数学

一类奇摄动时滞反应扩散方程的空间对照结构

  • 甘清照 ,
  • 倪明康
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  • 华东师范大学 数学科学学院, 上海 200241

收稿日期: 2021-05-19

  网络出版日期: 2023-03-23

基金资助

国家自然科学基金(11471118); 上海市科学技术委员会基金(18dz2271000)

Contrast structure in a singularly perturbed delay reaction-diffusion equation

  • Qingzhao GAN ,
  • Mingkang NI
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  • School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

Received date: 2021-05-19

  Online published: 2023-03-23

摘要

研究了一类具有非线性反应项的奇摄动时滞反应扩散方程的Neumann边值问题. 运用边界层函数法、空间对照结构理论和压缩映射原理构造该问题解的渐近展开式并证明了解的存在性. 最后给出一个具体的例子说明了结果的有效性.

本文引用格式

甘清照 , 倪明康 . 一类奇摄动时滞反应扩散方程的空间对照结构[J]. 华东师范大学学报(自然科学版), 2023 , 2023(2) : 34 -47 . DOI: 10.3969/j.issn.1000-5641.2023.02.006

Abstract

This paper considers a Neumann boundary value problem of a singularly perturbed delay reaction-diffusion equation with a nonlinear reactive term. By using the boundary layer function method, contrast structure theory, and contraction mapping principle, the asymptotic expansion of the solution is constructed, and the existence of a uniformly valid solution is proven. Finally, an example is presented to show the effectiveness of our result.

参考文献

1 HUANG C X, TAN Y X. Global behavior of a reaction-diffusion model with time delay and Dirichlet condition. Journal of Differential Equations, 2021, 271, 186- 215.
2 PAO C V. Global attractors of some predator-prey reaction-diffusion systems with density-dependent diffusion and time-delays. Journal of Mathematical Analysis and Applications, 2018, 464 (1): 164- 187.
3 WU C F, YANG Y, WU Z H. Existence and uniqueness of forced waves in a delayed reaction-diffusion equation in a shifting environment. Nonlinear Analysis: Real World Applications, 2021, 57, 103198.
4 WU K N, REN M Z, LIU X Z. Exponential input-to-state stability of stochastic delay reaction–diffusion neural networks. Neurocomputing, 2020, 412, 399- 405.
5 WU J H. Theory and Applications of Partial Functional Differential Equations [M]. New York: Springer, 1996.
6 PAO C V. Dynamics of nonlinear parabolic systems with time delays. Journal of Mathematical Analysis and Applications, 1996, 198 (3): 751- 779.
7 PAO C V. Convergence of solutions of reaction–diffusion systems with time delays. Nonlinear Analysis, 2002, 48 (3): 349- 362.
8 PAO C V. Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays. Journal of Mathematical Analysis and Applications, 2003, 281, 186- 204.
9 PAO C V. Periodic solutions of parabolic systems with time delays. Journal of Mathematical Analysis and Applications, 2000, 251 (1): 251- 263.
10 HE L H, LIU A P. Existence and uniqueness of solutions for nonlinear impulsive partial differential equations with delay. Nonlinear Analysis: Real World Applications, 2010, 11 (2): 952- 958.
11 LI S Y, MA Z E. Estimation of a stability region on uncertain large-scale nonlinear systems with time delay. International Journal of Systems Science, 2002, 33 (5): 1187- 1193.
12 WANG C Y. Existence and stability of periodic solutions for parabolic systems with time delays. Journal of Mathematical Analysis and Applications, 2008, 339 (2): 1354- 1361.
13 WANG C Y, WANG S. Oscillation of partial population model with diffusion and delay. Applied Mathematics Letters, 2009, 22 (12): 1793- 1797.
14 WANG C Y, WANG S, YAN X P, et al. Oscillation of a class of partial functional population model. Journal of Mathematical Analysis and Applications, 2010, 368 (1): 32- 42.
15 WANG Z C, LI W T, RUAN S G. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. Journal of Differential Equations, 2006, 222 (1): 185- 232.
16 YAN X P, ZHANG C H. Direction of Hopf bifurcation in a delayed Lotka–Volterra competition diffusion system. Nonlinear Analysis: Real World Applications, 2009, 10, 2758- 2773.
17 YAN Z M. Existence of solutions for nonlocal impulsive partial functional integro-differential equations via fractional operators. Journal of Computational and Applied Mathematics, 2011, 235 (8): 2252- 2262.
18 YE R P. Existence of solutions for impulsive partial neutral functional differential equation with infinite delay. Nonlinear Analysis, 2010, 73 (1): 155- 162.
19 AMIRALIYEVA I G, AMIRALIYEV G M. Uniform difference method for parameterized singularly perturbed delay differential equations. Numerical Algorithms, 2009, 52 (4): 509- 521.
20 LANGE C G, MIURA R M. Singular perturbation analysis of boundary-value problems for differential-difference equations (Ⅵ): Small shifts with rapid oscillations. Siam Journal on Applied Mathematics, 1994, 54 (1): 273- 283.
21 TIAN H J. Asymptotic expansion for the solution of singularly perturbed delay differential equations. Journal of Mathematical Analysis and Applications, 2003, 281 (2): 678- 696.
22 KUEHN C. Multiple Time Scale Dynamics [M]. New York: Springer, 2015.
23 VASIL’EVA A B. An equation of neutral type with small lag. Doklady Akademii Nauk SSSR, 1962, 145, 495- 497.
24 VASIL’EVA A B, BUTUZOV V F. Asymptotic Methods in the Theory of Singular Perturbations [M]. Moscow, Russia: High School, 1990.
25 NI M K. On the internal layer for a singularly perturbed system of second-order delay differential equations. Differential Equations, 2013, 49 (8): 941- 954.
26 倪明康, 林武忠. 具有内部层的奇摄动微分差分方程的渐近解. 数学物理学报, 2010, 30 (6): 1413- 1423.
27 WANG A F, NI M K. The interior layer for a nonlinear singularly perturbed differential-difference equation. Acta Mathematica Scientia, 2012, 32 (2): 695- 709.
28 NI M K, NEFEDOV N N, LEVASHOVA N T. Asymptotics of the solution of a singularly perturbed second-order delay differential equation. Differential Equations, 2020, 56 (3): 290- 303.
29 MO J Q, FENG M C. The nonlinear singularly perturbed problems for reaction diffusion equations with time delay. Acta Mathematica Scientia, 2001, 21 (2): 254- 258.
30 MO J Q. A class of singularly perturbed differential-difference reaction diiffusion equations. Advances in Mathematics, 2009, 38 (2): 227- 232.
31 MO J Q. Singularly perturbed differential-difference reaction diffusion equations with time delay. Journal of Shanghai Jiaotong University (Science), 2009, 14 (5): 629- 631.
32 倪明康. 一类含时滞奇摄动抛物型方程的初边值问题. 华东师范大学学报(自然科学版), 1992, (3): 1- 7.
33 MO J Q, WEN Z H. Singularly perturbed reaction diffusion equations with time delay. Applied Mathematics and Mechanics (English Edition), 2010, 31 (6): 769- 774.
34 DENISOV A I, DENISOV I V. Corner boundary layer in boundary value problems for singularly perturbed parabolic equations with nonlinearities. Computational Mathematics and Mathematical Physics, 2019, 59 (1): 96- 111.
35 VASIL’EVA A B, BUTUZOV V F, KALACHEV L V. The Boundary Function Method for Singular Perturbation Problems [M]. Pennsylvania, USA: Society for Industrial and Applied Mathematics, 1995.
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