带有指数型非线性项的离散泊松方程和热方程

Abstract

Abstract:

This paper mainly study the relations between the solution of the discrete Poisson equation and the solution of the discrete heat equation with exponential nonlinear term by monotone iterative method and comparison principle. When the solutions of the discrete Poisson equation exist, we discuss the asymptotic stability of the solutions to the discrete heat equation with exponential nonlinear term.

Key words: Discrete Poisson equation, Discrete heat equation, Graph, Comparison principle, Blow-up

CLC Number:

O175.26

Yafeng Li,Qiao Xin,Chunlai Mu. The Discrete Possion Equation and the Heat Equation with the Exponential Nonlinear Term[J].Acta mathematica scientia,Series A, 2019, 39(4): 773-784.

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References

1	Chung S Y , Berenstein C A . ω-Harmonic functions and inverse conductivity problems on networks. SIAM J Appl Math, 2005, 65 (4): 1200- 1226
2	Chung S Y , Chung Y S , Kim J H . Diffusion and elastic equations on networks. Publ Res Inst Math Sci, 2007, 43 (3): 699- 726
3	Xin Q , Xu L , Mu C . Blow-up for the ω-heat equation with Dirichlet boundary conditions and a reaction term on graphs. Appl Anal, 2014, 93 (8): 1691- 1701
4	Xin Q , Liu D . Blow-up of solutions for the heat equations with variable source on graphs. J Nonlinear Sci Appl, 2016, 9 (4): 1685- 1692
5	Chung S Y , Choi M J . A new condition for blow-up solutions to discrete semilinear heat equations on networks. Comp Math Appl, 2017, 74 (12): 2929- 2939
6	Chung S Y , Lee Y S , Chung S Y . Extinction and positivity of the solutions of the heat equation with absorption on networks. J Math Anal Appl, 2011, 380 (2): 642- 652
7	辛巧, 许璐, 王安平. 无限图上带有吸收项的解的熄灭和正性. 云南大学学报(自然科学版), 2013, 35 (6): 727- 730
7	Xin Q , Xu L , Wang A P . Extinction and positivity of solutions of the heat equation on infinite graphs. J Yunnan Univ (Natural Sciences Edition), 2013, 35 (6): 727- 730
8	Liu W , Chen K , Yu J . Extinction and asymptotic behavior of solutions for the ω-heat equation on graphs with source and interior absorption. J Math Anal Appl, 2016, 435 (1): 112- 132
9	Xin Q , Liu D . Quenching for the discrete heat equation with a singular absorption term on finite graphs. J Comput Anal Appl, 2017, 23 (7): 1265- 1280
10	Elmoataz A , Lezoray O , Bougleux S . Nonlocal dicrete regularization on weighted graphs:a framework for image and manifold processing. IEEE Tans Image Process, 2008, 17 (7): 1047- 1060
11	Ta V T , Elmoataz A , Lezoray O . Graph-based tools for microscopic cellular image segmentation. Pattern Recognit, 2009, 42 (6): 1113- 1125
12	Bendito E , Carmona A , Encinas A M . potential theory for schr?dinger operators on finite networks. Rev Mat Iberoam, 2005, 21 (3): 771- 813
13	Bendito E , Carmona A , Encinas A M . Solving Dirichlet and poisson problems on graphs by means of equilibrium measure. European J Combin, 2003, 24 (4): 365- 375
14	Chung S Y , Kim D , Lee N K . Poisson equations with nonlinear source terms on networks. Publ RIMS, Kyoto Univ, 2007, 43 (4): 945- 961
15	Grigor'Yan A , Lin Y , Yang Y . Kazdan-Warner equations on graphs. Calc Var Partial Diff, 2016, 55 (4): 1- 13
16	Gel'fand I M . Some problems of the theory of quasilinear equations. Uspekhi Mat Nauk, 1959, 14 (2): 87- 158
17	Chandrasekar S . An Introduction to the Study of Stellar Structure. New York: Dover, 1985
18	Frank-Kamenetskii D A . Diffusion and Heat Transfer in Chemical Kinetics. New York: Plenum, 1969
19	Azorero J G , PAlonso I , Puel J P . Quasilinear problems with exponential growth in the reaction term. Nonlinear Anal, 1994, 22 (4): 481- 498
20	Fujita H . On the nonlinear equations △u + e^u=0 and $\frac{\partial v}{\partial t}=\Delta v+e^{v}$ . B Am Math Soc, 1969, 75 (1): 132- 135
21	Suzuki T , Nagasaki K . On the nonlinear eigenvalue problem △u +λe^u=0. Trans Amer Math Soc, 1988, 309 (2): 591- 608
22	Zhou S , Mu C , Mi Y , Zhang F . Blow-up for a non-local diffusion equation with exponential reaction term and neumann boundary condition. Commun Pure Appl Anal, 2013, 12 (6): 2935- 2946
23	Tello J L . Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation. J Math Anal Appl, 2006, 324 (1): 381- 396
24	Park J H. Chung S Y . The Dirichlet boundary value problems for p-Sch?dinger operators on finite networks. J Differ Equ Appl, 2011, 17 (5): 795- 811
25	郭大钧. 非线性算子方程的正解及其对非线性积分方程的应用. 数学进展, 1984, 13 (4): 294- 310
25	Guo D J . Positive solution of nonlinear operator equation and application to nonlinear integral equation. Adv Math (China), 1984, 13 (4): 294- 310
26	孙经先. 非连续的增算子的不动点定理及其对含有间断项的非线性方程的应用. 数学学报, 1988, 31 (1): 101- 107
26	Sun J X . Fixed point theorems of increasing operators and applications to nonlinear equations with discontinuous terms. Acta Math Sin, 1988, 31 (1): 101- 107
27	Amann H . On the number of solution of nonlinear equation in ordered Banach space. J Funct Anal, 1972, 11 (3): 346- 384

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The Discrete Possion Equation and the Heat Equation with the Exponential Nonlinear Term

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