A free boundary problem for a diffusion-convection equation

TL;DR

This paper studies a one-dimensional free boundary problem involving nonlinear diffusion-convection equations, transforming it into a diffusion problem and proving local existence of solutions using fixed point theorems.

Contribution

It introduces a novel approach by reducing a nonlinear diffusion-convection free boundary problem to an integral formulation and establishes local existence of solutions.

Findings

Existence of solutions for small time established.

Transformation reduces problem to a diffusion equation.

Integral formulation enables fixed point analysis.

Abstract

One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, the existence of at least a solution, for small time, to a system of coupled nonlinear integral equations is obtained.