BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

doi:10.1016/S0252-9602(17)30052-8

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (4): 974-984.doi: 10.1016/S0252-9602(17)30052-8

BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

夏安银^1,2, 樊明书³, 李珊⁴

1. School of Science, Xihua University, Chengdu 610039, China;
2. Department of Mathematics, Sichuan University, Chengdu 610065, China;
3. Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China;
4. Business School, Sichuan University, Chengdu 610065, China

收稿日期:2016-05-13 修回日期:2016-08-07 出版日期:2017-08-25 发布日期:2017-08-25
通讯作者: Shan LI,E-mail:lishan@scu.edu.cn E-mail:lishan@scu.edu.cn
作者简介:Anyin XIA,E-mail:xay004@163.com;Mingshu FAN,E-mail:fanmingshu@hotmail.com
基金资助:
This work was supported in part by NSFC (11571243), Sichuan Youth Science and Technology Foundation (2014JQ0003), Fundamental Research Funds for the Central Universities (2682016CX116) and Project of Education Department of Sichuan Province (14226423).

BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

Anyin XIA^1,2, Mingshu FAN³, Shan LI⁴

1. School of Science, Xihua University, Chengdu 610039, China;
2. Department of Mathematics, Sichuan University, Chengdu 610065, China;
3. Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China;
4. Business School, Sichuan University, Chengdu 610065, China

Received:2016-05-13 Revised:2016-08-07 Online:2017-08-25 Published:2017-08-25
Contact: Shan LI,E-mail:lishan@scu.edu.cn E-mail:lishan@scu.edu.cn
About author:Anyin XIA,E-mail:xay004@163.com;Mingshu FAN,E-mail:fanmingshu@hotmail.com
Supported by:
This work was supported in part by NSFC (11571243), Sichuan Youth Science and Technology Foundation (2014JQ0003), Fundamental Research Funds for the Central Universities (2682016CX116) and Project of Education Department of Sichuan Province (14226423).

摘要/Abstract

摘要：

This paper deals with the singularity and global regularity for a class of nonlinear porous medium system with time-dependent coefficients under homogeneous Dirichlet boundary conditions. First, by comparison principle, some global regularity results are established. Secondly, using some differential inequality technique, we investigate the blow-up solution to the initial-boundary value problem. Furthermore, upper and lower bounds for the maximum blow-up time under some appropriate hypotheses are derived as long as blow-up occurs.

关键词: porous medium systems, Dirichlet boundary conditions, global existence, blowup, upper and lower bounds

Abstract:

Key words: porous medium systems, Dirichlet boundary conditions, global existence, blowup, upper and lower bounds

夏安银, 樊明书, 李珊. BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS[J]. 数学物理学报(英文版), 2017, 37(4): 974-984.

Anyin XIA, Mingshu FAN, Shan LI. BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS[J]. Acta mathematica scientia,Series B, 2017, 37(4): 974-984.

