Bilateral Estimates of Solutions with Blow up Regime of the Nonlinear Heat Equation with a Quadratic Source
DOI:
https://doi.org/10.52575/2687-0959-2023-55-3-273-284Keywords:
Approximation of Solutions, Compact Support, Nonlinear Equation of Heat Transfer, Blow-up Regime, Etalon SolutionAbstract
Solutions u(x, t) ≥ 0, x ∈ R, t ≥ 0 with compact support of one-dimensional quasilinear heat transfer equation
degenerated at u(x, t) = 0 is studied. The equation has the linear on u transport coefficient and self-consistent source αu + βu2 of general type. Bulateral estimates of the blow-up time for solutions with a compact support are established, functionally depended on the initial conditions u(x, t).
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Andreucci D., Tedeev A. F. 2005. Universal bounds at the blow-up time for nonlinear parabolic equations. Advances in Differential Equations. 10(1): 89–120.
Barenblatt G. I. 1956. Automodel solutions of Cauchy problem of nonlinear parabolic equation of the gas nonstationary filtration in porous medium. Applied mathematics and Mechanics. 20(6): 761–763.
Carrillo J. 1999. Entropy solutions for nonlinear degenerate problems. Archive for Rational Mechanics and Analysis 147: 269—361.
Danilov V. G., Maslov V. P., Volosov K. A. 1995. Mathematical Modelling of Heat and Mass Transfer Processes. Mathematics and Its Applications (MAIA, volume 348), 324.
Galaktionov V. A., Samarskii A. A. 1983. Methods of constructing approximate self-similar solutions of nonlinear heat equations. I. Mathematics of the USSR-Sbornik, 46(3): 291–321.
Galaktionov V.A., Samarskii A.A. 1983. Methods of constructing approximate self-similar solutions of nonlinear heat equations. II. Mathematics of the USSR-Sbornik, 46(4): 439–458.
Kalashnikov A. S. 1986. On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity. Mathematics of the USSR-Sbornik, 53(2): 399–410.
Kolmogorov A. N., Fomin S. V. 1961. Elements of the Theory of Functions and Functional Analysis II. New York, Graylock Press, 486.
Kruzkov S. N. 1969. Generalized solutions of the Cauchy problem in the large for first order nonlinear equations. Doklady Akademii nauk SSSR, 187: 29–32.
Leibenzon L. S. 1930. The Motion of a Gas in a Porous Medium. M., Russian Academy of Sciences, 348.
Mikhailov A. P. 2002. Classification of unbounded solutions to a quasilinear transport equation. Computational Mathematics and Mathematical Physics, 42(6): 802–813.
Panov E. Yu. 2019. To the theory of entropy sub-solutions of degenerate nonlinear parabolic equations. arXiv:1910.08739v1 [math.AP] 19 Oct 2019.
Panov E. Yu. 2020. To the theory of entropy sub-solutions of degenerate nonlinear parabolic equations. Journal of Mathematical Sciences, 66(2): 292–313.
Panov E. Yu. 2020. To the theory of entropy sub-solutions of degenerate nonlinear parabolic equations. Mathematical Methods in the Applied Sciences. DOI: 10.1002/mma.6262.
Potemkina E.V. 1996. Peaking modes in the Cauchy problem for the inhomogeneous heat equation. Russian Mathematical Surveys, 51(6): 1223–1224.
Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P. 1979. Localization of the diffusion processes in media with constant properties. Proceedings of the Academy of Sciences, 247(2): 349–353.
Samarskii A. A., Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P. 1975. Effect of the met as table localization of heat in a medium with nonlinear heat conduction. Doklady Akademii nauk SSSR, 223(6): 1344–1347.
Samarskii A. A., Sobol’ I. M. 1963. Examples of the numerical calculation of temperature waves. USSR Computational Mathematics and Mathematical Physics, 3(4): 945-970.
Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P. 2011. Blow-Up in Quasilinear Parabolic Equations. Berlin, Walter de Gruyter, 554.
Tedeev A. F. 2004. Conditions for the time-global existence and nonexistence of a compact support of solutions of the Cauchy problem for quasilinear degenerate parabolic equations. Siberian Mathematical Journal, 45(1): 155–164.
Virchenko Yu. P., Vodyanitskii A. A. 1996. Semiconductors materials heat breakdown under action of the penetrating electromagnetic radiation. II. One-dimensional model analysis. Functional Materials, 3(3): 312–319.
Virchenko Yu. P., Vodyanitskii A. A. 2002. Heat localization and formation of secondary breakdown structure in semiconductor materials. II. Mathematical analysis of the model. Functional Materials, 9(4): 601–607.
Volosov K. A., Danilov V. G., Maslov V. P. 1988. Structure of a weak discontinuity of solutions of quasilinear degenerate parabolic equations. Mathematical Notes, 43(6): 479–485.
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