一类具有时变系数梯度源项的弱耦合反应-扩散方程组解的爆破分析

Abstract

Abstract:

This paper investigate the blow-up phenomena for a weakly coupled reaction-diffusion system with gradient sources terms and time-dependent coefficients subject to null Dirichlet boundary condition. By virtue of the differential inequality technique and comparison principle, we derive some sufficient conditions to guarantee that the solutions exist globally or blow up in finite time under several different measure sense. Moreover, the bounds for the blow-up time of the blow-up solution are obtained in higher dimensional space.

Key words: Reaction-diffusion system, Gradient sources terms, Time dependent-coefficients, Bounds of blow-up time

CLC Number:

O175.2

Yadong Zheng,Zhongbo Fang. Blow-Up Analysis for a Weakly Coupled Reaction-Diffusion System with Gradient Sources Terms and Time-Dependent Coefficients[J].Acta mathematica scientia,Series A, 2020, 40(3): 735-755.

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References 30

1	Amann H . Dynamic theory of quasilinear parabolic systems. Ⅲ. Global existence. Math Z, 1989, 202, 219- 250 doi: 10.1007/BF01215256
2	Ladyzenskaja O A, Solonnikov V A, Uralceva N N. Linear and Quasilinear Equations of Parabolic Type. Providence, RI: Amer Math Soc, 1968
3	Ben-Artzi M , Souplet P , Weissler F B . The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J Math Pures Appl, 2002, 81 (4): 343- 378 doi: 10.1016/S0021-7824(01)01243-0
4	Gilding B H , Guedda B H , Kersner R . The Cauchy problem for u_t=Δu+u^q. J Math Anal Appl, 2003, 284, 733- 755 doi: 10.1016/S0022-247X(03)00395-0
5	Bebernes J , Eberly D . Mathematical Problems from Combustion Theory. New York: Springer-Verlag, 1989
6	Straughan B . Explosive Instabilities in Mechanics. Berlin: Springer, 1998
7	Quittner R , Souplet P . Superlinear Parabolic Problems:Blow-Up, Global Existence and Steady States. Basel: Bikhäuser, 2007
8	Hu B . Blow-up Theories for Semilinear Parabolic Equations. Heidelberg: Springer, 2011
9	Levine H A . The role of critical exponents in blow-up theorems. SIAM Rev, 1990, 32 (2): 262- 288 doi: 10.1137/1032046
10	Levine H A . Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics:the method of unbounded Fourier coefficients. Math Ann, 1975, 214 (3): 205- 220 doi: 10.1007/BF01352106
11	Payne L E , Schaefer P W . Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl Anal, 2006, 85 (10): 1301- 1311
12	Payne L E , Schaefer P W . Lower bounds for blow-up time in parabolic problems under Dirichlet conditions. J Math Anal Appl, 2007, 328 (2): 1196- 1205 doi: 10.1016/j.jmaa.2006.06.015
13	Fang Z B , Wang Y X . Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Z Angew Math Phys, 2015, 66 (5): 2525- 2541
14	Baghaei K , Hesaaraki M . Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations. C R Acad Paris, 2013, 351 (19-20): 731- 735 doi: 10.1016/j.crma.2013.09.024
15	Hesaaraki M , Moameni A . Blow-up of positive solutions for a family of nonlinear parabolic equations in general in R^N. Michigan Math J, 2004, 52 (2): 375- 389 doi: 10.1307/mmj/1091112081
16	Payne L E , Song J C . Lower bounds for blow-up time in a nonlinear parabolic problem. J Math Anal, 2009, 354 (1): 394- 396 doi: 10.1016/j.jmaa.2009.01.010
17	Liu Y , Luo S G , Ye Y H . Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions. Comput Math Appl, 2013, 65 (8): 1194- 1199 doi: 10.1016/j.camwa.2013.02.014
18	Li H X , Gao W J , Han Y Z . Lower bounds for the blowup time of solutions to a nonlinear parabolic problem. Electron J Differ Equa, 2014, 2014 (20): 1- 6
19	Marras M , Piro S V , Viglialoro G . Lower bounds for blow-up time in a parabolic problem with a gradient team under various boundary conditions. Kodai Math J, 2014, 37 (3): 532- 543 doi: 10.2996/kmj/1414674607
20	Ding J . Global and blow-up solutions for nonlinear parabolic problems with a gradient term under Robin boundary conditions. Bound Value Probl, 2013, 2013 (237): 1- 12
21	Zhang Q Y , Jiang Z X , Zheng S N . Blow-up time estimate for a degenerate diffusion equation with gradient absorption. Appl Math Comput, 2015, 251, 331- 335
22	Payne L E , Song J C . Lower bounds for blow-up in a model of chemotaxis. J Math Anal Appl, 2012, 385 (2): 672- 676 doi: 10.1016/j.jmaa.2011.06.086
23	Xu X J , Ye Z . Life span of solutions with large initial data for a class of coupled parabolic systems. Z Angew Math Phys, 2013, 64 (3): 705- 717 doi: 10.1007/s00033-012-0255-3
24	Payne L E , Philippin G A . Blow-up phenomena for a class of parabolic systems with time dependent coefficients. Appl Math, 2012, 3 (4): 325- 330 doi: 10.4236/am.2012.34049
25	Tao X Y , Fang Z B . Blow-up phenomenon for a nonlinear reaction-diffusion system with time dependent coefficients. Comput Math Appl, 2017, 74 (10): 2520- 2528 doi: 10.1016/j.camwa.2017.07.037
26	Bao A G , Song X F . Bounds for the blow-up time of the solution to a parabolic system with nonlocal factors in nonlinearities. Comput Math Appl, 2016, 71 (3): 723- 729 doi: 10.1016/j.camwa.2015.12.029
27	Wang N , Song X F , Lv X S . Estimates for the blow-up time of a combustion model with nonlocal heat sources. J Math Anal Appl, 2016, 436 (2): 1180- 1195 doi: 10.1016/j.jmaa.2015.12.025
28	Petersson J H . On global existence for semilinear parabolic systems. Nonlinear Anal, 2005, 60 (2): 337- 347 doi: 10.1016/j.na.2004.08.035
29	Rasheed M A, Chlebik M. The blow-up rate estimates for a reaction diffusion system with gradient terms. 2012, arXiv: 1211.6500
30	Souplet P , Weissler F B . Poincaré's inequality and global solutions of a nonlinear parabolic equation. Ann Inst H Poincaré Anal Nonlin, 1999, 16 (3): 337- 373 doi: 10.1016/s0294-1449(99)80017-1

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Blow-Up Analysis for a Weakly Coupled Reaction-Diffusion System with Gradient Sources Terms and Time-Dependent Coefficients

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