Step-like contrast structure for a quasilinear system of singularly perturbed differential equations with a zero characteristic number

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• 作者：Ni Mingkang ; Wang Aifeng ; Cheng Huaxiong
• 刊名：Differential Equations
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：52
• 期：2
• 页码：186-196
• 全文大小：408 KB
• 参考文献：1.Vasil’eva, A.B. and Butuzov, V.F., Singulyarno vozmushchennye uravneniya v kriticheskikh sluchayakh (Singularly Perturbed Equations in Critical Cases), Moscow: Moskov. Gos. Univ., 1978.
  2.Schmesiser, C. and Weiss, R., Asymptotic Analysis of Singular Singularly Perturbed Boundary Value Problems, SIAM J. Math. Anal., 1986, vol. 17, pp. 560–579.MathSciNet CrossRef MATH
  3.Vasil’eva, A.B., Butuzov, V.F., and Kalachev, L.V., The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadephia, 1995.CrossRef MATH
  4.Butuzov, V.F., Vasil’eva, A.B., and Nefedov, N.N., Asymptotic Theory of Contrast Structures (A Survey), Avtomat. i Telemekh., 1997, no. 7, pp. 4–32.MathSciNet MATH
  5.Vasil’eva, A.B., Butuzov, V.F., and Nefedov, N.N., Singularly Perturbed Problems with Boundary and Internal Layers, Proc. Steklov Inst. Math., 2010, vol. 268, pp. 258–273.MathSciNet CrossRef MATH
  6.Ni, Mingkang and Dmitriev, M.G., Contrast Structures in the Simplest Vector Variational Problem and Their Asymptotics, Avtomat. i Telemekh., 1998, no. 5, pp. 41–52.MathSciNet
  7.Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of the Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.
  8.Butuzov, V.F., Levashova, A.A., and Mel’nikova, A.A., A Step-Like Contrast Structure in a Singularly Perturbed System of Elliptic Equations, Zh. Vychisl. Mat. Mat. Fiz., 2013, vol. 53, no. 9, pp. 1427–1447.MathSciNet MATH
  9.Wang, Aifeng and Ni, Mingkang, Spike-Type Contrast Structures for a Nonlinear Singularly Perturbed Second-Order Equation, Acta Math. Sci. Ser. A Chin. Ed., 2009, vol. 29, no. 1, pp. 208–216.MathSciNet MATH
  10.Volkov, V.T. and Nefedov, N.N., Development of the Asymptotic Method of Differential Inequalities for Studying Periodic Contrast Structures in Reaction–Diffusion Equations, Zh. Vychisl. Mat. Mat. Fiz., 2006, vol. 46, no. 4, pp. 615–623.MathSciNet MATH
  11.Nefedov, N.N., The Method of Differential Inequalities for Nonlinear Singularly Perturbed Problems with Contrast Structures of Step Type in the Critical Case, Differ. Uravn., 1996, vol. 32, no. 11, pp. 1529–1537.MathSciNet
  12.Nefedov, N.N., Two-Dimensional Contrast Structures of Step Type: Asymptotics, Existence and Stability, Dokl. Akad. Nauk, 1996, vol. 349, no. 5, pp. 603–605.MathSciNet
  13.Ni, Mingkang and Wu, Limeng, A Step-Type Solution for the Affine Singularly Perturbed Problem of Optimal Control, Avtomat. i Telemekh., 2013, no. 12, pp. 104–118.MathSciNet
  
• 作者单位：Ni Mingkang (1) (2)
  Wang Aifeng (1) (2)
  Cheng Huaxiong (1) (2)
  
  1. East China Normal University, Shanghai, China
  2. Huaiying Normal Institute, Shanghai, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Ordinary Differential Equations
  Partial Differential Equations
  Difference and Functional Equations
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1608-3083

文摘

We consider a system of singularly perturbed first-order differential equations with a zero characteristic number. The solution of such a problem is characterized by the presence of a contrast structure, that is, of an internal transition layer on a given interval. We prove the existence of an exact solution with a step-like contrast structure and construct its uniform asymptotic expansion. An example is given.