheng S N, Song X F. Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions. Appl. Math. Lett., 2004, 17:193-199.
[54] Fila M, Kawohl B. Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition. Adv. Math. Sci. Appl., 2001, 11:113— 126.
[55] Li H L, Wang M X. Properties of positive solutions to a nonlinear parabolic problem. Sci. China Ser. A, 2007, 50:590-608.
[56] Jiang Z X, Zheng S N. Blow-up rate for a nonlinear diffusion equation with absorption and nonlinear boundary flux. Adv. Math. (China), 2004, 33:615-620.
[57] Song X F, Zheng S N. Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities. J. Math. Anal. Appl., 2003, 281:739-756.
[58] Deng K. Blow-up rates for parabolic systems. Z. Angew. Math. Phys., 1996, 47:132-143.
[59] Rossi J D. The blow-up rate for a system of heat equations with non-trival coupling at the boundary. Math. Methods Appl. Sci., 1997, 20:1-11.
[60] Pedersen M, Lin Z G. Blow-up estimates of the positive solution of a parabolic system. J. Math. Anal. Appl., 2001, 255:551-563.
[61] Chen W Y. The blow-up estimate for heat equations with non-linear boundary conditions. Appl. Math. Comput., 2004, 156:355-366.
[62] Wang M X. Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition. Sci. China Ser. A, 2001, 44:1465-1468.
[63] Wang M X. Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions. Appl. Math. Lett., 2003, 16:543-549.
[64] Fu S C, Guo J S. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Anal. Appl., 2002, 276:458-475.
[65] Zheng S N, Liu B C, Li F J. Blow-up rate estimates for a doubly coupled reactiondiffusion system. J. Math. Anal. Appl., 2005, 312:576-595.
[66] 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.
[67] Zheng S N, Li F J. Critical exponents for a reaction-diffusion system with absorptions and coupled boundary flux. Proc. Edinburgh Math. Soc, 2005, 48:241-252.
[68] Kawarada H. On solutions of initial-boundary problem for u_t = u_(xx) + 1/(1 - u). Publ. Res. Inst. Math. Sci., 1975, 10:729-736.
[69] Acker A, Walter W. The quenching problem for nonlinear parabolic differential equations. Lecture Notes in Mathematics, No. 564, pp. 1-12, Springer-Verlag, New York, 1976.
[70] Levine H A, Montgomery J T. The quenching of solutions of some nonlinear parabolic equations. SIAM J. Math. Anal., 1980, 11:842-847.
[71] Chan C Y, Kwong M K. Quenching phenomena for singular nonlinear parabolic euqations. Nonlinear Anal., 1988, 12:1377-1383.
[72] Acker A, Kawohl B. Remarks on quenching. Nonlinear Anal., 1989, 13:53-61.
[73] Deng K, Levine H A. On the blow-up of u_t at quenching. Proc. Amer. Math. Soc, 1989, 106:1049-1056.
[74] Chan C Y, Ke L. Parabolic quenching for nonsmooth convex domains. J. Math. Anal. Appl., 1994, 186:52-65.
[75] Levine H A. Quenching, nonquenching, and beyond quenching for solutions of some parabolic euqations. Annadi di Mat. Pure et Appl., 1990, 155:243-260.
[76] Guo J S. On the quenching behavior of the solution of a semilinear parabolic equation. J. Math. Anal. Appl., 1990, 151:58-79.
[77] Guo J S. On the quenching rate estimate. Quart. Appl. Math., 1991, 49:747-752.
[78] Fila M, Hulshof J. A note on the quenching rate. Proc. Amer. Math. Soc., 1991, 112:473-477.
[79] Fila M, Hulshof J, Quittner P. The quenching problem on the N-dimensional ball. in: N.G. Lloyd, W.N. Ni, L.A. Peletier, J. Serrin (Eds.), in: Nonlinear Diffusion Equations and Their Equilibrium States: III, Birkhauser, Boston, MA, 1992, pp. 183-196.
[80] Salin T. On quenching with logarithmic singularity. Nonlinear Anal., 2003, 52:261-289.
