一类非线性微分-积分时滞反应扩散系统奇摄动问题的广义解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 297-306.

一类非线性微分-积分时滞反应扩散系统奇摄动问题的广义解

韩祥临¹(),汪维刚²,莫嘉琪³

¹ 湖州师范学院求真学院浙江湖州 313000
² 合肥幼儿师范高等专科学校基础部合肥 230011
³ 安徽师范大学数学与统计学院安徽芜湖 230011

收稿日期:2017-10-31 出版日期:2019-04-26 发布日期:2019-05-05
作者简介:韩祥临, E-mail: xlhan@zjhu.edu.cn
基金资助:
国家自然科学基金(11771005);浙江省自然科学基金(LY13A010005);安徽省高校自然科学研究重点项目(KJ2017A901);安徽省教育科学规划课题(JG10068)

Generalized Solution to the Singular Perturbation Problem for a Class of Nonlinear Differential-Integral Time Delay Reaction Diffusion System

Xianglin Han¹(),Weigang Wang²,Jiaqi Mo³

¹ Qiuzhen School of Huzhou Teachers College, Zhejiang Huzhou 313000
² Department of Basic, Hefei Preschool Education College, Hefei 230011
³ School of Mathematics & Statistics, Anhui Normal University, Anhui Wuhu 241003

Received:2017-10-31 Online:2019-04-26 Published:2019-05-05
Supported by:
the NSFC(11771005);the Natural Science Foundation of Zhejiang Province(LY13A010005);the Natural Science Foundation of the Education Department of Anhui Province(KJ2017A901);the Education Science Programming Foundation of Anhui Province(JG10068)

摘要/Abstract

摘要：

该文研究了一类非线性微分-积分时滞广义反应扩散系统奇摄动问题.在适当的条件下，利用奇摄动方法构造了初始-边值问题广义解的渐近展开式.建立了广义解的微分不等式理论，并证明了相应解的存在性及其解的渐近展开式的一致有效性.

关键词: 反应扩散, 奇摄动, 非线性系统

Abstract:

A class of nonlinear differential-integral system for the singular perturbation generalized reaction diffusion equations with time delay is considered. Under suitable conditions, the asymptotic expansions of generalized solution to the initial boundary problem is obtained by using the singular perturbation method. And the theory of differential inequality for generalized solution is constructed. Corresponding existence and the uniformly validity of the asymptotic expansion for the solution are proved.

Key words: Reaction diffusion, Singular perturbation, Nonlinear system

中图分类号:

O175.29

韩祥临,汪维刚,莫嘉琪. 一类非线性微分-积分时滞反应扩散系统奇摄动问题的广义解[J]. 数学物理学报, 2019, 39(2): 297-306.

Xianglin Han,Weigang Wang,Jiaqi Mo. Generalized Solution to the Singular Perturbation Problem for a Class of Nonlinear Differential-Integral Time Delay Reaction Diffusion System[J]. Acta mathematica scientia,Series A, 2019, 39(2): 297-306.

参考文献 18

1	Baru L, Morosanu G. Singularly Perturbed Boundary-Value Problems. Basel: Birkhauser, 2007
2	De Jager E M, Jiang Furu. The Theory of Singular Perturbation. Amsterdam: North-Holland Publishing Co, 1996
3	Tian Canrong , Zhu Peng . Existence and asymptotic behavior of solutions for quasilinear parabolic systems. Acta Appl Math, 2012, 121 (1): 157- 173 doi: 10.1007/s10440-012-9701-7
4	Samusenko P F . Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations. J Math Sci, 2013, 189 (5): 834- 847 doi: 10.1007/s10958-013-1223-y
5	Kelloff R B , Kopteva N . A singularly perturbed semi-linear reaction-diffusion problem in a polygonal domain. J Differ Equations, 2010, 248 (1): 184- 208 doi: 10.1016/j.jde.2009.08.020
6	Skrynnkov Y . Solving initial value problem by matching asymptotic expansions. SIAM J Appl Math, 2012, 72 (1): 405- 416
7	Martinez S , Wolanshi N . A singular perturbation problem for a quasi-linear operator satisfying the natural condition of Lieberman. SIAM Math Anal, 2009, 41 (1): 318- 359 doi: 10.1137/070703740
8	Mo Jiaqi . Singular perturbation for a class of nonlinear reaction diffusion systems. Science in China Ser A, 1989, 32 (11): 1306- 1315
9	Mo J Q . Approximate solution of homotopic mapping to solitary wave for generalized nonlinear KdV system. Chin Phys Lett, 2009, 26 (1): 010204 doi: 10.1088/0256-307X/26/1/010204
10	Mo J Q , Lin W T . Generalized variation iteration Solution of an atmosphere-ocean oscillator model for global climate. J Sys Sci & Complexity, 2011, 24 (2): 271- 276
11	Han X L , Wang W G , Mo J Q . Solution of singularly perturbed boundary value peoblem for nonlinear higher order elliptic partial differential equations with two parameters. Adv Math, 2015, 44 (6): 931- 938
12	Han X L , Lin W T , Du Z J , Mo J Q . Shock wave solution for a class of nonlocal nonlinear singularly oerturbed boundary value problems with turning point. Acta Math Appl Sinica, 2015, 31 (3): 701- 705 doi: 10.1007/s10255-015-0496-y
13	Han X L , Wang W G , Mo J Q . Solution of singularly perturbed boundary value problem for nonlinear higher order elliptic partial differential equations with two parameters. Adv Math, 2015, 44 (6): 931- 938
14	韩祥临, 赵振江, 汪维刚, 莫嘉琪. 分数阶广义扰动热波方程的泛函映射解. 高校应用数学学报, 2016, 21 (1): 101- 108 doi: 10.3969/j.issn.1000-4424.2016.01.012
	Han X L , Zhou Z J , Wang W G , Mo J Q . The functional mapping solution for fractional generalized disturbed thermal wave equation. Appl Math J Chin Univ, 2016, 21 (1): 101- 108 doi: 10.3969/j.issn.1000-4424.2016.01.012
15	韩祥临, 石兰芳, 莫嘉琪. 双参数非线性非局部奇摄动问题的广义解. 数学进展, 2016, 45 (1): 95- 101
	Han X L , Shi L F , Mo J Q . Generalized solution of nonlinear nonlocal singularly perturbed problems with two parameters. Adv Math, 2016, 45 (1): 95- 101
16	韩祥临, 石兰芳, 许永红, 莫嘉琪. 分数阶双参数奇摄动非线性微分方程的渐近解. 应用数学学报, 2015, 38 (4): 721- 729
	Han X L , Shi L F , Xu Y H , Mo J Q . Asymptotic solution for the fractional order singularly perturbed nonlinear differential equation with two parameters. Acta Math Appl Sinica, 2015, 38 (4): 721- 729
17	韩祥临, 石兰芳, 莫嘉琪. 一类海-气振子模型的微扰解. 物理学报, 2014, 63 (6): 060205
	Han X L , Shi L F , Mo J Q . Small perturbed solution for a class of the sea-air oscillator. Acta Phys Sinica, 2014, 63 (6): 060205
18	韩祥临, 陈贤峰, 莫嘉琪. 一类量子等离子体类孤波的近似解析解. 物理学报, 2014, 63 (3): 030202
	Han X L , Chen X F , Mo J Q . Approximate analytic solution of solitary-like waves in a class of quantum plasma. Acta Phys Sinica, 2014, 63 (3): 030202

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