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

TL;DR

This work addresses local null controllability for a free-boundary three-dimensional radially symmetric semilinear heat equation with Stefan boundary conditions. The authors reduce the problem to a one-dimensional formulation, derive an adapted Carleman inequality to obtain a crucial observability estimate for the linearized system, and then extend the result to the nonlinear problem via Schauder's fixed-point theorem. The combination of radial reduction, Carleman-based observability, and a fixed-point argument yields the first known local null controllability result for this free-boundary Stefan problem in more than one spatial dimension. The paper also discusses boundary controllability in two dimensions and outlines open problems such as general diffusion, nonlocal terms, and quasi-linear variants, signaling directions for future research.

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.

Theorems & Definitions (9)

• Theorem 1
• Remark 1: On the $N$-dimensional case
• Proposition 2
• Theorem 3
• Lemma 4
• Proposition 5
• proof
• Proposition 6
• proof