具有非局部源的p-Laplace方程解的爆破时间下界估计

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 911-923.

具有非局部源的p-Laplace方程解的爆破时间下界估计

孙宝燕()

南京大学数学系南京 210093

收稿日期:2017-09-30 出版日期:2018-10-26 发布日期:2018-11-09
作者简介:孙宝燕, E-mail: bysun@smail.nju.edu.cn
基金资助:
南京大学研究生科研创新基金(2016CL01)

Lower Bound of Blow-Up Time for a p-Laplacian Equation with Nonlocal Source

Baoyan Sun()

Department of Mathematics, Nanjing University, Nanjing 210093

Received:2017-09-30 Online:2018-10-26 Published:2018-11-09
Supported by:
the Scientific Research Foundation of Graduate School of Nanjing University(2016CL01)

摘要/Abstract

摘要：

该文考虑了三维空间中具有非局部源的p-Laplace方程分别在Dirichlet边界条件和Robin边界条件下解的爆破性质.通过构造辅助函数并利用微分不等式的技巧，得到了两种边界条件下方程解的爆破时间下界估计.另外，给出了方程解在L²-范数下不会发生爆破的充分条件.

关键词: p-Laplace方程, Dirichlet边界条件, Robin边界条件, 爆破, 爆破时间下界

Abstract:

In this paper, we consider an initial boundary value problem for a p-Laplacian equation under Dirichlet boundary condition or Robin boundary condition in three dimensional space. We use a differential inequality technique to determine a lower bound of blow-up time for the blow-up solution. In addition, we also give a sufficient condition which implies that blow-up does not occur.

Key words: p-Laplacian equation, Dirichlet boundary condition, Robin boundary condition, Blow up, Lower bound of blow-up time

中图分类号:

O175

孙宝燕. 具有非局部源的p-Laplace方程解的爆破时间下界估计[J]. 数学物理学报, 2018, 38(5): 911-923.

Baoyan Sun. Lower Bound of Blow-Up Time for a p-Laplacian Equation with Nonlocal Source[J]. Acta mathematica scientia,Series A, 2018, 38(5): 911-923.

参考文献

1	Li F C , Xie C H . Global and blow up solutions to a p-Laplacian equation with nonlocal source. Comput Math Appl, 2003, 46 (10/11): 1525- 1533
2	Niculescu C P , Roventa I . Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem. Nonlinear Anal, 2012, 75 (1): 270- 277
3	Alikakos N D , Evans L C . Continuing of the gradient for weak solutions of a degenerate parabolic equation. J Math Pures Appl, 1983, 62 (3): 253- 268
4	Zhao J N . Existence and nonexistence of solutions for u_t=div(\|▽u\|^p-2▽u) + f(▽u, u, x, t). J Math Anal Appl, 1993, 172 (1): 130- 146
5	Yin J X , Jin C H . Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math Meth Appl Sci, 2007, 30 (10): 1147- 1167
6	Fang Z B , Xu X H . Extinction behavior of solutions for the p-Laplacian equations with nonlocal sources. Nonlinear Anal Real World Appl, 2012, 13 (4): 1780- 1789
7	Yamashita Y , Yokota T . Existence of solutions to some degenerate parabolic equation associated with the p-Laplacian in the critical case. Nonlinear Anal, 2013, 93: 168- 180
8	Qu C Y , Bai X L , Zheng S N . Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions. J Math Anal Appl, 2014, 412 (1): 326- 333
9	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: 205- 220
10	Levine H A . The role of critical exponents in blow-up theorems. SIAM Rev, 1990, 32 (2): 262- 288
11	Bandle C , Brunner H . Blow-up in diffusion equations:A survey. J Comput Appl Math, 1998, 97 (1/2): 3- 22
12	Straughan B . Explosive Instabilities in Mechanics. Berlin: Springer-Verlag, 1998
13	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
14	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
15	Payne L E , Schaefer P W . Blow-up in parabolic problems under Robin boundary conditions. Appl Anal, 2008, 87 (6): 699- 707
16	Payne L E , Philippin G A , Schaefer P W . Bounds for blow-up time in nonlinear parabolic problems. J Math Anal Appl, 2008, 338 (1): 438- 447
17	Payne L E , Philippin G A , Schaefer P W . Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Anal, 2008, 69 (10): 3495- 3502
18	Li Y F , Liu Y , Lin C H . Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions. Nonlinear Anal Real World Appl, 2010, 11 (15): 3815- 3823
19	Mu C L , Zeng R , Chen B T . Blow-up phenomena for a doubly degenerate equation with positive initial energy. Nonlinear Anal, 2010, 72 (2): 782- 793
20	Enache C . Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition. Appl Math Lett, 2011, 24 (3): 288- 292
21	Li F S , Li J L . Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. J Math Anal Appl, 2012, 385 (2): 1005- 1014
22	Wang N , Song X F , Lv X S . Estimates for the blowup time of a combustion model with nonlocal heat sources. J Math Anal Appl, 2016, 436 (2): 1180- 1195
23	Di H F , Shang Y D , Peng X M . Blow-up phenomena for a pseudo-parabolic equation with variable exponents. Appl Math Lett, 2017, 64: 67- 73
24	Ding J T , Shen X H . Blow-up in p-Laplacian heat equations with nonlinear boundary conditions. Z Angew Math Phys, 2016, 67 (3): 125
25	Ma L W , Fang Z B . Bolw-up analysis for a reaction-diffusion with weighted nonlocal inner absorptions under nonlinear boundary flux. Nonlinear Anal Real World Appl, 2016, 32: 338- 354
26	Payne L E , Song J C . Lower bounds for blow-up time in a nonlinear parabolic problem. J Math Anal Appl, 2009, 354 (1): 394- 396
27	Song J C . Lower bounds for the blow-up time in a non-local reaction-diffusion problem. Appl Math Lett, 2011, 24 (5): 793- 796
28	Liu D M , Mu C L , Xin Q . Lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation. Acta Math Sci, 2012, 32 (3): 1206- 1212
29	Liu Y . Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin bondary condition. Comput Math Appl, 2013, 66 (10): 2092- 2095
30	Talenti G . Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353- 372
31	Daners D . A Faber-Krahn inequality for Robin problems in any space dimension. Math Ann, 2006, 335 (4): 767- 785
32	Payne L E , Philippin G A , Vernier Piro S . Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, Ⅱ. Nonlinear Anal, 2010, 73 (4): 971- 978

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