Local existence of solutions in one-phase Stefan problems for semilinear heat equations using the unified transform method

Abstract

We investigate the one-phase Stefan problem for a semilinear heat equation, which plays a pivotal role in understanding phase transitions across diverse media within nonlinear phenomena. Through the application of the unified transform method, commonly referred to as the Fokas method, we derive equivalent nonlinear integral equations to the Stefan problem for the semiliner heat equation. As a result, we establish the local existence and uniqueness of solutions within this framework. The proposed approach not only extends the applicability of the Fokas method to effectively handle Stefan problems, but also provides a direct and efficient analytical method for addressing free boundary value problems.