Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 104, pp. 1-10.
Title: Existence of scale invariant solutions to horizontal flow with a
Fujita type diffusion coefficient
Authors: Gastao A. Braga (UFMG, Belo Horizonte, MG, Brazil)
Paulo C. Carriao (UFMG, Belo Horizonte, MG, Brazil)
Antonio A. G. Ruas (UFMG, Belo Horizonte, MG, Brazil)
Abstract:
In this article, we study a boundary-initial value problem on the half-line
for the diffusion equation with a Fujita type diffusion coefficient
that carries a parameter $\alpha $. The equation models
the flow of water in soil within an approximation where gravitational
effects are not taken into account and, when $\alpha = 1$, an explicit
self-similar solution $\psi(x/\sqrt t)$ can be found.
We prove that if $\alpha > 1$ then the above problem, with
uniform boundary conditions, posses self-similar solutions.
This is the first step towards a multiscale (renormalization group)
asymptotic analysis of solutions to more general equations than the ones
studied here.
Submitted October 11, 2011. Published June 21, 2012.
Math Subject Classifications: 34E10, 34E13, 35C06, 35Q86.
Key Words: Water infiltration; nonlinear diffusion; self-similar solutions;
Fujita diffusion coefficient.