## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 1-2 (2004), 99-118.

### Singular perturbations for parabolic equations with unbounded coefficients leading to ultraparabolic equations

Denis R. Akhmetov, Mikhail M. Lavrentiev, and Renato Spigler

#### Abstract

Linear parabolic equations with * coefficients* of the lower-order terms *
unbounded*, and with a *small parameter* multiplying some of the second (highest)
space derivatives are considered, in the limiting case when such a parameter goes to zero.
This yields a degenerate parabolic *(ultraparabolic)* equation with *one*
space-like variable, $x$, and *two* time-like variables, $y$ and $t$. No
boundary-layer is found to be needed in the case of the boundary-value problem on the
$x$-*unbounded* domain $\mathcal{Q}_T=\{(x,y,t)\in \mathbb{R}\times
[0,1]\times[0,T]\}$ with a *periodic* boundary condition in the variable $y$ and
initial data at $t=0$.

#### Article information

**Source**

Differential Integral Equations, Volume 17, Number 1-2 (2004), 99-118.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060474

**Mathematical Reviews number (MathSciNet)**

MR2035497

**Zentralblatt MATH identifier**

1164.35312

**Subjects**

Primary: 35K70: Ultraparabolic equations, pseudoparabolic equations, etc.

Secondary: 35B25: Singular perturbations 35F30: Boundary value problems for nonlinear first-order equations 35K20: Initial-boundary value problems for second-order parabolic equations

#### Citation

Akhmetov, Denis R.; Lavrentiev, Mikhail M.; Spigler, Renato. Singular perturbations for parabolic equations with unbounded coefficients leading to ultraparabolic equations. Differential Integral Equations 17 (2004), no. 1-2, 99--118. https://projecteuclid.org/euclid.die/1356060474