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Internal Layer for a Singularly Perturbed Equation with Discontinuous Right-Hand Side

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Abstract

We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and an asymptotic representation for the solution are to be determined. The existence theorem is proved by the method of matching asymptotic expansions. To this end, we use theorems on existence of solutions of boundary value problems for singularly perturbed equations and methods for constructing asymptotic approximations to these solutions.

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Funding

This work was supported by the National Natural Science Foundation of China, project no. 11871217, with assistance from the Moscow Center for Fundamental and Applied Mathematics. The research by M.K. Ni was supported by the Science and Technology Commission of Shanghai Municipality, project no. 18dz2271000, School of Mathematical Sciences & Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice. The research by N.T. Levashova was supported by the Russian Foundation for Basic Research, project no. 19-01-00327.

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Correspondence to M. K. Ni, X. T. Qi or N. T. Levashova.

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Translated by V. Potapchouck

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Ni, M.K., Qi, X.T. & Levashova, N.T. Internal Layer for a Singularly Perturbed Equation with Discontinuous Right-Hand Side. Diff Equat 56, 1276–1284 (2020). https://doi.org/10.1134/S00122661200100031

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  • DOI: https://doi.org/10.1134/S00122661200100031

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