[81] Fila M, Levine H A. Quenching on the boundary. Nonlinear Anal., 1993, 21:795-802.
[82] Deng K, Xu M X. Quenching for a nonlinear diffusion equation with a singular boundary condition. Z. Angew. Math. Phys., 1999, 50:574-584.
[83] de Pablo A, Quiros F, Rossi J D. Non-simultaneous quenching. Appl. Math. Lett., 2002, 15:265-269.
[84] Ferreira R, de Pablo A, Quiros F, Rossi J D. Non-simultaneous quenching in a system of heat equations coupled at the boundary. Z. Angew. Math. Phys., 2006, 57:586-594.
[85] Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm. Partial Differential Equations, 1991, 16:105-143.
[86] Wu Y H, Wang M X. Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary condition. Chin. Ann. of Math., 1995, 16B:371-378.
[87] Boni T. Sur l'explosion et le comportement asymptotique de la solution d'une equation parabolique semi-lineaire du second ordre. C. R. Acad. Sci. Paris, Ser. I, Math., 1998, 326:317-322.
[88] Escher J. Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions. Math. Ann., 1989, 284:285-305.
[89] Egorov Y V, Kondratiev V A. Two theorems on blow-up solutions for semilinear parabolic equations of second order. C. R. Acad. Sci. Paris, Ser. I, Math., 1998, 327:47-52.
[90] Galaktionov V A, Levine H A. On critical Fujita exponents for heat equations with nonlinear flux boundary conditions on the boundary. Israel J. Math., 1996, 94:125-146.
[91] Lopez-Gomez J, Marquez V, Wolanski N. Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Un. Mat. Argent., 1993, 38:196-209.
[92] Lacey A A, Ockendon J R, Sabina J, Salazar D. Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions. Rocky Mountain J. Math., 1996, 26:195-212.
[93] Leung A W, Zhang Q. Reaction diffusion equations with non-linear boundary conditions, blowup and steady states. Math. Methods Appl. Sci., 1998, 21:1593-1617.
[94] Rodriguez-Bernal A, Tajdine A. Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differential Equations, 2001, 169:332-372.
[95] Wang M X. Fast-slow diffusion systems with nonlinear boundary conditions. Nonlinear Anal., 2001, 46:893-908.
[96] Wang M X, Wang S. Quasilinear reaction-diffusion systems with nonlinear boundary conditions. J. Math. Anal. Appl., 1999, 231:21-33.
[97] Lacey A A, Ockendon J R, Sabina J. Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math., 1998, 58:1622-1647.
[98] Alfonsi L, Weissler F B. Blow up in R~N for a parabolic equation with a damping nonlinear gradient term. Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989), pp. 1-20, Progr. Nonlin. Differential Equations Appl., 7, Birkhauser, Boston, MA, 1992.
[99] Souplet P, Weissler F B. Poincare inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. H. Poincare, Analyse non lineaire, 1999, 16:337-373.
[100] Levine H A, Payne L N, Sacks P E, Straughan B. Analysis of convective reaction-diffusion equation (II). SIAM J. Math. Anal., 1989, 20:133-147.
[101] Aguirre J, Escobedo M. On the blow-up of solutions for a convective reaction diffusion equation. Proc. Roy. Soc. Edinburgh, 1993, 123A:433-460.
[102] Guo J S, Souplet P. Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Math. Ann., 2005, 331:651-667.
[103] Souplet P. The influence of gradient perturbations on blow-up asymptotics in semi-linear parabolic problems: a survey. Progr. Nonlinear Differential Equations Appl., 2005, 64:473-495.
[104] Ferreira R, Quiros F, Rossi J D. The balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation. J. Differential Equations, 2002, 184:259-282.
[105] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd edn. Berlin: Springer-Verlag, 1983.
[106] Deng K, Xu M X. On solutions of a singular diffusion equation. Nonlinear Anal., 2000, 41:489-500.
[107] Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992.
[108] Lieberman G M. Second Order Parabolic Differential Equations. River Edge: World Scientific, 1996.
