Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 50. no. 1. P. 62 – 77. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-50-1-62-77
Research Article
Full text in English
MSC 35B45, 35K20, 35K57, 35K59

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A Diffusive Predator-Prey System with a Free Boundary

M. S. Rasulov¹²^{\ast}, Sh. M. Jamoldinova²

¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 100174, Tashkent, University street, 9, Uzbekistan
²National Research University “TIIAME 100000, Tashkent, K. Niyoziy str., 39, Uzbekistan

Abstract. In this paper, we consider a problem with a free boundary for a diffusive predator-prey system in the onedimensional case. Nonlinear problems with free boundary are studied using a method based on constructing a priori estimates. Therefore, first, using a method based on constructing a priori estimates, we will determine the constraints on the parameters of the problem, under which it is globally solvable. The first, fundamental estimate, gives the initial information, starting from which one can receive step by step, moving up the scale of Banach spaces. To do this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has timeand space-dependent 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 a solution to the problem was proved using the Leray-Schauder fixed point theorem.

Key words: free boundary, predator-prey, reaction-diffusion, parabolic equation, aprior bounds, existence and uniqueness.

Received: 03.04.2025; Revised: 16.04.2025; Accepted: 17.04.2025; First online: 18.04.2025

For citation. Rasulov M. S., Jamoldinova Sh. M. A diffusive predator-prey system with a free boundary. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 62-77. EDN: GKCHHF. https://doi.org/10.26117/2079-6641-2025-50-1-62-77.

Funding. The study was conducted without the support of foundations.

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

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}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., Jamoldinova Sh. M., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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