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

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.