On Self-Similar Solutions of a Multi-Phase Stefan Problem in a Moving Ray

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/p6735-7356-6252-u On Self-Similar Solutions of a Multi-Phase Stefan Problem in a Moving Ray Panov, E. Yu. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 2. Abstract: We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the moving ray \(x>\alpha\sqrt{t}\) with Dirichlet or Neumann boundary conditions at the boundary \(x=\alpha\sqrt{t}\). In the case of Dirichlet condition we prove that an algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, which determines free boundaries and provides the solution. In the case of Neumann condition solutions with different numbers (called types) of phase transitions appear. For each fixed type the system for determination of the free boundaries is again gradient with a strictly convex potential. This allows to find precise conditions for existence and uniqueness of a solution. In the last section we study Stefan-Dirichlet problem on the half-line \(x>0\) with infinitely many phase transitions. Using again a variational approach, we find sufficient conditions of existence and uniqueness of a solution to the problem under consideration. Keywords: heat equation, Stefan problem, free boundaries, Dirichlet and Neumann boundary conditions, self-similar solutions, variational formulation Language: English Download the full text Download JATS XML For citation: Panov, E. Yu. On Self-Similar Solutions of a Multi-Phase Stefan Problem in a Moving Ray, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 112-128. DOI 10.46698/p6735-7356-6252-u + References 1. Kamenomostskaya, S. L. On Stefan's Problem, Matematicheskii Sbornik. Novaya Seriya, 1961, vol. 53(95), no. 4, pp. 489-514 (in Russian). 2. Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural'tseva, N. N. Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, Providence, RI, American Mathematical Society, 1968, 648 p. 3. Carslaw, H. S. and Jaeger, J. C. Conduction of Heat in Solids, 2nd edition, Oxford, Oxford University Press, 1959, 510 p. 4. Panov, E. Yu. Solutions of an Ill-Posed Stefan Problem, Journal of Mathematical Sciences, 2023, vol. 274, no. 4, pp. 534-543. DOI: 10.1007/s10958-023-06618-4. 5. Panov, E. Yu. On the Structure of Weak Solutions of the Riemann Problem for a Degenerate Nonlinear Diffusion Equation, Journal of Mathematical Sciences, 2024, vol. 285, no. 6, pp. 846-854. DOI: 10.1007/s10958-024-07479-1. 6. Panov, E. Yu. On Self-Similar Solutions of a Multi-Phase Stefan Problem in the Half-Line, Journal of Mathematical Sciences, 2025, vol. 287, no. 4, pp. 620-629. DOI: 10.1007/s10958-025-07627-1. 7. Panov, E. Yu. On the Structure of Entropy Solutions to the Riemann Problem for a Degenerate Nonlinear Parabolic Equation, Journal of Dynamics and Differential Equations, 06 April 2024, DOI: 10.1007/s10884-024-10361-y. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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