Geodesic
The asymptotics of a solution of the multidimensional heat equation with unbounded initial data
Ural mathematical journal, Tome 7 (2021) no. 1, pp. 168-177 Cet article a éte moissonné depuis la source Math-Net.Ru

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For the multidimensional heat equation, the long-time asymptotic approximation of the solution of the Cauchy problem is obtained in the case when the initial function grows at infinity and contains logarithms in its asymptotics. In addition to natural applications to processes of heat conduction and diffusion, the investigation of the asymptotic behavior of the solution of the problem under consideration is of interest for the asymptotic analysis of equations of parabolic type. The auxiliary parameter method plays a decisive role in the investigation.
Keywords: multidimensional heat equation, Сauchy problem, asymptotics, auxiliary parameter method. 
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Sergey V. Zakharov. The asymptotics of a solution of the multidimensional heat equation with unbounded initial data. Ural mathematical journal, Tome 7 (2021) no. 1, pp. 168-177. http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a12/

[1] Danilin A. R., “Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity”, Sb. Math., 189:11 (1998), 1611–1642 | DOI | MR | Zbl

[2] Denisov V. N., “On the behavior of solutions of parabolic equations for large values of time”, Russian Math. Surveys, 60:4 (2005), 721–790 | DOI | MR | Zbl

[3] Erdélyi A., Wyman M., “The asymptotic evaluation of certain integrals”, Arch. Rational Mech. Anal., 14 (1963), 217–260 | DOI | MR | Zbl

[4] Fourier J., Théorie Analytique de la Chaleur, Firmin Didot Père et Fils, Paris, 1822, 639 pp. (in French) | MR

[5] Friedman A., “Asymptotic Behavior of solutions of parabolic equations of any order”, Acta Math., 106:1–2 (1961), 1–43 | DOI | MR | Zbl

[6] Friedman A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964, 347 pp. | MR | Zbl

[7] Gilkey P. B., Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, v. 11, Math. Lect. Ser., Publish or Perish, Inc., Wilmington, Delaware, 1984, 512 pp. | MR | Zbl

[8] Il'in A. M., Khas'minskii R. Z., “Asymptotic behavior of solutions of parabolic equations and an ergodic property of non-homogeneous diffusion processes”, Math. Sb. (N. S.), 60:3 (1963), 366—392 (in Russian)

[9] Il'in A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, v. 102, Transl. Math. Monogr., Am. Math. Soc., 1992, 281 pp. | DOI | MR

[10] Narasimhan T. N., “Fourier's heat conduction equation: History, influence, and connections”, Rev. Geophys., 37:1 (1999), 151–172 | DOI

[11] Kamin S., Peletier L. A., Vazquez J. L., “A nonlinear diffusion-absorption equation with unbounded initial data”, Nonlin. Diff. Eq. Equilib. Stat. Progr. Nonlinear Differential Equations Appl., v. 3, eds. Lloyd N.G., Ni W.M., Peletier L.A., Serrin J., Birkhäuser, Boston, MA, 1992, 243–263 | DOI | MR

[12] Krzyżański M., “Sur l'allure asymptotique des solutions d'équations du type parabolique”, Bull. Acad. Polon. Sci., 4:5 (1956), 247–251 | MR | Zbl

[13] Lacey A. A., Tzanetis D. E., “Global, unbounded solutions to a parabolic equation”, J. Differential Equations, 101:1 (1993), 80–102 | DOI | MR | Zbl

[14] Li J., “Heat equation in a model matrix geometry”, C. R. Math. Acad. Sci. Paris, 353:4 (2015), 351–355 | DOI | MR | Zbl

[15] Lieberman G. M., Second Order Parabolic Differential Equations, World Scientific, River Edge, 1996, 452 pp. | DOI | MR | Zbl

[16] Poincaré H., “Sur les intégrales irrégulières des équations linéaires”, Acta Math., 8:1 (1886), 295–344 (in French) | DOI | MR

[17] Reynolds A., “Asymptotic behavior of solutions of nonlinear parabolic equations”, J. Differential Equations, 12:2 (1972), 256–261 | DOI | MR | Zbl

[18] Tychonoff A., “Théorèmes d'unicité pour l'équation de la chaleur”, Math. Sb., 42:2 (1935), 199—216 (in French) | Zbl

[19] Widder D. V., The Heat Equation, Academic Press, New York, 1976, 267 pp. | MR

[20] Zakharov S. V., “Heat distribution in an infinite rod”, Math. Notes, 80:3 (2006), 366–371 | DOI | MR | Zbl

[21] Zakharov S. V., “Two-parameter asymptotics in the Cauchy problem for a quasi-linear parabolic equation”, Asympt. Anal., 63:1–2 (2009), 49–54 | DOI | MR | Zbl