[109] Ferreira R, de Pablo A, Vazquez J L. Classification of blow-up with nonlinear diffusion and localized reaction. J. Differential Equations, 2006, 231:195-211.
[110] Song X F, Zheng S N. Multi-nonlinear interactions in quasilinear reaction-diffusion equations with nonlinear boundary flux. Math. Comput. Modelling, 2004, 39:133-144.
[111] Fila M, Quittner P. The blow-up rate for the heat equation with a nonlinear boudary condition. Math. Methods Appl. Sci., 1991, 14:197-205.
[112] Gidas B, Spruck J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations, 1981, 6:883-901.
[113] Ladyzhenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type, in: Transl. Math. Mono., vol. 23. Povidence RI: AMS, 1968.
[114] Ziemer W P. Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc, 1982, 271:733-748.
[115] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1983.
[116] Ke L, Ning S. Quenching for degenerate parabolic equations. Nonlinear Anal., 1998, 34:1123-1135.
[117] Guo J S. On the semilinear elliptic equation △w - (1/2)y▽w + λw - w~(-β) = 0 in R~n. Chinese J. Math., 1991, 19:355-377.
[118] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Comm. Pure Appl. Math., 2005, 4:523-536.
[119] Quiros F, Rossi J D. Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud., 2003, 3:397-418.
[120] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential and Integral Equations, 2005, 18:405-418.
[121] Friedman A. Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice-Hall, 1964.
[122] Cortazar C, Elgueta M, Rossi J D. The blow-up problem for a semilinear parabolic equation with a potential. J. Math. Anal. Appl., 2007, 335:418-427.
[123] Giga Y, Kohn R V. Characterizing blow-up using similarity variables. Indiana Univ. Math. J., 1987, 36:1-40.
[124] Giga Y, Kohn R V. Nondegeneracy of blow-up for semilinear heat equations. Comm. Pure Appl. Math., 1989, 42:845-884.
[125] Friedman A, Lacey A A. The blow-up time for solutions of nonlinear heat equations with small diffusion. SIAM J. Math. Anal., 1987, 18:711-721.
[126] Ishige K, Yagisita H. Blow-up problems for a semilinear heat equation with large diffusion. J. Differential Equations, 2005, 212:114-128.
[127] Mizoguchi N, Yanagida E. Life span of solutions for a semilinear parabolic problem with small diffusion. J. Math. Anal. Appl., 2001, 261:350-368.
[128] Mizoguchi N, Yanagida E. Life span of solutions with large initial data in a semilinear parabolic equation. Indiana Univ. Math. J., 2001, 50:591-610.
[2] Aronson D, Weinberger H F. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math., 1978, 30:33-76.
[3] Hayakawa K. On the nonexistence of global solutions of some semilinear parabolic equations. Proc. Japan Acad., 1973, 49:503-525.
[4] Kobayashi K, Sirao T, Tanaka H. On the blowing up problem for semilinear heat equations. J. Math. Soc. Japan, 1977, 29:407-424.
[5] Weissler F B. Existence and non-existence of global solutions for semilinear equations. Israel J. Math., 1981, 6:29-40.
[6] Galaktionov V A. Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc. Roy. Soc. Edinburgh Sect. A, 1994, 124:517-525.
[7] Galaktionov V A, Kurdjumov S P, Mikhailov A P, Samarskii A A. On unbounded solutions of the Cauchy problem for the parabolic equation u_t = ▽(u~σ▽u) + u~β. Dokl. Akad. Nauk SSSR, 1980, 252:1362-1364 (in Russian); English translation Sov. Phys. Dokl., 1980, 25:458-459.
[8] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J. Differential Equations, 1981, 89:176-202.
[9] Qi Y W, Levine H A. The critical exponent of degenerate parabolic systems. Z. Angew. Math. Phys., 1993, 44:249-265.
[10] Huang Q, Mochizuki K. A note on the global solutions of a degenerate parabolic system. Tokyo J. Math., 1997, 20:63-66.
[11] Levine H A. The role of critical exponents in blow-up theorems. SIAM Reviews, 1990, 32:262-288.
