BLOW-UP PHENOMENA AND SPATIAL LOCALIZATION IN NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE AMBIENT DENSITY IN MULTIDIMENSIONAL DOMAINS

Authors

  • Aliyev N.M. Urgench State Pedagogical Institute Student of the Uzbek Language and Literature Department 3rd year student

DOI:

https://doi.org/10.55640/

Keywords:

blow-up; spatial localization; variable density; double-nonlinear parabolic equation; self-similar solution; finite speed of propagation; Cauchy problem; reaction-diffusion; nonlinear splitting; numerical simulation.

Abstract

This paper is devoted to the investigation of blow-up phenomena and spatial localization effects arising in double-nonlinear parabolic equations with variable ambient density and power-type source terms in multidimensional domains. The Cauchy problem for the equation  is considered, where  is the variable density, m > 1 and q > 1 are nonlinearity exponents. Using the method of nonlinear splitting and self-similar substitution, approximate compactly supported solutions are constructed and their qualitative properties are analyzed. Two comparison lemmas are proved that provide explicit upper and lower bounds for the solution. A main theorem on blow-up and localization is derived, which gives sharp estimates for the front position and the blow-up time in terms of the system parameters. An implicit alternating-direction numerical scheme combined with Newton iterations is implemented with visualization. The theoretical predictions are validated through extensive numerical experiments confirming the finite-speed effect and the localization of thermal disturbances.

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References

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Published

2026-06-10

Issue

Vol. 5 No. 6 (2026): Journal of Multidisciplinary Sciences and Innovations

Section

Articles

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This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors retain the copyright of their manuscripts, and all Open Access articles are disseminated under the terms of the Creative Commons Attribution License 4.0 (CC-BY), which licenses unrestricted use, distribution, and reproduction in any medium, provided that the original work is appropriately cited. The use of general descriptive names, trade names, trademarks, and so forth in this publication, even if not specifically identified, does not imply that these names are not protected by the relevant laws and regulations.

How to Cite

BLOW-UP PHENOMENA AND SPATIAL LOCALIZATION IN NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE AMBIENT DENSITY IN MULTIDIMENSIONAL DOMAINS. (2026). Journal of Multidisciplinary Sciences and Innovations, 5(6), 655-660. https://doi.org/10.55640/

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