# Singularly Perturbed System of Parabolic Equations in the Critical Case

## Abstract

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

- Authors:

- Kyrgyz Turkish Manas University (Kyrgyzstan)
- Naryn State University (Kyrgyzstan)

- Publication Date:

- OSTI Identifier:
- 22771277

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 230; Journal Issue: 5; Conference: International symposium on differential equations, Perm (Russian Federation), 17-18 May 2016; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ASYMPTOTIC SOLUTIONS; BOUNDARY LAYERS; DISTURBANCES; EQUATIONS; FUNCTIONS; MATHEMATICAL OPERATORS

### Citation Formats

```
Omuraliev, A. S., E-mail: asan.omuraliev@mail.ru, and Kulmanbetova, S., E-mail: sagynkulmanbetova@mail.ru.
```*Singularly Perturbed System of Parabolic Equations in the Critical Case*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3778-0.

```
Omuraliev, A. S., E-mail: asan.omuraliev@mail.ru, & Kulmanbetova, S., E-mail: sagynkulmanbetova@mail.ru.
```*Singularly Perturbed System of Parabolic Equations in the Critical Case*. United States. doi:10.1007/S10958-018-3778-0.

```
Omuraliev, A. S., E-mail: asan.omuraliev@mail.ru, and Kulmanbetova, S., E-mail: sagynkulmanbetova@mail.ru. Tue .
"Singularly Perturbed System of Parabolic Equations in the Critical Case". United States. doi:10.1007/S10958-018-3778-0.
```

```
@article{osti_22771277,
```

title = {Singularly Perturbed System of Parabolic Equations in the Critical Case},

author = {Omuraliev, A. S., E-mail: asan.omuraliev@mail.ru and Kulmanbetova, S., E-mail: sagynkulmanbetova@mail.ru},

abstractNote = {We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.},

doi = {10.1007/S10958-018-3778-0},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 5,

volume = 230,

place = {United States},

year = {2018},

month = {5}

}