[12] Deng K, Levine H A. The role of crictical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl., 2000, 243:85-126.
[13] Friedman A. Blow-up of solutions of nonlinear parabolic equations. in: W.M. Ni, P.L. Peletier, J. Serrin (Eds.), Nonlinear Diffusion Equations and Their Equilibrium States I, Springer, New York, 1988.
[14] Friedman A, McLeod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J., 1985, 34:425-447.
[15] Giga Y, Kohn R V. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 1985, 38:297-319.
[16] Chipot M, Weissler F B. Some blow-up results for a nonlinear parabolic equation with a gradient term. SIAM J. Math. Anal., 1989, 20:886-907.
[17] Kawohl B, Peletier L A. Observations on blow-up and dead cores for nonlinear parabolic equations. Math. Z., 1989, 202:207-217.
[18] Fila M. Remarks on blow-up for a nonlinear parabolic equation with a gradient term. Proc. Amer. Math. Soc., 1991, 111:795-801.
[19] Quittner P. Blow-up for semilinear parabolic equations with a gradient term. Math. Methods Appl. Sci., 1991, 14:413-417.
[20] Quittner P. On global existence and stationary solutions for two classes of semilinear parabolic problems. Comment. Math. Univ. Carolinae, 1993, 34:105-124.
[21] Souplet P. Resultats d'explosion en temps fini pour une equation de la chaleur non lineaire. C. R. Acad. Sci. Paris Ser. I, 1995, 321:721-726.
[22] Souplet P. Finite time blow-up for a nonlinear parabolic equation with a gradient term and applications. Math. Methods Appl. Sci., 1996, 19:1317-1333.
[23] Souplet P, Weissler F B. Self-similar subsolutions and blow-up for nonlinear parabolic equations. J. Math. Anal. Appl., 1997, 212:60-74.
[24] Chlebik M, Fila M. From critical exponents to blow-up rates for parabolic problems. Rend. Mat. Appl., Ser. VII, 1999, 19:449-470.
[25] Souplet P, Tayachi S. Blow-up rates for nonlinear heat equations with gradient terms and for parabolic inequalities. Colloq. Math., 2001, 88:135-154.
[26] Chlebik M, Fila M, Quittner P. Blow-up of positive solutions of a semilinear parabolic equation with a gradient term. Dyn. Contin. Discrete Impulsive Syst. Ser. A Math. Anal., 2003, 10:525-537.
[27] Fila M, Souplet P. The blow-up rate for semilinear parabolic problems on general domains. Nonlinear Differ. Equ. Appl., 2001, 8:473-480.
[28] Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Ann. Mat. Pura Appl., 1993, 165:315-336.
[29] Zheng S N. Global existence and global non-existence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39:327-340.
[30] Zheng S N. Global boundedness of solutions to a reaction-diffusion system. Math. Methods Appl. Sci., 1999, 22:43-54.
[31] Zheng S N. Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl., 1999, 232:293-311.
[32] Wang M X. Blow-up estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl., 2001, 257:46-51.
[33] Wang MX. Blow-up rate estimates for semilinear parabolic systems. J. Differential Equations, 2001, 170:317-324.
[34] Galaktionov V A, Kurdjumov S P, Samarskii A A. A parabolic system of quasilinear equations I. Differential Equations, 1983, 19:1558-1571.
[35] Galaktionov V A, Kurdjumov S P, Samarskii A A. A parabolic system of quasilinear equations II. Differential Equations, 1985, 21:1049-1062.
[36] Deng W B. Global existence and finite time blow-up for a degenerate reaction-diffusion system. Nonlinear Anal., 2005, 60:977-991.
[37] Lei P D, Zheng S N. Global and nonglobal weak solutions to a degenerate parabolic system. J. Math. Anal. Appl., 2006, 324:177-198.
[38] 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.
[39] Lopez-Gomez J, Marquez V, Wolanski N. Blow-up results and localization of blowup points for the heat equation with a nonlinear boundary condition. J. Differential Equations, 1991, 92:384-401.
