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Information × Registration Number 0225U002038, (0120U100178) , R & D reports Title Properties of singular solutions of differential and difference systems that model complex nonlinear physical processes popup.stage_title Властивості cингулярних розв’язків диференціальних та різницевих систем, які моделюють складні нелінійні фізичні процеси Head Skrypnik Ihor I., Доктор фізико-математичних наук Registration Date 16-02-2025 Organization Institute of Applied Mathematics and Mechanics National Academy of Science of Ukraine popup.description1 A study of the geometry of solutions of quasilinear parabolic equations with localized limiting data having singular peakings at a finite point in time. Development of potential theory for nonlinear elliptic and parabolic equations and further application of these developed methods for solving of urgent problems of the theory of nonlinear partial differential equations. popup.description2  The object of the research is elliptic and parabolic equations with non-standard growth conditions and non-logarithmic conditions on coefficients, as well as anisotropic elliptic and parabolic equations. The aim of the research is to investigate the geometry of solutions to quasilinear parabolic equations with localized boundary data that exhibit singular sharpening at a finite time. The development of potential theory for nonlinear elliptic and parabolic equations, along with the further application of the developed methods to solve real problems in the theory of nonlinear partial differential equations, is also a key goal. Research methods include: the theory of differential and integral equations, potential theory, elements of Sobolev and Gelder space theory, the method of monotone operators, topological methods in nonlinear analysis, various methods for obtaining a priori estimates, proving the regularity of solutions to quasilinear equations of arbitrary order, and the method of energy estimates. The results obtained are fundamental in nature, meet high international standards, make a significant contribution to the development of science, and hold potential for practical application. The developed methods and new approaches to the analysis of elliptic and parabolic equations with non-standard growth conditions allow for a more accurate characterization of the behavior of solutions under conditions typical for modeling physical systems with anisotropic properties. These results can be applied to improve mathematical models in thermodynamics, mechanics, and materials science. Product Description popup.authors Yevhenieva Yevheniia O. Voitovych Mykhailo V. Savchenko Mariia O. popup.nrat_date 2025-02-16 Close