Free boundary problem governed by a non-linear diffusion-convection equation with Neumann condition

TL;DR

This paper studies a one-dimensional free boundary problem involving nonlinear diffusion and convection, establishing existence of solutions over time using integral representations and fixed point theorems.

Contribution

It introduces a novel integral formulation for the nonlinear diffusion-convection free boundary problem and proves the existence of solutions for all times.

Findings

Existence of solutions established for all times.

Integral representation involving coupled nonlinear integral equations.

Application of fixed point theorems to prove solution existence.

Abstract

We consider a one-dimensional free boundary problem governed by a nonlinear diffusion - convection equation with a Neumann condition at fixed face $x=0$, which is variable in time and a like Stefan convective condition on the free boundary. Through successive transformations, an integral representation of the problem is obtained that involves a system of coupled nonlinear integral equations. Existence of the solution is obtained for all times by using fixed point theorems.