具有非局部源的p-Laplace方程解的爆破时间下界估计

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 32

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 doi: 10.1016/j.na.2011.08.031
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 doi: 10.1006/jmaa.1993.1012
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 doi: 10.1002/(ISSN)1099-1476
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 doi: 10.1016/j.nonrwa.2011.12.008
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 doi: 10.1016/j.na.2013.07.035
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 doi: 10.1016/j.jmaa.2013.10.040
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 doi: 10.1007/BF01352106
10	Levine H A . The role of critical exponents in blow-up theorems. SIAM Rev, 1990, 32 (2): 262- 288 doi: 10.1137/1032046
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 doi: 10.1016/j.jmaa.2006.06.015
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 doi: 10.1080/00036810802189662
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 doi: 10.1016/j.jmaa.2007.05.022
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 doi: 10.1016/j.na.2007.09.035
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 doi: 10.1016/j.na.2009.07.020
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 doi: 10.1016/j.aml.2010.10.006
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 doi: 10.1016/j.jmaa.2011.07.018
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 doi: 10.1016/j.jmaa.2015.12.025
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 doi: 10.1016/j.aml.2016.08.013
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 doi: 10.1016/j.nonrwa.2016.05.005
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 doi: 10.1016/j.aml.2010.12.042
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 doi: 10.1016/S0252-9602(12)60092-7
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 doi: 10.1016/j.camwa.2013.08.024
30	Talenti G . Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353- 372 doi: 10.1007/BF02418013
31	Daners D . A Faber-Krahn inequality for Robin problems in any space dimension. Math Ann, 2006, 335 (4): 767- 785 doi: 10.1007/s00208-006-0753-8
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 doi: 10.1016/j.na.2010.04.023

[1]	赵元章, 马相如. 一类具有变扩散系数的非局部反应-扩散方程解的爆破分析[J]. 数学物理学报, 2018, 38(4): 750-769.
[2]	刘灯明, 曾宪忠, 穆春来. 具非线性记忆的高阶阻尼双曲系统的一个爆破结果[J]. 数学物理学报, 2018, 38(2): 304-312.
[3]	苏晓, 王书彬. 任意正初始能量状态下半线性波动方程解的有限时间爆破[J]. 数学物理学报, 2017, 37(6): 1085-1093.
[4]	段双双, 黄守军. Keller-Segel生物学方程组周期解的爆破[J]. 数学物理学报, 2017, 37(2): 326-341.
[5]	马羚未, 方钟波. 具有加权非局部源和Robin边界条件的反应-扩散方程解的爆破时间下界[J]. 数学物理学报, 2017, 37(1): 146-157.
[6]	杨凌燕, 李晓光, 陈樱. 一类Schrödinger-Hartree方程爆破解的门槛条件[J]. 数学物理学报, 2016, 36(6): 1117-1123.
[7]	刘洋. 位势井及其对具临界初始能量的非牛顿渗流方程的应用[J]. 数学物理学报, 2016, 36(6): 1211-1220.
[8]	凌征球, 王泽佳. 带具有耗散梯度项的p-Laplace方程解的爆破问题[J]. 数学物理学报, 2016, 36(5): 958-964.
[9]	黄守军, 王蕊. 等熵Chaplygin气体的平面波解[J]. 数学物理学报, 2016, 36(4): 681-689.
[10]	边东芬, 唐童. 可压缩磁流体方程光滑解的爆破性[J]. 数学物理学报, 2016, 36(4): 715-721.
[11]	雷倩, 李宁, 杨晗. 梁方程解的爆破及渐近性行为[J]. 数学物理学报, 2015, 35(4): 738-747.
[12]	狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性[J]. 数学物理学报, 2015, 35(3): 618-633.
[13]	凌征球, 覃思乾. 非线性非局部边界条件的半线性耦合抛物型方程组[J]. 数学物理学报, 2015, 35(2): 234-244.
[14]	王华, 贺艺军. 一类具变指数和正能量的半线性双曲方程解的爆破[J]. 数学物理学报, 2015, 35(2): 288-293.
[15]	陈翔英. 具有弱阻尼项的广义IMBq方程初值问题解的整体存在性[J]. 数学物理学报, 2015, 35(1): 68-82.