参考文献

[1] Deng W B. Global existence and finite time blow up for a degenerate reaction-diffusion system. Nonlinear Anal, 2005, 60:977-991
[2] Du L L, Mu C L, Fan M S. Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source. Dyn Syst, 2005, 20:401-412
[3] Du L L. Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources. J Math Anal Appl, 2006, 324:304-320
[4] Du L L. Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources. J Comput Appl Math, 2007, 202:237-247
[5] Du L L, Mu C L. Global existence and blow-up analysis to a degenerate reaction-diffusion system with nonlinear memory. Nonlinear Anal Real World Appl, 2008, 9:303-315
[6] Du L L, Xiang Zhaoyin. A further blow-up analysis for a localized porous medium equation. Appl Math Comput, 2006, 179:200-208
[7] Du L L, Yao Z A. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Comm Pure Appl Anal, 2007, 6:183-190
[8] Du L L, Yao Z A. Note on non-simultaneous blow-up for a reaction-diffusion system. Appl Math Lett, 2008, 21:81-85
[9] Fan M S, Mu C L, Du L L. Uniform blow-up profiles for a nonlocal degenerate parabolic system. Appl Math Sci, 2007, 1:13-23
[10] Friedman A, Mcleod B. Blow-up of solutions of nonlinear degenerate parabolic equations. Arch Rational Mech Anal, 1986, 96(1):55-80
[11] Fujita H. On the blowing up of solutions of the Cauchy problem for u_t=△u+u^1+α. J Fac Sci Univ Tokyo Sect, 1966, 113:109-124
[12] Hernndez J, Mancebo F, Vega J. Positive solutions for singular nonlinear elliptic equation. Proc Roy Soc Edinburgh Sect, 2007, 137(1):41-62
[13] Hu Y, Wang L W, Chen X C. Lower bounds for blow-up time of porous medium equation with nonlinear flux on boundary. Int J Math Anal (Ruse), 2013, 7:2671-2676
[14] Li F C, Xie C H. Global existence and blow-up for a nonlinear porous medium equation. Appl Math Lett, 2003, 16:185-192
[15] Marras M, Vernier Piro S. Explicit estimates for blow-up solutions to parabolic systems with time dependent coefficients. Comtes Rendus de l'Academie Bulgare des Sciences, 2014, 67:459-466
[16] Mu C L, Hu X G, Li Y H, Cui Z J. Blow-up and global existence for a coupled system of degenerate parabolic equations in a bounded clomain. Acta Math Sci, 2007, 27B(1):92-106
[17] Payne L E, Schaefer P W. Lower bound for blow-up time in parabolic System under Dirichlet conditions. J Math Anal Appl, 2007, 328:1196-1205
[18] Payne L E, Philippin G A. Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions. Proc Amer Math Soc, 2013, 141(7):2309-2318
[19] Payne L E, Philippin G A. Blow-up phenamena for a class of Parabolic System with time dependent coefficients. Appl Math, 2012, 3:325-330
[20] Quittner P, Souplet P. Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Basel:Birkhäuser, 2007
[21] Talenti G. Best constant in Sobolev inequality. Annali di Matematica Pura ed Applicata, 1976, 110:352-372
[22] Viglialoro G. Explicit blow-up time for complete porous medium problems. Proceeddings of the XXIV Congress on Equations and Applications, 2015, 7(8/12):821-826
[23] Weissler F B. Existence and nonexistence of global solutions for a heat equation. Israël J Math, 1981, 38(1/2):29-40
[24] Weigner M. A degenerate difussion equation with a nonlinear source term. Nonlinear Anal TMA, 1997, 28:1977-1995
[25] Xia A Y, Pu X X, Li S. Global existence and nonexistence for a class of parabolic systems with timedependent coefficients. Boundary Value Problems, 2016, 1:1-12

BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	孔德兴, 刘琦. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J]. 数学物理学报(英文版), 2018, 38(3): 745-755.
[2]	张晶, 张永前. INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS[J]. 数学物理学报(英文版), 2018, 38(1): 34-56.
[3]	刘颖博, Ingo WITT. SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS[J]. 数学物理学报(英文版), 2018, 38(1): 125-150.
[4]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[5]	夏莉, 李敬娜, 刘强. EXISTENCE OF WEAK SOLUTIONS FOR SOME SINGULAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(6): 1651-1661.
[6]	叶嵎林. GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(5): 1419-1432.
[7]	丁时进, 黄丙远, 卢友波. BLOWUP CRITERION FOR THE COMPRESSIBLE FLUID-PARTICLE INTERACTION MODEL IN 3D WITH VACUUM[J]. 数学物理学报(英文版), 2016, 36(4): 1030-1048.
[8]	蒋毅, 罗川. OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 872-880.
[9]	王忠, 崔尚斌. ON THE CAUCHY PROBLEM OF A COHERENTLY COUPLED SCHRÖDINGER SYSTEM[J]. 数学物理学报(英文版), 2016, 36(2): 371-384.
[10]	吴舜堂. GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND NONLINEAR BOUNDARY DAMPING-SOURCE INTERACTIONS[J]. 数学物理学报(英文版), 2015, 35(5): 981-994.
[11]	罗兰, 喻洪俊. BOUNDED SOLUTION OF THE RELATIVISTIC BOLTZMANN EQUATION[J]. 数学物理学报(英文版), 2015, 35(4): 777-786.
[12]	冯斌华, 赵敦, 孙春友. HOMOGENIZATION FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH PERIODIC NONLINEARITY AND DISSIPATION IN FRACTIONAL ORDER SPACES[J]. 数学物理学报(英文版), 2015, 35(3): 567-582.
[13]	雷远杰. THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE[J]. 数学物理学报(英文版), 2014, 34(5): 1519-1539.
[14]	秦玉明, 李海燕. GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SOLUTIONS TO THE QUASILINEAR THERMO-DIFFUSION EQUATIONS WITH SECOND SOUND[J]. 数学物理学报(英文版), 2014, 34(3): 759-778.
[15]	张映辉, 吴国春. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING[J]. 数学物理学报(英文版), 2014, 34(2): 424-434.