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
Received date: 2021-05-19
Online published: 2023-03-23
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
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.
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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