[40] Fila M. Boundedness of global solutions for the heat equation with nonlinear boundary conditions. Comment. Math. Univ. Carolinae, 1989, 30:479-484.
[41] Levine H A, Payne L E. Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time. J. Differential Equations, 1974, 16:319-334.
[42] Levine H A, Smith R A. A potential well theory for the heat equation with a nonlinear boundary condition. Math. Methods Appl. Sci., 1987, 9:127-136.
[43] Hu B, Yin H M. The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition. Trans. Amer. Math. Soc., 1994, 346(1):117-135.
[44] Hu B. Remarks on the blow-up estimate for solution of the heat equation with a nonlinear boundary condition. Differential Integral Equations, 1996, 9:891-901.
[45] 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.
[46] 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. Comenianae (N.S.), 1991, 60:35-103.
[47] Andreu F, Mazon J M, Toledo J, Rossi J D. Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal., 2002, 49:541-563.
[48] Rossi J D. The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math. Univ. Comenianae (N.S.), 1998, 67:343-350.
[49] 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, 19:942-948.
[50] Filo J. Diffusivity versus absorption through the boundary. J. Differential Equations, 1992, 99:281-305.
[51] Deng K, Xu M X. Remark on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions. Proc. Amer. Math. Soc., 1999, 127:167-172.
[52] Song X F, Zheng S N. Blow-up and blow-up rate for a reaction-diffusion model with multiple nonlinearities. Nonlinear Anal., 2003, 54:279-289.
[53] Jiang Z X, Zheng S N, Song X F. Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions. Appl. Math. Lett., 2004, 17:193-199.
[54] Fila M, Kawohl B. Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition. Adv. Math. Sci. Appl., 2001, 11:113— 126.
[55] Li H L, Wang M X. Properties of positive solutions to a nonlinear parabolic problem. Sci. China Ser. A, 2007, 50:590-608.
[56] Jiang Z X, Zheng S N. Blow-up rate for a nonlinear diffusion equation with absorption and nonlinear boundary flux. Adv. Math. (China), 2004, 33:615-620.
[57] Song X F, Zheng S N. Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities. J. Math. Anal. Appl., 2003, 281:739-756.
[58] Deng K. Blow-up rates for parabolic systems. Z. Angew. Math. Phys., 1996, 47:132-143.
[59] Rossi J D. The blow-up rate for a system of heat equations with non-trival coupling at the boundary. Math. Methods Appl. Sci., 1997, 20:1-11.
[60] Pedersen M, Lin Z G. Blow-up estimates of the positive solution of a parabolic system. J. Math. Anal. Appl., 2001, 255:551-563.
[61] Chen W Y. The blow-up estimate for heat equations with non-linear boundary conditions. Appl. Math. Comput., 2004, 156:355-366.
[62] Wang M X. Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition. Sci. China Ser. A, 2001, 44:1465-1468.
[63] Wang M X. Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions. Appl. Math. Lett., 2003, 16:543-549.
[64] Fu S C, Guo J S. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Anal. Appl., 2002, 276:458-475.
[65] Zheng S N, Liu B C, Li F J. Blow-up rate estimates for a doubly coupled reactiondiffusion system. J. Math. Anal. Appl., 2005, 312:576-595.
[66] 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.
[67] Zheng S N, Li F J. Critical exponents for a reaction-diffusion system with absorptions and coupled boundary flux. Proc. Edinburgh Math. Soc, 2005, 48:241-252.
[68] Kawarada H. On solutions of initial-boundary problem for u_t = u_(xx) + 1/(1 - u). Publ. Res. Inst. Math. Sci., 1975, 10:729-736.
[69] Acker A, Walter W. The quenching problem for nonlinear parabolic differential equations. Lecture Notes in Mathematics, No. 564, pp. 1-12, Springer-Verlag, New York, 1976.
[70] Levine H A, Montgomery J T. The quenching of solutions of some nonlinear parabolic equations. SIAM J. Math. Anal., 1980, 11:842-847.
[71] Chan C Y, Kwong M K. Quenching phenomena for singular nonlinear parabolic euqations. Nonlinear Anal., 1988, 12:1377-1383.
