What is it about?

This paper studies the properties of solutions for a double nonlinear parabolic equation with variable density, not in divergence form with a source. The problem is formulated as a partial differential equation with a nonlinear term. The main results are the existence of weak solutions in suitable function spaces; regularity and positivity of solutions; asymptotic behaviour of solutions as time goes to infinity; comparison principles; and maximum principles for solutions. The proofs are based on the energy method, comparison methods, and asymptotic techniques.

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Why is it important?

This research is crucial for advancing the study of nonlinear partial differential equations (PDEs), which model various natural and engineering phenomena. The findings contribute to a deeper understanding of critical problems such as global existence, blow-up behaviour, and asymptotic analysis of solutions in inhomogeneous mediums. These equations have applications in heat transfer, fluid dynamics, biological processes, and the modelling of infectious disease spread.

Perspectives

Mathematical: Offers new insights into the regularity, positivity, and energy methods for PDEs with nonlinear and non-divergent terms. Physical: The results simulate physical processes like diffusion in complex materials or mediums, which are not uniformly structured. Practical: Enhances computational models that predict behaviours in scenarios such as gas filtration or thermal dynamics in heterogeneous environments.

Mr. Makhmud Bobokandov
National University of Uzbekistan

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This page is a summary of: ANALYSIS OF A DOUBLE NONLINEAR PARABOLIC EQUATION WITH A SOURCE IN AN INHOMOGENEOUS MEDIUM, Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, January 2024, Trade Unions Republican Commiittee of Azerbaijan Water Economy Workers,
DOI: 10.30546/2409-4994.2024.50.2.285.
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