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.