Contrast structure in a singularly perturbed delay reaction-diffusion equation

doi:10.3969/j.issn.1000-5641.2023.02.006

Abstract

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.

Key words: delay, reaction-diffusion equation, singular perturbation, triangle layer, contrast structure, asymptotic expansion

CLC Number:

O175.26

Qingzhao GAN, Mingkang NI. Contrast structure in a singularly perturbed delay reaction-diffusion equation[J]. Journal of East China Normal University(Natural Science), 2023, 2023(2): 34-47.

Figures/Tables 1

References 35

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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