# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 108-122. ISSN 2079-6641

**MATHEMATICS **

https://doi.org/10.26117/2079-6641-2023-42-1-108-122

Research Article

Full text in Russian

MSC 35K20, 35K59, 35R35

**Two Free Boundaries Problem for a Parabolic Equation**

**M. S. Rasulov^{*, 1, 2}**

¹Institute of Mathematics named after V. I. Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, 100174, Tashkent, University str., 9, Uzbekistan

²Tashkent Institute of Irrigation and Agricultural Mechanization Engineers-National Research University, 100000, Tashkent, Kori Niyazov str., 39, Uzbekistan

Abstract. This paper considers a two-free-boundary Stefan-type problem for a quasi-linear parabolic equation in one dimension. Nonlinear problems with free boundaries are studied using a method based on constructing a priori estimates. Therefore, some initial a priori estimates for the solution to the problem under consideration are first established. The main difficulty in constructing a theory for second-order quasi-linear parabolic equations is obtaining an a priori estimate for the solution’s derivative module, and additional arguments are required in problems with a free boundary. To address this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has time- and spacedependent coefficients with nonlinear terms. Next, Schauder-type a priori estimates are constructed for the equation with nonlinear terms and a fixed boundary. Based on these estimates, the uniqueness of the solution to the problem is proven. Then, the global existence of the solution to the problem is demonstrated using the Leray-Schauder fixed-point theorem.

*Key words: quasilinear parabolic equation, free boundary, a priori estimates, existence and uniqueness theorem.*

Received: 06.02.2023; Revised: 20.03.2023; Accepted: 25.03.2023; First online: 16.04.2023

**For citation.** Rasulov M. S. Two free boundaries problem for a parabolic equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 108-122. EDN: HFLTKL. https://doi.org/10.26117/2079-6641-2023-42-1-108-122.

**Funding.** The study was carried out without financial support from foundations.

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** The author participated in the writing of the article and is fully responsible for

submitting the final version of the article to print.

^**Correspondence: E-mail: rasulovms@bk.ru*

*The content is published under the terms of the Creative Commons Attribution 4.0 International License*

© Rasulov M. S., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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** Information about the author **

**Rasulov Mirojiddin Sobirjonovich** – Ph. D. (Phys. & Math.), Senior Researcher, Institute of Mathematics named after V. I. Romanovskiy, Tashkent, Uzbekistan, https://orcid.org/0000-0003-0704-6012.