## Advances in Differential Equations

- Adv. Differential Equations
- Volume 3, Number 3 (1998), 361-386.

### Asymptotic behaviour for a diffusion-convection equation with rapidly decreasing initial data

#### Abstract

We study the large-time behaviour of nonnegative solutions of the problem $$ \begin{cases} u_t - (u^m)_x = u_{xx} \quad& \hbox{in }\; \mathbb{R}^+\times\mathbb{R}^+ \cr u_x=0 & \hbox{in }\; \{0\}\times\mathbb{R}^+ \cr u=u_0 & \hbox{in }\; \mathbb{R}^+\times\{0\}, \end{cases} $$ where $m>1$ and $u_0$ is a nonnegative function in $L^\infty(\mathbb{R}^+)$. We investigate the competition between the diffusion and the convection terms with respect to the concentration of the initial data controlled by the condition $$ \lim_{x\to +\infty} x^\alpha u_0(x) = A >0. $$ Convergence results are proved rescaling the equation and using Bernstein-type methods to obtain the necessary estimates.

#### Article information

**Source**

Adv. Differential Equations Volume 3, Number 3 (1998), 361-386.

**Dates**

First available in Project Euclid: 19 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366399846

**Mathematical Reviews number (MathSciNet)**

MR1751949

**Zentralblatt MATH identifier**

0954.35088

**Subjects**

Primary: 35K60: Nonlinear initial value problems for linear parabolic equations

Secondary: 35B40: Asymptotic behavior of solutions 76R99: None of the above, but in this section

#### Citation

Claudi, S. Asymptotic behaviour for a diffusion-convection equation with rapidly decreasing initial data. Adv. Differential Equations 3 (1998), no. 3, 361--386. https://projecteuclid.org/euclid.ade/1366399846.