Local controllability of free boundary three-dimensional semilinear radial parabolic equations

TL;DR

This paper proves local null controllability for a 3D free boundary semilinear heat equation with Stefan boundary condition, using reduction to 1D and a Carleman inequality, marking a novel result in multiple dimensions.

Contribution

It introduces the first controllability result for 3D free boundary semilinear heat equations with Stefan boundary conditions.

Findings

Established local null controllability for the problem.

Reduced the 3D problem to a 1D formulation.

Applied a Carleman inequality and Schauder fixed-point theorem.

Abstract

We prove that a free boundary semilinear heat equation with Stefan boundary condition and radially symmetric data is locally null controllable. The strategy involves reducing the problem to the corresponding one-dimensional formulation and adapting a Carleman inequality in that setting. The local null controllability of the free-boundary problem is then established via the Schauder fixed-point theorem. To the best of our knowledge, this is the first controllability result for this problem with Stefan boundary condition in more than one spatial dimension.