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Article information
2026 , Volume 31, ¹ 1, p.66-75
Khaydarov A., Begulov O.U.
Self-similar solution of a parabolic equation in divergence form with a source and double nonlinearity
The Cauchy problem for the heat conduction equation with double nonlinearity, variable density, and nonlinear source is considered. Such equations degenerate into first-order equations; therefore, the Cauchy problem usually does not have a classical solution. It is necessary to consider a generalized (weak) solution that satisfies the equation in the sense of distributions. Using the methods of reference equations and nonlinear splitting, a self-similar solution of the equation is constructed. A theorem on the global solvability of the problem for small initial data is proved. To prove the theorem, an upper solution is constructed for certain parameter values. The asymptotic behaviour of the weak solution with compact support is obtained. Conditions on the parameters of the medium are derived under which the self-similar solution serves as the asymptotic representation.
Keywords: weak solution, self-similar solution, generalized solution, self-similar equation, global solvability, asymptotic behavior, finite solution, mathematical model
Author(s):
Khaydarov Abdugappar
PhD. , Associate Professor
Position: Professor
Office: National university of Uzbekistan named after Mirzo Ulugbek
Address: 100174, Uzbekistan, Tashkent, University Street, 4th House
Phone Office: (998) 71 227 12 24
E-mail: abdugapparxaydarov1953@gmail.com
Begulov Olmas Uktam ugli
Position: Senior Fellow
Office: Karshi branch of the Tashkent University of Information Technologies named after Muhammad al-Khwarizmi
Address: 180101, Uzbekistan, Karshi, Beshkent highway, third kilometer
E-mail: olmos90begulov@gmail.com
SPIN-code: 8160-1290
Bibliography link:
Khaydarov A., Begulov O.U. Self-similar solution of a parabolic equation in divergence form with a source and double nonlinearity // Computational technologies. 2026. V. 31. ¹ 1. P. 66-75
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