Electron. J. Diff. Equ., Vol. 2012 (2012), No. 104, pp. 1-10.

Existence of scale invariant solutions to horizontal flow with a Fujita type diffusion coefficient

Gastao A. Braga, Paulo C. Carriao, Antonio A. G. Ruas

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.

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Gastão A. Braga
Departamento de Matemática - UFMG, Caixa Postal 1621
30161-970 Belo Horizonte, MG, Brazil
email: gbraga@mat.ufmg.br
Paulo C. Carrião
Departamento de Matemática - UFMG, Caixa Postal 1621
30161-970 Belo Horizonte, MG, Brazil
email: carrion@mat.ufmg.br
Antonio A. G. Ruas
Departamento de Matemática - UFMG, Caixa Postal 1621
30161-970 Belo Horizonte, MG, Brazil
email: gaspar@mat.ufmg.br

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