[72] Acker A, Kawohl B. Remarks on quenching. Nonlinear Anal., 1989, 13:53-61.
[73] Deng K, Levine H A. On the blow-up of u_t at quenching. Proc. Amer. Math. Soc, 1989, 106:1049-1056.
[74] Chan C Y, Ke L. Parabolic quenching for nonsmooth convex domains. J. Math. Anal. Appl., 1994, 186:52-65.
[75] Levine H A. Quenching, nonquenching, and beyond quenching for solutions of some parabolic euqations. Annadi di Mat. Pure et Appl., 1990, 155:243-260.
[76] Guo J S. On the quenching behavior of the solution of a semilinear parabolic equation. J. Math. Anal. Appl., 1990, 151:58-79.
[77] Guo J S. On the quenching rate estimate. Quart. Appl. Math., 1991, 49:747-752.
[78] Fila M, Hulshof J. A note on the quenching rate. Proc. Amer. Math. Soc., 1991, 112:473-477.
[79] Fila M, Hulshof J, Quittner P. The quenching problem on the N-dimensional ball. in: N.G. Lloyd, W.N. Ni, L.A. Peletier, J. Serrin (Eds.), in: Nonlinear Diffusion Equations and Their Equilibrium States: III, Birkhauser, Boston, MA, 1992, pp. 183-196.
[80] Salin T. On quenching with logarithmic singularity. Nonlinear Anal., 2003, 52:261-289.
[81] Fila M, Levine H A. Quenching on the boundary. Nonlinear Anal., 1993, 21:795-802.
[82] Deng K, Xu M X. Quenching for a nonlinear diffusion equation with a singular boundary condition. Z. Angew. Math. Phys., 1999, 50:574-584.
[83] de Pablo A, Quiros F, Rossi J D. Non-simultaneous quenching. Appl. Math. Lett., 2002, 15:265-269.
[84] Ferreira R, de Pablo A, Quiros F, Rossi J D. Non-simultaneous quenching in a system of heat equations coupled at the boundary. Z. Angew. Math. Phys., 2006, 57:586-594.
[85] Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm. Partial Differential Equations, 1991, 16:105-143.
[86] Wu Y H, Wang M X. Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary condition. Chin. Ann. of Math., 1995, 16B:371-378.
[87] Boni T. Sur l'explosion et le comportement asymptotique de la solution d'une equation parabolique semi-lineaire du second ordre. C. R. Acad. Sci. Paris, Ser. I, Math., 1998, 326:317-322.
[88] Escher J. Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions. Math. Ann., 1989, 284:285-305.
[89] Egorov Y V, Kondratiev V A. Two theorems on blow-up solutions for semilinear parabolic equations of second order. C. R. Acad. Sci. Paris, Ser. I, Math., 1998, 327:47-52.
[90] Galaktionov V A, Levine H A. On critical Fujita exponents for heat equations with nonlinear flux boundary conditions on the boundary. Israel J. Math., 1996, 94:125-146.
[91] Lopez-Gomez J, Marquez V, Wolanski N. Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Un. Mat. Argent., 1993, 38:196-209.
[92] Lacey A A, Ockendon J R, Sabina J, Salazar D. Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions. Rocky Mountain J. Math., 1996, 26:195-212.
[93] Leung A W, Zhang Q. Reaction diffusion equations with non-linear boundary conditions, blowup and steady states. Math. Methods Appl. Sci., 1998, 21:1593-1617.
[94] Rodriguez-Bernal A, Tajdine A. Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differential Equations, 2001, 169:332-372.
[95] Wang M X. Fast-slow diffusion systems with nonlinear boundary conditions. Nonlinear Anal., 2001, 46:893-908.
[96] Wang M X, Wang S. Quasilinear reaction-diffusion systems with nonlinear boundary conditions. J. Math. Anal. Appl., 1999, 231:21-33.
[97] Lacey A A, Ockendon J R, Sabina J. Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math., 1998, 58:1622-1647.
