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<p>We study a nonlinear three-phase Stefan problem in an axisymmetric cylindrical setting with temperature-dependent thermal coefficients and an internal source term depending on the unknown temperature. Two formulations are treated: (P1) with a Dirichlet condition at the vapor–liquid interface and (P2) with a Neumann-type flux condition at that interface; both include a transmission condition at the liquid–solid interface and a far-field condition in the solid. After non-dimensionalization, a similarity substitution reduces the free-boundary PDE system to coupled boundary-value problems for nonlinear ordinary differential equations posed on fixed, dimensionless domains with unknown interface parameters. These problems are then recast as nonlinear Volterra integral equations built from explicitly defined integral operators. Working in appropriate Banach spaces, we establish existence and uniqueness of the integral solutions via the Banach fixed point theorem under structural assumptions on the coefficient functions. In addition, we prove solvability of the accompanying nonlinear algebraic system determining the interface parameters and derive a priori bounds that ensure well-posedness. The results furnish a rigorous fixed point approach for multiphase Stefan problems with variable coefficients and internal sources in one-dimensional cylindrical geometry.</p>

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Theoretical analysis of nonlinear Stefan problem including internal Joule heat source in cylindrical geometry by fixed point theory

  • 期刊论文

  • 发表日期：

    2025年9月19日

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  • Targyn A. Nauryz

摘要

We study a nonlinear three-phase Stefan problem in an axisymmetric cylindrical setting with temperature-dependent thermal coefficients and an internal source term depending on the unknown temperature. Two formulations are treated: (P1) with a Dirichlet condition at the vapor–liquid interface and (P2) with a Neumann-type flux condition at that interface; both include a transmission condition at the liquid–solid interface and a far-field condition in the solid. After non-dimensionalization, a similarity substitution reduces the free-boundary PDE system to coupled boundary-value problems for nonlinear ordinary differential equations posed on fixed, dimensionless domains with unknown interface parameters. These problems are then recast as nonlinear Volterra integral equations built from explicitly defined integral operators. Working in appropriate Banach spaces, we establish existence and uniqueness of the integral solutions via the Banach fixed point theorem under structural assumptions on the coefficient functions. In addition, we prove solvability of the accompanying nonlinear algebraic system determining the interface parameters and derive a priori bounds that ensure well-posedness. The results furnish a rigorous fixed point approach for multiphase Stefan problems with variable coefficients and internal sources in one-dimensional cylindrical geometry.