在本篇論文，我們首先討論一個半線性熱方程解的爆破行為，其中我們所考慮的邊界條件是非線性的。在某種假設下，我們得到解只會在邊界爆破。接著，利用Giga-Kohn轉換，我們得到解趨近爆破時間的漸進行為。此外，我們得到這種爆破是徹底的(complete)。
接著，我們討論一個非線性反應擴散系統解的爆破行為。我們主要的目的是要瞭解反應項和吸收項對爆破性質的影響。在某種假設下，我們得到爆破的充分必要條件，爆破速度的上下界估計，及爆破集合。

In this thesis, we first study the blow-up
behaviors of solutions of a semilinear heat
equation with a nonlinear boundary condition.
Under certain conditions, we prove that the
blow-up point occurs only at the boundary.
Then, by applying the well-known method of
Giga-Kohn, we derive the time asymptotic of
solutions near the blow-up time. In addition,
we prove that the blow-up is complete.
Next, we study the blow-up behavior for
a semilinear reaction-diffusion
system coupled in both equations and boundary conditions.
The main purpose is to understand how the reaction terms
and the absorption terms affect the blow-up properties.
We obtain a necessary and sufficient condition for blow-up,
derive the upper bound and lower bound for the blow-up rate,
and find the blow-up set under certain assumptions.

[1] S. Angenent, The zeroset of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79--96.
[2] N. Bedjaoui and P. Souplet, Critical blowup exponents for a system of reaction-diffusion equations with absorption, Z. Angew. Math. Phys. (to appear).
[3] G. Caristi and E. Mitidieri, Blow-up estimates of positive
solutions of a parabolic system, J. Diff. Equ. 113 (1994), 265-271.
[4] M. Chlebik and M. Fila, From critical exponents to blow-up rate for parabolic problems, Rend. Mat. 19 (1999), 449-470.
[5] M. Chlebik and M. Fila, Some recent results on blow-up on the boundary for the heat equation\publ B. Bojarski et al.
(ed.), Proceedings of the Minisemester, Warsaw, Poland,
(Warsaw, 1998), Banach Center Publ. 52, 61--71, Polish Acad. Sci., Warsaw, 2000.
[6] K. Deng, Global existence and blow-up for a system of heat equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences 18 (1995), 307-315.
[7] K. Deng, Blow-up rates for parabolic systems, Z. Angew. Math. Phys. 47 (1996), 132-143.
[8] K. Deng, M. Fila, and H.A. Levine, On critical exponents
for a system of heat equations coupled in the boundary conditions, Acta Math. Univ. Comenianae 63 (1994), 169-192.
[9] K. Deng and H.A. Levine, The role of critical exponents
in blow-up theorems, J. Math. Anal. Appl. 243 (2000), 85-126.
[10] K. Deng and C.L. Zhao, Blow-up for a parabolic system
coupled in an equation and a boundary condition, Proceedings of the Royal Society of Edinburgh 131A (2001)\pages 1345-1355.
[11] M. Escobedo and M.A. Herrero, Boundedness and blow up
for a semilinear reaction-diffusion system, J. Diff. Equ.
89 (1991), 176-202.
[12] M. Fila and J. Filo, Blow-up on the boundary: A survey, S. Janeczko et al. (ed.), Singularities and differential equations, (Warsaw, 1996), Banach Center Publ. 33, 67-78, Polish Acad. Sci., Warsaw, 1996.
[13] M. Fila and J.-S. Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition, Nonlinear Anal. 48 (2002), 995-1002.
[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), 494-521.
[15] M. Fila and P. Quittner, The blow-up rate for the
heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 14 (1991), 197--205.
[16] A. Friedman and Y. Giga, A single point blow-up for
solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 44 (1987), 65-79.
[17] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations\jour Indiana Univ. Math. J. 34 (1985), 425--447.
[18] S.C. Fu and J.S. Guo, Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Appl. 276 (2002), 458-475.
[19] S.-C. Fu, J.-S. Guo, and J.-C. Tsai, Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoko Math. J. (to appear).
[20] Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297--319.
[21] B. Gidas and J. Spruck, A priori bounds for positive
solutions of nonlinear elliptic equations, Comm. Part. Diff. Equ. 6 (1981), 883-901.
[22] J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl. 151 (1990), 58-79.
[23] J.-S.Guo and J.-C. Tsai, Structure of positive solutions
to semilinear initial value problem, Funkcialaj Ekvacioj (to appear).
[24] B. Hu, Remark on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Jifferential and Integral Equations 9 (1996), 891-901.
[25] B. Hu and H.M. 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.
[26] B. Hu and H.M. Yin, Critical exponents for a system
of heat equations coupled in a non-linear boundary
condition, Math. Meth. Appl. Sci. 19 (1996), 1099-1120.
[27] O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva,
Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968.
[28] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.
[29] Z. Lin and M. Wang, The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys. 50 (1999), 361-374.
[30] Z. Lin and C. Xie, The blow-up rate for a system of heat equations with nonlinear boundary conditions, Nonlinear Analysis 34 (1998), 767-778.
[31] A. de Pablo, F. Quiros, and J.D. Rossi, Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition, IMA J. Appl. Math. 67 (2002), 69-98.
[32] M. Pedersen and Z. Lin, Blow-up estimates of the positive solution of a parabolic system, J. Math. Anal. Appl. 255 (2001), 551-563.
[33] M. Pedersen and Z. Lin, Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition, Appl. Math. Letters 14 (2001), 171-176.
[34] M. Pedersen and Z. Lin, Coupled diffusion systems with localized nonlinear reactions, Comp. Math. Appl. 42 (2001), 807-816.
[35] M.H. Protter and H.F. Weinberger, Maximum Principles
in Differential Equations, Springer-Verlag, New York, 1984.
[36] F. Quiros and J.D. Rossi, blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear
boundary conditions\jour Indiana Univ. Math. J. (to appear).
[37] F. Quiros and J.D. Rossi, Non-simutaneous blow-up
in a semilinear parabolic system, Z. Angew. Math. Phys. 52 (2001), 342-346.
[38] M.X. Wang, Blow-up estimates for semilinear parabolic
systems coupled in an equation and a boundary condition,
Science in China, Series A 44 (2001), 1465-1468.
[39] M.X. Wang, Blow-up properties of solutions to parabolic systems coupled in equations and boundary conditions, J. Lond. Math. Soc. (to appear).
[40] S. Wang, M.X. Wang, and C. Xie, Reaction-diffusion systems with nonlinear boundary conditions, Z. Angew. Math. Phys. 48(6) (1997), 994-1001.
[41] S. Wang, C. Xie, and M.X. Wang, Note on critical exponents for a system of heat equations coupled in the boundary conditions, J. Math. Anal. Appl. 218 (1998), 313-324.
[42] S. Zheng, Global existence and global non-existence
of solutions to a reaction-diffusion system, Nonlinear Analysis 39 (2000), 327-340.