[98] Alfonsi L, Weissler F B. Blow up in R~N for a parabolic equation with a damping nonlinear gradient term. Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989), pp. 1-20, Progr. Nonlin. Differential Equations Appl., 7, Birkhauser, Boston, MA, 1992.
[99] Souplet P, Weissler F B. Poincare inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. H. Poincare, Analyse non lineaire, 1999, 16:337-373.
[100] Levine H A, Payne L N, Sacks P E, Straughan B. Analysis of convective reaction-diffusion equation (II). SIAM J. Math. Anal., 1989, 20:133-147.
[101] Aguirre J, Escobedo M. On the blow-up of solutions for a convective reaction diffusion equation. Proc. Roy. Soc. Edinburgh, 1993, 123A:433-460.
[102] Guo J S, Souplet P. Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Math. Ann., 2005, 331:651-667.
[103] Souplet P. The influence of gradient perturbations on blow-up asymptotics in semi-linear parabolic problems: a survey. Progr. Nonlinear Differential Equations Appl., 2005, 64:473-495.
[104] Ferreira R, Quiros F, Rossi J D. The balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation. J. Differential Equations, 2002, 184:259-282.
[105] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd edn. Berlin: Springer-Verlag, 1983.
[106] Deng K, Xu M X. On solutions of a singular diffusion equation. Nonlinear Anal., 2000, 41:489-500.
[107] Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992.
[108] Lieberman G M. Second Order Parabolic Differential Equations. River Edge: World Scientific, 1996.
[109] Ferreira R, de Pablo A, Vazquez J L. Classification of blow-up with nonlinear diffusion and localized reaction. J. Differential Equations, 2006, 231:195-211.
[110] Song X F, Zheng S N. Multi-nonlinear interactions in quasilinear reaction-diffusion equations with nonlinear boundary flux. Math. Comput. Modelling, 2004, 39:133-144.
[111] Fila M, Quittner P. The blow-up rate for the heat equation with a nonlinear boudary condition. Math. Methods Appl. Sci., 1991, 14:197-205.
[112] Gidas B, Spruck J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations, 1981, 6:883-901.
[113] Ladyzhenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type, in: Transl. Math. Mono., vol. 23. Povidence RI: AMS, 1968.
[114] Ziemer W P. Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc, 1982, 271:733-748.
[115] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1983.
[116] Ke L, Ning S. Quenching for degenerate parabolic equations. Nonlinear Anal., 1998, 34:1123-1135.
[117] Guo J S. On the semilinear elliptic equation △w - (1/2)y▽w + λw - w~(-β) = 0 in R~n. Chinese J. Math., 1991, 19:355-377.
[118] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Comm. Pure Appl. Math., 2005, 4:523-536.
[119] Quiros F, Rossi J D. Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud., 2003, 3:397-418.
[120] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential and Integral Equations, 2005, 18:405-418.
[121] Friedman A. Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice-Hall, 1964.
[122] Cortazar C, Elgueta M, Rossi J D. The blow-up problem for a semilinear parabolic equation with a potential. J. Math. Anal. Appl., 2007, 335:418-427.
[123] Giga Y, Kohn R V. Characterizing blow-up using similarity variables. Indiana Univ. Math. J., 1987, 36:1-40.
[124] Giga Y, Kohn R V. Nondegeneracy of blow-up for semilinear heat equations. Comm. Pure Appl. Math., 1989, 42:845-884.
[125] Friedman A, Lacey A A. The blow-up time for solutions of nonlinear heat equations with small diffusion. SIAM J. Math. Anal., 1987, 18:711-721.
[126] Ishige K, Yagisita H. Blow-up problems for a semilinear heat equation with large diffusion. J. Differential Equations, 2005, 212:114-128.
[127] Mizoguchi N, Yanagida E. Life span of solutions for a semilinear parabolic problem with small diffusion. J. Math. Anal. Appl., 2001, 261:350-368.
[128] Mizoguchi N, Yanagida E. Life span of solutions with large initial data in a semilinear parabolic equation. Indiana Univ. Math. J., 2001, 50:591-610.